Logic, Computation, Hierarchies 9781614518044, 9781614517832

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Logic, Computation, Hierarchies

Ontos Mathematical Logic

Edited by Wolfram Pohlers, Thomas Scanlon, Ernest Schimmerling, Ralf Schindler, Helmut Schwichtenberg

Volume 4

Logic, Computation, Hierarchies Edited by Vasco Brattka, Hannes Diener, Dieter Spreen

ISBN 978-1-61451-783-2 e-ISBN 978-1-61451-804-4 ISSN 2198-2341 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2014 Walter de Gruyter Inc., Boston/Berlin Printing: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

Victor Selivanov, Würzburg, 2013¹

1 Reproduced with permission of Ernestina Selivanova.

Preface Computability and hierarchies are the leading themes of Victor Selivanov’s scientific work. As a student he got interested in mathematical logic mainly because of being dissatisfied with the level of rigour in the mathematical analysis lectures he followed. Today, he is regarded as a world-expert in hierarchy theory, with his publications having been highly influential. This volume is dedicated to Victor on the occasion of his 60th birthday. The idea for this Festschrift was born during a workshop in Novosibirsk in honour of him and the splendid birthday party afterwards. We asked collaborators and friends of his to contribute to a book that also should reflect the width of his scientific interests. The result is a collection of 17 refereed articles written by 23 authors. We would like to thank all that helped to produce this volume of high scientific standard. These are of course the authors, but also the referees. Their careful reading of the manuscripts and critical remarks made a crucial contribution. Special thanks go to Svetlana Selivanova for answering questions on bibliographical data.

Cape Town, Christchurch, Munich, Pretoria, Siegen, Winter 2013/14 Vasco Brattka¹ Hannes Diener² Dieter Spreen³

1 Faculty of Computer Science, Universität der Bundeswehr, Munich, Germany, and Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa 2 Department of Mathematics, University of Siegen, Germany, and Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand 3 Department of Mathematics, University of Siegen, Germany, and Department of Decision Sciences, University of South Africa, Pretoria, South Africa

Contents Preface | VII Dieter Spreen The Life and Work of Victor L. Selivanov | 1 Collins Amburo Agyingi, Paulus Haihambo, and Hans-Peter A. Künzi Tight Extensions of T0 -Quasi-Metric Spaces | 9 Klaus Ambos-Spies On the Strongly Bounded Turing Degrees of Simple Sets | 23 Matthew de Brecht Levels of Discontinuity, Limit-Computability, and Jump Operators | 79 Jacques Duparc, Olivier Finkel, and Jean-Pierre Ressayre The Wadge Hierarchy of Petri Nets ω-Languages | 109 Willem L. Fouché Diophantine Properties of Brownian Motion: Recursive Aspects | 139 Sy-David Friedman The Completeness of Isomorphism | 157 Peter Hertling, Victor Selivanov Complexity Issues for Preorders on Finite Labeled Forests | 165 Anton Konovalov Boolean Algebras of Regular Quasi-aperiodic Languages | 191 Eryk Kopczyński, Damian Niwiński A Simple Indeterminate Infinite Game | 205 Luca Motto Ros, Philipp Schlicht Lipschitz and Uniformly Continuous Reducibilities on Ultrametric Polish Spaces | 213

X | Contents Sergey Odintsov On the Equivalence of Paraconsistent and Explosive Versions of Nelson Logic | 259 Svetlana Selivanova Computing Clebsch-Gordan Matrices with Applications in Elasticity Theory | 273 Nikolay V. Shilov An Approach to Design of Automata-Based Axiomatization for Propositional Program and Temporal Logics (by Example of Linear Temporal Logic) | 297 Dieter Spreen Partial Numberings and Precompleteness | 325 Dieter Spreen An Isomorphism Theorem for Partial Numberings | 341 Ludwig Staiger Two theorems on the Hausdorff measure of regular ω-languages | 383 Anton V. Zhukov Some Notes on the Universality of Three-Orders on Finite Labeled Posets | 393 Index | 411

Dieter Spreen

The Life and Work of Victor L. Selivanov || Dieter Spreen: Department of Mathematics, University of Siegen, Germany, and Department of Decision Sciences, University of South Africa, Pretoria, South Africa

Victor L. Selivanov was born on the 25th of August, 1952, in Oktyabr’sky, a small industrial city in the Republic of Bashkortostan (Russian Federation), near the border with Tatarstan. He was the fourth (and youngest) child in his family. His parents Lev N. Selivanov and Lidya G. Selivanova both worked as mathematics teachers at a college in Oktyabr’sky. He graduated with honours from Kazan State University (1974) and became a postgraduate student of M.M. Arslanov and R.G. Bukharayev in Kazan. In 1979 he was awarded a Ph.D. in Mathematics from Novosibirsk University. The title of the thesis was “On computable numberings”. Victor defended his habilitation thesis “Hierarchical classification of arithmetical sets and index sets” at the Institute of Mathematics of the Siberian Devision of the Russian Academy of Sciences in Novosibirsk in 1989. After spending a few years (1978–82) as Assistant Professor, first at Ulyanovsk Polytechnic Institute and then at Kazan Institute of Chemical Technology, he moved to Novosibirsk State Pedagogical University, where he was Senior Lecturer (1982–86), Docent (1986–90), Professor (1990–91) and Head of the Chair of Informatics and Discrete Mathematics (1991–2009). In 1993 he was appointed Full Professor. Since 2009 he is Chief Research Fellow at the A.P. Ershov Institute of Informatics Systems of the Siberian Devision of the Russian Academy of Sciences. Selivanov’s research started in 1974 when he became a postgraduate student at Kazan State University. It is devoted to Mathematical Logic, Computer Science, and Didactics of Mathematics and Informatics. The main focus of his research is computability, ranging from general computability theory to automata theory. In the beginning his main contributions were in numbering theory. Later he shifted to the study of hierarchies, which is still his central interest.

Numbering Theory In his Ph.D. thesis he (i) proved that any nontrivial semilattice of computable numberings is not a lattice; (ii) constructed an example of a discrete, but not effectively discrete family of total recursive functions having a unique (up to equivalence)

2 | Dieter Spreen computable numbering; (iii) constructed an example of a non-discrete family of computably enumerable sets having a unique (up to equivalence) computable numbering. The results answers well known questions on computable numberings [1; 2; 3; 4; 5]. Result (ii) was later applied in learning theory. In addition, he proved several facts on degrees of unsolvability, among others he classified the possible versions of tt-reducibilities (a result that was independently obtained by Bulitko) and completely described the relationship between the Ershov hierarchy and the high-low hierarchy of degrees of unsolvability [9; 16; 18; 25]. This was independently discovered by Jockusch and Shore. Selivanov also developed a relativized version of the theory of precomplete and complete numberings which led to priority-free proofs of several known results about the structure of tt-type degrees and about m-degrees of index sets in such structures [17; 19; 26; 29]. In the beginning of the 1990s, Selivanov obtained general results on positively numbered Boolean algebras, part of them jointly with Odintsov [21; 24]. The main results are that this class of Boolean algebras always has a good computable numbering (an analog of the numbering of the computably enumerable sets) and that there is a unique (up to effective isomorphism) universal positive Boolean algebra (an analog of the creative sets). He also characterized many natural and important index sets of positive Boolean algebras. Later some of these results were extended by him to the very general context of recursively axiomatizable quasivarieties [29; 34]. This research has different applications, e.g. to the classification of many natural index sets in the lattice of computably enumerable sets, in the semilattice of computably enumerable m-degrees and in the Lindenbaum algebra of propositions [21; 22; 24; 28; 29; 34].

Hierarchies in Computability Theory As is well known, the Ershov difference hierarchy is closely related to the so called m-jump operator. In his habilitation thesis Selivanov defined and investigated several generalizations of this operator. It turned out that the generalized operators are closely related to the theory of complete numberings developed by Mal’cev and Ershov, and to the study of structures of m-degrees of index sets of the computably enumerable sets as well as the partial recursive functions. The results led to a simplification and generalization of results by Hay about index sets [8; 11; 13; 15]. Selivanov, moreover, defined a refinement of the arithmetic hierarchy called the fine hierarchy and proved that it contains many known hierarchies and has some rather strong closure properties with respect to refinements. These results informally mean that the fine hierarchy is in some sense the finest possible. Later,

The Life and Work of Victor L. Selivanov | 3

using set-theoretic operations introduced by Wadge, he gave a set-theoretical description of the fine hierarchy. This yields the possibility of defining the hierarchy in very different contexts, e.g. in the context of logic and descriptive set theory. In the latter case, he could show that the fine hierarchy is, in an exact sense, the finite version of the Wadge hierarchy of Borel sets [13; 15; 20; 23; 24; 27; 28]. These results are also part of his habilitation thesis. He extended the fine hierarchy from the case of sets to the case of k-partitions [52]. With heavy use of Priestley duality he showed that the methods of alternating trees and of m-reducibilities developed earlier by Addison, by him, and by others for concrete examples of fine hierarchies apply to a broad class of fine hierarchies. Selivanov gave a complete description of analogs of the famous Rice-Shapiro theorem for all levels of the arithmetic hierarchy (as well as of some of its refinements). Another principal result is a complete description of m-degrees of index sets of the predicates which are first-order definable in the Lindenbaum algebra of statements of nontrivial signature. Note that this classification needs all levels of the fine hierarchy, i.e., it could not be achieved without the invention of the fine hierarchy (see [6; 7; 8; 10; 11; 12; 14; 20; 22; 24]).

Hierarchies in Complexity Theory In the context of structural complexity theory Selivanov defined and considered analogs of the hierarchies mentioned above. He was able to extend the wellknown result of Kadin about the non-collapse of the Boolean hierarchy over NP to a much richer hierarchy, called the plus-hierarchy [27; 31]. Further work in this direction, essentially developed jointly with Glasser and Reitwiessner [51], led to the solution of long-standing open problems such as a question by Blass and Gurevich on the status of the shrinking property (also known as the reduction property) for levels of the polynomial hierarchy.

Hierarchies in Language Theory Selivanov also used the fine hierarchy to classify regular ω-languages. It turned out that the resulting classification coincides with the Wagner hierarchy. This result establishes a close relationship between descriptive set theory and the theory of regular ω-languages. It leads to rather different proofs of deep and complicated results of Wagner [30]. This investigation seems to have been the first applications of modern descriptive set theory to the theory of ω-languages, a line which was also developed by French mathematicians such as Perrin, Carton, Du-

4 | Dieter Spreen parc and Finkel. Recently, Selivanov classified the Wadge degrees of ω-languages of deterministic Turing machines [35]. In this way he answered a question raised by Duparc. In pursuing this research Selivanov developed a complete analog of the Wagner hierarchy for the class of regular star-free ω-languages [42]. He also extended the Wagner hierarchy from the case of sets to the case of k-partitions which, for k > 2, leads to a much more complicated but still tractable structure of degrees [53; 56]. Jointly with Wagner, he comprehensively investigated complexity questions related to his hierarchy [41]. Further on, he proposed a new, logical approach to the well known problem of the decidability of the dot-depth hierarchy and of some its refinements. In this way, he obtained quite different and shorter proofs of important results, as well as new results on analogs of the dot-depth problem. Some of these results were obtained independently and by other methods by Glasser, Schmitz and Wagner. Using some previous results on the so called leaf languages, he established a deep connection of these automata-theoretic hierarchies with the complexity-theoretic ones mentioned above [32; 33]. In joint work with Wagner [38] the above-mentioned results were applied to defining and investigating a reducibility on star-free languages which fits perfectly to the dot-depth hierarchy. Both established a close relationship of this reducibility with the leaf-language approach to complexity classes. This research was extended to quasi-periodic languages, which form another important class of regular languages. [47].

Hierarchies in Descriptive Set Theory Selivanov picked up an old problem of Scott and developed a rather comprehensive theory of Borel and difference hierarchies in so called φ-spaces, which are topological counterparts of algebraic directed-complete partial orderings, and a theory of Wadge reducibility in such spaces. In a sense this theory parallels the corresponding classical theory for Polish spaces. The theory has non-trivial applications to classical ω-ary Boolean operations of Kantorovich and Livenson and is closely related to the theory of infinitary languages, i.e. languages that may contain both, finite and infinite words [36; 37; 39; 44]. The extension is very natural and suggests new interesting ways of extending classical descriptive set theory. Recently, the descriptive set theory for φ-spaces was extended by de Brecht, Becher and Grigorieff to some broader natural classes of spaces. The theory was partially extended from the case of sets to the case of k-partitions [40; 43; 44; 48]. Jointly with Motto Ros and Schlicht, Selivanov extended

The Life and Work of Victor L. Selivanov | 5

the classical Wadge theory to the broader class of quasi-Polish spaces [57], and together with Schröder [58] he introduced and studied hierarchies of topological spaces that include well known classes of spaces as the quasi-Polish spaces and the Kleene-Kreisel continuous functionals. New methods to study first order definability in countable structures were developed which apply to structures on words and labeled trees [45; 46; 49; 50; 54; 55]. As a consequence comprehensive definability results for degree structures naturally arising in topology and computable analysis were obtained, in particular, for initial segments of the Wadge and Weihrauch degrees of k-partitions. This provides first steps in the development of degree theory for topological structures in parallel to the classical degree theory for discrete structures. Many of Selivanov’s mathematical ideas have been taken up and developed further by other mathematicians. His results found their way into monographs and textbooks in computability and formal language theory, as well as in handbooks. In addition to his interest in the foundations of mathematics and computer science Selivanov has developed a strong interest in didactics. Here, his main aim was to create new and more efficient methods of teaching mathematics and informatics to students of different specialties and different qualifications. He produced methods for teaching basic skills of office software, and found ways of introducing algorithmics to school children, and conveying fundamentals of visual programming as well as elements of computer modeling in secondary and high school. Selivanov has supervised 9 Ph.D. theses and many master dissertations. He is the author of about 100 publications as well as several textbooks and didactic materials of different kind in Algebra, Logic, Informatics, and Didactics of Informatics. His work was honoured by a Humboldt Fellowship, Visiting Professorships in Siegen and Paris, as well as two Mercator Professorships by the German Research Foundation. He was awarded the honorary title “Merited Worker of the High School” of the Russian Government in 1999 and the honorary title “Merited Worker of Science and Technology of the Russian Federation” by the Ministry of Science and Education of the Russian Federation in 2012. Thanks to generous funding by the Austrian Science Fund, the German Academic Exchange Service, the German Research Foundation, the Russian Foundation for Basic Research and, more recently, the European Union Selivanov was an ever-welcome guest in Aachen, Cambridge, Darmstadt, Heidelberg, Munich, Siegen, Swansea, Vienna, and Würzburg.

6 | Dieter Spreen

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [18] [19] [20] [21]

[22] [23] [24] [25]

On numberings of families of total recursive functions. Algebra and Logic, 15, N 2 (1976), 128–141. Two theorems on computable numberings. Algebra and Logic, 15, N 4 (1976), 297–306. On computability of some classes of numberings. Prob. Methods and Cybernetics, v.12-13, Kazan University, Kazan,1976, 157–170 (Russian). Numberings of canonically computable families of finite sets. Sib. Math. J., 18, N 6 (1977), 1373–1381 (Russian, there is an English translation). Some remarks on classes of recursively enumerable sets. Sib. Math. J., 19, N 1 (1978), 109–115. On index sets of classes of numberings. Prob. Methods and Cybernetics, v.14, Kazan University, Kazan,1978, 90–103 (Russian). On index sets of computable classes of finite sets. In: Algorithms and Automata, Kazan University, Kazan,1978, 95–99 (Russian). On the structure of degrees of index sets. Algebra and Logic, 18, N 4 (1979), 286–299. On a class of reducibilities in recursion theory. Prob. Methods and Cybernetics, v.14, Kazan University, Kazan, v. 18 (1982), 83–101 (Russian). On index sets in the Kleene-Mostowski hierarchy. Trans. Inst. Math., Novosibirsk, N 2 (1982), 135–158 (Russian). On the structure of degrees of generalized index sets. Algebra and Logic, 21, N 4 (1982), 316–330. Effective analogs of A-, B-, and Cs-sets with applications to index sets. Prob. Methods and Cybernetics, v.14, Kazan University, Kazan, v. 19 (1983), 112–128 (Russian). Hierarchies of hyperarithmetical sets and functions. Algebra and Logic, 22, N 6 (1983), 473—491. Index sets in the hyperarithmetical hierarchy. Sib. Math. J., N 3 (1984), 474–488. On a hierarchy of limiting computations. Sib. Math. J., 25, No 5 (1984), 798–806. On Ershov hierarchy. Sib. Math. J., 26, N 1 (1985), 105–116. Index sets of factor-objects of the Post numbering. Algebra and Logic, 27, N 3 (1988), 215–224. Ershov hierarchy and Turing jump. Algebra and Logic, 27 N 4 (1988), 292–301. On algorithmic complexity of algebraic systems. Math. Notes, 44, No 5–6 (1988) p.944– 950. Applications of precomplete numberings to tt-type degrees and to index sets. Algebra and Logic, 28, N 1 (1989), 51–56. Fine hierarchies of arithmetical sets and definable index sets. Trans. Inst. Math., Novosibirsk, 12 (1989), 165–185 (Russian). Arithmetical hierarchy and ideals of numbered boolean algebras (jointly with S.P. Odintsov). Sib. Math. J., 30, N 6 (1989), 140–149 (Russian, there is an English translation). Index sets of classes of hyperhypersimple sets. Algebra and Logic, 29, N 2 (1990), 155– 168. A fine hierarchy of formulas. Algebra and Logic, 30, N 5 (1991), 368–378. Fine hierarchies and definable index sets. Alg. and Logic, 30, N 6 (1991), 463–475. Jumps of some classes of sets. Math. Notes, 50, No 6 (1991), p.1299–1300.

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[26] Precomplete numberings and functions without fixed points. Math. Notes, Math. Notes, 51, No 1 (1992), p.95–99. [27] Fine hierarchies and Boolean terms. The Journal of Symbolic Logic, 60, N 1 (1995), 289– 317. [28] Fine hierarchy and definability in the Lindenbaum algebra. In: Logic: from foundations to applications, Proceedings of the Logic Colloquium-93 in Keele. Oxford, 1996, 425–452. [29] On recursively enumerable structures. Annals of pure and applied logic, 78 (1996), 243– 258. [30] Fine hierarchy of regular omega-languages. Theor. Computer Science, 191 (1998), 37–59. [31] Refining the polynomial hierarchy. Algebra and Logic, 38 (1999), N 4, 248–258. [32] Relating automata-theoretic hierarchies to complexity-theoretic hierarchies. Theoret. Informatics Appl., 36 (2002), 29–42. [33] On decidability of classes of hierarchies of regular aperiodic languages. Algebra and Logic,41, N 5 (2002), 337–348. [34] Positive structures. In: Computability and Models, Perspectives East and West, S. Barry Cooper and Sergei S. Goncharov, eds., Kluwer Academic / Plenum Publishers, New York, 2003, 321–350. [35] Wadge degrees of ω-languages of deterministic Turing machines. Theoretical Informatics and Applications, 37 (2003), 67–83. [36] Difference hierarchy in φ-spaces. Algebra and Logic, 43, N 4 (2004), 238–248. [37] Hierarchies in φ-spaces and applications. Math. Logic Quarterly, 51, N 1 (2005), 45–61. [38] A reducibility for the dot-depth hierarchy (jointly with K.W. Wagner). Theoretical Computer Science, 345, N 2-3 (2005), 448–472. [39] Towards a descriptive set theory for domain-like structures. Theoretical Computer Science, 365 (2006), 258–282. [40] The quotient algebra of labeled forests modulo h-equivalence. Algebra and Logic, 46, N 2 (2007), 120–133. [41] Complexity of topological properties of regular ω-languages (jointly with K.W. Wagner). Fundamenta Informaticae, 83(1-2): 197–217 (2008). [42] Fine hierarchy of regular aperiodic ω-languages. International Journal of Foundations of Computer Science, 19, No 3 (2008) 649–675. [43] Wadge reducibility and infinite computations. Mathematics in Computer Science, 2 (2008), 5–36 doi: 10.1007/ s11786-008-0042-x [44] On the difference hierarchy in countably based T0 -spaces. Electronic Notes in Theoretical Computer Science, V. 221 (2008), 257-269, doi: 10.1016/ j.entcs. 2008.12.022. [45] Definability in the h-quasiorder of labeled forests (jointly with O.V. Kudinov and A.V. Zhukov). Annals of Pure and Applied Logic, 159(3): 318–332 (2009). doi: 10.1016/ j.apal. 2008.09.026. [46] Undecidability in Some Structures Related to Computation Theory. Journal of Logic and Computation, 19, No 1 (2009), 177–197; doi: 10.1093/ logcom/exn023 [47] Hierarchies and reducibilities on regular languages related to modulo counting. RAIRO Theoretical Informatics and Applications, 41 (2009), 95–132. DOI: 10.1051/ ita:2007063 [48] On the Wadge reducibility of k-partitions. Journal of Logic and Algebraic Programming, 79, No 1, 2010, 92–102. PII: S1567-8326(09)00024-1 DOI: 10.1016/ j.jlap.2009. 02.008 [49] Definability of closure operations in the h-quasiorder of labeled forests (jointly with Oleg Kudinov and Anton Zhukov). Algebra and Logic, 49, No 2 (2010), 181–194.

8 | Dieter Spreen [50] Definability in the subword order (jointly with O.V. Kudinov and L.V. Yartseva). Siberian Mathematical Journal, 51, No 3 (2010), 575–583. [51] The Shrinking Property for NP and coNP (jointly with C. Glasser and C. Reitwiessner). Theoretical Computer Science 412 (2011), 853-864. [52] Fine hierarchies via Piestley duality. Annals of Pure and Applied Logic, 163 (2012) 10751107, doi:10.1016/j.apal.2011.12.029 [53] Complexity of aperiodicity for topological properties of regular ω-languages (jointly with K.W. Wagner). Conf. Computability in Europe-2008 Lecture Notes in Computer Science, v. 5028. Berlin: Springer, 2008, 533–543. [54] A Gandy theorem for abstract structures and applications to first-order definability (jointly with O.V. Kudinov). Proc. Of CiE-2009 (K. Ambos-Spies, B. Löwe and W. Merkle, eds.), Lecture Notes in Computer Science, v. 5635. Berlin: Springer, 2009, 290–299. [55] Undecidability in Weihrauch degrees (jointly with O. Kudinov and A. Zhukov). Proc. of CiE-2010 (F. Ferreira, B. Löwe, E. Mayordomo and L.M. Gomes, eds.), Lecture Notes in Computer Science, v. 6158. Berlin: Springer, 2010, 256–265. [56] A fine nierarchy of ω-regular k-partitions. B. Löwe et.al. (Eds.): CiE 2011, LNCS 6735, pp. 260–269. Springer, Heidelberg (2011). [57] Wadge-like reducibilities on arbitrary quasi-Polish spaces (jointly with L. Motto Ros and P. Schlicht). Accepted by Mathematical Structures in Computer Science, arXiv:1204.5338 v1 [mathLO] 24 Apr 2012. [58] Some hierarchies of QCB0 -spaces (jointly with M. Schröder). Mathematical Structures in Computer Science, to appear.

Collins Amburo Agyingi, Paulus Haihambo, and Hans-Peter A. Künzi

Tight Extensions of T0-Quasi-Metric Spaces¹ Abstract: Dress introduced and studied the concept of the tight span of a metric space. It is known that Dress’s theory is equivalent to the theory of the injective hull of a metric space independently discussed by Isbell some years earlier. In a paper by Kemajou et al. it was shown that Isbell’s approach can be modified to work similarly for T0 -quasi-metric spaces and nonexpansive maps. Continuing that work we show in the present paper that large parts of the theory developed by Dress do not use the symmetry of the metric and - when appropriately modified - hold essentially unchanged for T0 -quasi-metric spaces. Keywords: hyperconvexity; T0 -quasi-metric space; injective hull; q-hyperconvex, tight extension, tight span Mathematics Subject Classification 2010: AMS (2010) Subject Classifications: 54D35; 54E15; 54E35; 54E55; 54E50 || Collins Amburo Agyingi, Paulus Haihambo, Hans-Peter A. Künzi: Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa

1 Introduction In [7] Isbell showed that every metric space X has an injective hull T X , which is compact if X is compact. Let us also recall that a metric space is called hyperconvex (see e.g. [9, p. 78]) if and only if it is injective in the category of metric spaces and nonexpansive maps. Dress [4] later gave an independent, but equivalent approach to Isbell’s theory that is based on the concept of a tight extension. In analogy to Isbell’s theory Kemajou et al. [8] proved that each T0 -quasimetric space X has a q-hyperconvex hull Q X , which is joincompact if X is joincompact. They called a T0 -quasi-metric space q-hyperconvex if and only if it is injective in the category of T0 -quasi-metric spaces and nonexpansive maps. In this paper we intend to generalize results due to Dress [4] on tight extensions of met-

1 The authors would like to thank the National Research Foundation of South Africa for partial financial support. This research was also supported by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme.

10 | C. A. Agyingi, P. Haihambo, and H.-P.A. Künzi ric spaces to the category of T0 -quasi-metric spaces and nonexpansive maps. In particular we show that large parts of the theory of tight extensions do not use the symmetry of the metric and under appropriate modifications still hold essentially unchanged for T0 -quasi-metrics. The results of the present paper will be applied in further investigations of the authors about endpoints in T0 -quasi-metric spaces [2]. Hence it was necessary to develop the discussed theory below carefully in detail, although it sometimes closely follows the classical metric theory.

2 Preliminaries This section recalls the most important definitions that we shall use in the following. Definition 1. Let X be a set and d : X × X → [0, ∞) be a function mapping into the set [0, ∞) of the nonnegative reals. Then d is a quasi-pseudometric on X if (a) d(x, x) = 0 whenever x ∈ X, and (b) d(x, z) ≤ d(x, y) + d(y, z) whenever x, y, z ∈ X. We shall say that (X, d) is a T0 -quasi-metric space provided that d also satisfies the following condition: For each x, y ∈ X, d(x, y) = 0 = d(y, x) implies that x = y. Let d be a quasi-pseudometric on a set X. Then d−1 : X × X → [0, ∞) defined by d−1 (x, y) = d(y, x) whenever x, y ∈ X is also a quasi-pseudometric, called the conjugate quasi-pseudometric of d. Note that if d is a T0 -quasi-metric on X, then d s = max{d, d−1 } = d ∨ d−1 is a metric on X. Let (X, d) be a quasi-pseudometric space. For each x ∈ X and ϵ > 0, B d (x, ϵ) = {y ∈ X : d(x, y) < ϵ} denotes the open ϵ-ball at x. The collection of all “open” balls yields a base for a topology τ(d). It is called the topology induced by d on X. A map f : (X, d) → (Y , e) between quasi-pseudometric spaces is called isometric provided that d(x, y) = e(f (x), f (y)) whenever x, y ∈ X. Note that each isometric map with a T0 -quasi-metric domain is a one-to-one map. A map f : (X, d) → (Y , e) between quasi-pseudometric spaces is called nonexpansive provided that e(f (x), f (y)) ≤ d(x, y) whenever x, y ∈ X. ˙ b for max{a − Given two nonnegative real numbers a and b we shall write a− b, 0}, which in a more lattice-theoretic terminology we shall also denote by (a − ˙ y with x, y ∈ [0 , ∞) defines the standard T0 -quasib) ∨ 0. Note that u(x, y) = x− metric on [0, ∞).

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For further basic concepts used from the theory of asymmetric topology we refer the reader to [5] and [10]. Some recent work about quasi-pseudometric spaces can be found in [1; 3; 11; 12; 14]. In the preprint [3] de Brecht for instance investigates connections between the theory of T0 -quasi-metric spaces and Victor Selivanov’s work in descriptive set theory (see e.g. [15]).

3 q-hyperconvex hulls of T0 -quasi-metric spaces Next we recall some results mainly from [8] belonging to the theory of the qhyperconvex hull of a T0 -quasi-metric space. Let (X, d) be a T0 -quasi-metric space. We shall say that a function pair f = (f1 , f2 ) on (X, d) where f i : X → [0 , ∞) (i = 1, 2) is ample provided that d(x, y) ≤ f2 (x) + f1 (y) whenever x, y ∈ X. Let P X denote the set of all ample function pairs on (X, d). (In such situations we may also write P(X,d) in cases where d is not obvious.) For each f , g ∈ P X we ˙ g1 (x))∨supx∈X (g2 (x)− ˙ f2 (x)). Then D is an extended² set D(f , g ) = supx∈X (f1 (x)− T0 -quasi-metric on P X . We shall call a function pair f minimal on (X, d) (among the ample function pairs on (X, d)) if it is ample and whenever g is ample on (X, d) and for each x ∈ X we have g1 (x) ≤ f1 (x) and g2 (x) ≤ f2 (x),³ then g = f . It is well known that Zorn’s Lemma implies that below each ample function pair there is a minimal ample pair (for a more constructive and global approach, essentially due to Dress [4], see Proposition 2 below). By Q X we shall denote the set of all minimal ample pairs on (X, d) equipped with the restriction of D to Q X × Q X , which we shall also denote by D. Recall that D is indeed a (real-valued) T0 -quasi-metric on Q X × Q X [8, Remark 6]. ˙ f2 (y) : Furthermore f ∈ P X belongs to Q X if and only if f1 (x) = sup{d(y, x)− ˙ y ∈ X } and f2 (x) = sup{d(x, y)−f1 (y) : y ∈ X } whenever x ∈ X (see [11, Remark 2]). It is known (see [8, Lemma 3]) that f ∈ Q X implies that f1 (x) − f1 (y) ≤ d(y, x) and f2 (x) − f2 (y) ≤ d(x, y) whenever x, y ∈ X :

2 If we replace in the definition of a quasi-pseudometric [0, ∞) by [0, ∞] we obtain the definition of an extended quasi-pseudometric. Of course, the triangle inequality for extended quasipseudometrics is interpreted in the self-explanatory way. 3 For any function pairs f and g satisfying this relation we shall write g ≤ f .

12 | C. A. Agyingi, P. Haihambo, and H.-P.A. Künzi Indeed (compare [13, remark before Proposition 3.1]) given x ∈ X we have that ˙ f1 (y) : y ∈ X } − d(x, x ) ≤ sup{ d(x , y)− ˙ f1 (y) : f2 (x) − d(x, x ) = sup{d(x, y)−  y ∈ X } = f2 (x ). The inequality for f1 is verified similarly. ˙ g1 (x)) = supx∈X (g2 (x)− ˙ f2 (x)) whenever f , g ∈ Q X Moreover supx∈X (f1 (x)− (compare [8, Lemma 7]). For each x ∈ X we can define the minimal function pair f x (y) = (d(x, y), d(y, x)) (whenever y ∈ X) on (X, d). The map j defined by x → f x whenever x ∈ X defines an isometric embedding of (X, d) into (Q X , D) (see [8, Lemma 1]). We recall that (Q X , D) is called the q-hyperconvex hull of (X, d). A T0 -quasimetric space X is said to be q-hyperconvex if f ∈ Q X implies that there is an x ∈ X such that f = f x (compare [8, Corollary 4]). For an intrinsic characterization of q-hyperconvexity see [8, Definition 2]. We also note that D(f , f x ) = f1 (x) and D(f x , f ) = f2 (x) whenever x ∈ X and f ∈ Q X [8, Lemma 8].

4 T0 -quasi-metric tight extensions In this section we generalize some crucial results about tight extensions of metric spaces from [4] to our quasi-metric setting. Proposition 2. (compare [4, Section 1.9]) Let (X, d) be a T0 -quasi-metric space. There exists a retraction map p : P X → Q X , i.e., a map that satisfies the conditions (a) D(p(f ), p(g )) ≤ D(f , g ) whenever f , g ∈ P X . (b) p(f ) ≤ f whenever f ∈ P X . (In particular p(f ) = f whenever f ∈ Q X , since each f in Q X is minimal, and thus p indeed is a retraction.) ˙ f2 (x ) : x ∈ X } whenever Proof. Given a pair f ∈ P X , we set f1∗ (y) = sup{d(x , y)− ∗  ˙   y ∈ X and f2 (x) = sup{d(x, y )−f1 (y ) : y ∈ X } whenever x ∈ X. Claim 1: f ∗ ≤ f . Note that for any x, y ∈ X, we have d(x, y ) ≤ f2 (x)+ f1 (y ) and thus d(x, y )− ˙ f1 (y ) : y ∈ X } ≤ f2 (x). In a f1 (y ) ≤ f2 (x). Therefore f2∗ (x) = sup{d(x, y )− ∗ similar manner, we can show that f1 (y) ≤ f1 (y) whenever y ∈ X. Thus f ∗ ≤ f . Claim 2: d(x, y) ≤ f2∗ (x)+ f1 (y) and d(x, y) ≤ f2 (x)+ f1∗ (y) whenever x, y ∈ X. ˙ f1 (y ) : y ∈ X } + f1 (y) ≥ d(x, y) − Let x, y ∈ X. We see that sup{d(x, y )− ˙ f2 (x ) : f1 (y) + f1 (y) = d(x, y). Similarly we have d(x, y) ≤ f2 (x) + sup{d(x , y)−

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x ∈ X }. Therefore we get d(x, y) ≤ f2∗ (x) + f1 (y) and d(x, y) ≤ f2 (x) + f1∗ (y) whenever x, y ∈ X. Define q : P X → P X by f → q(f ) = ( 12 (f1 + f1∗ ), 12 (f2 + f2∗ )) whenever f ∈ P X . Claim 3: q(f ) is indeed ample and q(f ) ≤ f . In fact (q(f ))2 (x)+(q(f ))1 (y) = 12 (f1 (y)+ f1∗ (y))+ 12 (f2 (x)+ f2∗ (x)) = 12 (f1 (y)+ f2∗ (x)) + 12 (f1∗ (y) + f2 (x)) ≥ 12 d(x, y) + 12 d(x, y) = d(x, y) whenever x, y ∈ X. This shows that q(f ) is ample. Obviously q(f ) ≤ f , since f ∗ ≤ f . Claim 4: We have that D(q(f ), q(g )) ≤ D(f , g ) whenever f , g ∈ P X : Let f , g ∈ P X and x ∈ X. Then ˙ f2 (y) : y ∈ X } ≤ f1∗ (x) = sup{d(y, x)−

˙ g 2 ( y ) + g 2 ( y )− ˙ f2 (y) : y ∈ X } ≤ sup{d(y, x)−

˙ g2 (y) : y ∈ X } + sup{g2 (y)− ˙ f2 (y) : y ∈ X } ≤ g1∗ (x) + D(f , g ). sup{ d(y, x)−

Therefore

˙ q(g ))1 (x)) ≤ sup((q(f ))1 (x)−( x∈X

1 ˙ g1 (x)) + 1 sup(f1∗ (x)− ˙ g1∗ (x)) sup(f1 (x)− 2 x∈X 2 x∈X

1 1 D(f , g) + D(f , g) = D(f , g). 2 2 ˙ q(f ))2 (x)) ≤ D(f , g ) whenever Similarly we can show that supx∈X ((q(g ))2 (x)−( f , g ∈ P X . Thus D(q(f ), q(g )) ≤ D(f , g ) whenever f , g ∈ P X . In the rest of the proof, given f ∈ P X , we obtain a minimal ample function pair below f as the pointwise limit of the sequence (q n (f ))n∈N where q n is the n th iteration of q : Fix f ∈ P X . Obviously for n ∈ N the n-iteration q n of q yields a monotonically decreasing sequence (q n (f )) that is bounded below by the 0-pair. Hence the map p(f ) := limn→∞ q n (f ) exists, where we take the pointwise limit pair with respect to the usual topology τ(u s ) on [0, ∞). Note that obviously for each n ∈ N, q n (f ) belongs to P X and q n satisfies the ≤

conditions (a) and (b), too. Therefore p(f ) ∈ P X and p also satisfies condition (b). For each n ∈ N, f , g ∈ P X and x ∈ X we have

˙ q n (g ))1 (x)] ∨ [(q n (g ))2 (x)−( ˙ q n (f ))2 (x)] ≤ D(f , g ); [(q n (f ))1 (x)−(

thus D(p(f ), p(g )) ≤ D(f , g ) and p satisfies condition (a), too. We finally show that p(f ) ∈ Q X whenever f ∈ P X . Let f ∈ P X . For all n ∈ N we have p(f ) ≤ q n (f ) and hence p(f )∗ ≥ q n (f )∗ by definition of the ∗-operation.

14 | C. A. Agyingi, P. Haihambo, and H.-P.A. Künzi So 0 ≤ p(f ) − p(f )∗ ≤ q n (f ) − q n (f )∗ = 2(q n (f ) − q n+1 (f )) (compare [13, n n ∗ proof of Proposition 3.1]), since q n+1 (f ) = q (f )+2q (f ) . In particular this yields that p(f ) = p(f )∗ . But then h := p(f ) is minimal among the ample function pairs: Indeed let g ≤ h, that is g1 ≤ h1 and g2 ≤ h2 , and let g be an ample function pair. Then for each x ∈ X, ˙ h1 (y)) ≤ sup(d(x, y)− ˙ g1 (y)) ≤ g2 (x) h2 (x) = sup(d(x, y)− y∈X

y∈X

by ampleness of g. So g2 = h2 . Similarly g1 = h1 and therefore the pair h is minimal ample. Remark 1. We next note that Proposition 2 can also be proved by Zorn’s Lemma (compare [4, Section 1.9]). Proof. We only sketch the idea of this proof. Let P be the set of all maps p from P X to P X satisfying the following two conditions (1) p(f ) ≤ f and (2) D(p(f ), p(g )) ≤ D(f , g ) whenever f , g ∈ P X . Define a partial order  on P as follows: p  q iff [p(f ) ≤ q(f ) and D(p(f ), p(g )) ≤ D(q(f ), q(g ))] whenever f , g ∈ P X . Let K be a nonempty chain in (P, ). Define a map t : P X → P X by t(f )(x) := ( inf (k(f ))1 (x), inf (k(f ))2 (x)) k ∈K

k ∈K

whenever x ∈ X, where the infima are taken pointwise in [0, ∞). One verifies that t is a lower bound of K in (P, ). By Zorn’s Lemma P has a minimal element, say m. Let us now see that m(f ) ∈ Q X whenever f ∈ P X . We need the following lemma that is of independent interest. Lemma 3. (compare [4, Section 1.3]) Let (X, d) be a T0 -quasi-metric space and let f ∈ P X . For each x ∈ X set (p x (f ))1 (z) = f1 (z) if z ∈ X \ {x} and ˙ f2 (y) : y ∈ X } (p x (f ))1 (x) = sup{d(y, x)−

and (p x (f ))2 (z) = f2 (z) if z ∈ X \ {x} and ˙ f1 (y) : y ∈ X }. (p x (f ))2 (x) = sup{d(x, y)−

Then for each x ∈ X, p x ∈ P.

Proof. We leave the details of the proof of the lemma to the reader.

We now finish the proof of Remark 1. For each x ∈ X we obviously have that p x ◦ m ∈ P and p x ◦ m  m. Hence by minimality of m, p x ◦ m = m whenever x ∈ X.

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It follows that for each x ∈ X, p x (m(f )) = m(f ) whenever f ∈ P X . Thus by the definition of the elements of Q X we conclude that m(f ) ∈ Q X whenever f ∈ P X by [11, Remark 2]. Remark 2. (The general quasi-metric “segment I ab ”.) Let X = [0, 1]. Choose a, b ∈ [0, ∞) such that a + b = 0. Set d ab (x, y) = (x − y)a if x > y and d ab (x, y) = (y − x)b if y ≥ x. Then ([0, 1], d ab ) is a T0 -quasi-metric space, as it is readily checked, by considering the various cases for the underlying asymmetric norm n ab on R defined by n ab (x) = xa if x > 0 and n ab (x) = −xb if x ≤ 0. Remark 3. (compare [4, Section 1.10]) Using some p (as in Proposition 2) we can define for any f ∈ Q X a map Ψ : [0, 1] × Q X → Q X as follows: (t, g ) → p t (g ) = p(tf + (1 − t)g) from p0 , the identity on Q X , to p1 , the constant map Q X → {f } ⊆ Q X . Here we used the fact that tf + (1 − t)g ∈ P X . Note that moreover, for any s, t ∈ [0, 1] with t ≤ s one has D(p s (g), p t (g )) = D(p(sf + (1 − s)g ), p(tf + (1 − t)g )) ≤ D(sf + (1 − s)g, tf + (1 − t)g) = supx∈X ((s − ˙ s − t)g1 (x)) ∨ supx∈X ((s − t)g2 (x)−( ˙ s − t)f2 (x)) = (s − t)D(f , g ). t)f1 (x)−( Furthermore D(f , g ) ≤ D(f , p s (g)) + D(p s (g ), p t (g ))

Therefore and thus

+

D(p t (g), g) =

D(p1 (g ), p s (g )) + D(p s (g ), p t (g ))

+

D(p t (g ), p0 (g)) =

(1 − s)D(f , g ) + D(p s (g ), p t (g ))

+

tD(f , g ).

D(p s (g ), p t (g )) ≥ (s − t)D(f , g ) D(p s (g ), p t (g )) = (s − t)D(f , g )

whenever g ∈ Q X and 0 ≤ t ≤ s ≤ 1. If 0 ≤ s ≤ t ≤ 1 and g ∈ Q X , then a similar computation yields D(p s (g ), p t (g )) = D−1 (p t (g), p s (g )) = (t − s)D−1 (f , g ) = (t − s)D(g, f ). Set a = D(f , g ) and b = D(g, f ). Then the map ([0, 1], d ab ) → (Q X , D) defined by s → p s (g ) yields an isometric map connecting g to f .

We next show that if we equip the unit interval [0, 1] of the reals with its standard topology τ(u s ) (where, as usual, u s also denotes the restriction of the metric u s to [0, 1]) and Q X with the topology τ(D), then the map Ψ is continuous, that is, Ψ : [0, 1] × Q X → Q X is a homotopy and Q X is contractible in the classical sense. Indeed suppose that the sequence (s n )n∈N converges to s in ([0, 1], τ(u s )) and the sequence (g n )n∈N converges to g in (Q X , τ(D)), that is D(g, g n ) → 0 if n → ∞.

16 | C. A. Agyingi, P. Haihambo, and H.-P.A. Künzi Then for each n ∈ N, by the triangle inequality we have that D(p s (g ), p s n (g n )) ≤ D(p s (g ), p s n (g )) + D(p s n (g), p s n (g n )). Therefore for each n ∈ N we see that D(p s (g ), p s n (g )) = (s − s n )D(f , g ) if s ≥ s n and

D(p s (g ), p s n (g )) = (s n − s)D(g, f )

if s n ≥ s, according to the calculations completed above. Moreover for each n ∈ N, we get that

D(p s n (g ), p s n (g n )) = (1 − s n )D(g, g n ) by definition of D. By our assumptions we conclude that D(p s (g ), p s n (g )) → 0 and

D(p s n (g ), p s n (g n )) → 0

if n → ∞. Therefore D(p s (g ), p s n (g n )) → 0 if n → ∞, and hence Ψ is indeed continuous. Proposition 4. (compare [4, Section 1.11]) Let (Y , d) be a T0 -quasi-metric space and let X be a (nonempty) subspace of (Y , d). Then there exists an isometric embedding τ : Q X → Q Y such that τ(f )|X = f whenever f ∈ Q X . Proof. Let x0 ∈ X be fixed and choose a retraction p : P Y → Q Y satisfying the conditions (a) and (b) of Proposition 2. Furthermore let s : Q X → P Y be defined as s(f ) = f  where f1 (y) = f1 (y) whenever y ∈ X, and f1 (y) = f1 (x0 ) + d(x0 , y) whenever y ∈ Y \ X. Similarly f2 (y) = f2 (y) whenever y ∈ X, and f2 (y) = d(y, x0 )+ f2 (x0 ) whenever y ∈ Y \ X. We prove that f  ∈ P Y by considering four cases: Suppose x ∈ X and y ∈ X ; then f2 (x) + f1 (y) = f2 (x) + f1 (y) ≥ d(x, y). Suppose that x ∈ Y \ X and y ∈ Y \ X ; then f2 (x) + f1 (y) = d(x, x0 ) + f2 (x0 ) + f1 (x0 ) + d(x0 , y) ≥ d(x, x0 ) + d(x0 , y) ≥ d(x, y). Suppose that x ∈ X and y ∈ Y \ X ; then f2 (x) + f1 (y) = f2 (x) + f1 (x0 ) + d(x0 , y) ≥ d(x, x0 ) + d(x0 , y) ≥ d(x, y). Suppose that x ∈ Y \ X and y ∈ X ; then f2 (x) + f1 (y) = d(x, x0 ) + f2 (x0 ) + f1 (y) ≥ d(x, x0 ) + d(x0 , y) ≥ d(x, y). Thus f  ∈ P Y .

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Define τ = p ◦ s : Q X → Q Y . Note that τ(f )|X = p(f  )|X = f whenever f ∈ Q X , since p(f  ) ≤ f  , therefore p(f  )|X ≤ f  |X = f , and f is minimal on X. Furthermore for any f , g ∈ Q X , D(f , g ) = D(τ(f )|X , τ(g )|X ) D(p(f ), p(g )) 



≤ ≤

D(τ(f ), τ(g )) = D(f  , g ) = D(f , g)

where the last equality follows from the definition of f  and g . Definition 5. (compare [8, Remark 7]) Let X be a subspace of a T0 -quasi-metric space (Y , d). Then Y is called a tight extension of X if for any quasi-pseudometric e on Y that satisfies e ≤ d and agrees with d on X × X we have that e = d. In [8, Remark 7] it was shown that for any T0 -quasi-metric space (X, d) we have that the isometric embedding j : X → Q X is tight, that is, Q X is a tight extension of j(X ). Remark 4. (see [4, Section 1.12]) For any T0 -quasi-metric tight extension Y1 of X, any T0 -quasi-metric extension (Y2 , d) of X and any nonexpansive map ψ : Y1 → Y2 satisfying ψ(x) = x whenever x ∈ X, ψ is necessarily an isometric map. Proof. Otherwise the quasi-pseudometric ρ : Y1 × Y1 → [0, ∞) defined by (x, y) → ρ(x, y) = d(ψ(x), ψ(y)) would contradict the tightness of the extension Y1 of X.

Proposition 6. (compare [4, Section 1.13]) Let (Y , d) be a T0 -quasi-metric tight extension of X. Then the restriction map defined by f → f |X whenever f ∈ Q Y is a bijective isometric map Q Y → Q X . Proof. Choose a retraction p : P X → Q X satisfying the conditions (a) and (b) of Proposition 2 and let ψ : Q Y → Q X : f → p(f |X ) denote the composition of p with the restriction map. Then ψ is a nonexpansive map and Q Y and Q X are T0 -quasimetric extensions of X. According to Remark 4 ψ must be an isometric map, since Q Y is a tight extension of X, because Q Y is a tight extension of Y and Y is a tight extension of X. By Proposition 4 there is an isometric embedding τ : Q X → Q Y satisfying τ(f )|X = f for all f ∈ Q X . Then we have ψ(τ(f )) = p(τ(f )|X ) = p(f ) = f for all f ∈ Q X and thus ψ is necessarily surjective. But a surjective isometric map on a T0 -quasi-metric domain is necessarily bijective. So τ : Q X → Q Y has to be the map inverse to ψ and thus for any f ∈ Q Y we necessarily have the formula f |X = τ(ψ(f ))|X = ψ(f ) ∈ Q X , that is, the restriction map Q Y → P X : f → f |X maps

18 | C. A. Agyingi, P. Haihambo, and H.-P.A. Künzi Q Y already onto Q X , without having to be composed with the retraction map p. Hence we see that for any T0 -quasi-metric tight extension Y of X the restriction map Q Y → Q X : f → f |X yields a bijective isometric map between Q Y and Q X . Proposition 7. (compare [6, Theorem 2]) Let X be a subspace of the T0 -quasimetric space (Y , d). Then the following three conditions are equivalent: (a) Y is a tight extension of X. (b) d(y1 , y2 ) = sup{(d(x1 , x2 ) − d(x1 , y1 ) − d(y2 , x2 )) ∨ 0 : x1 , x2 ∈ X } whenever y1 , y2 ∈ Y . (c) f y |X (x) = (d(y, x), d(x, y)) with x ∈ X is minimal on X whenever y ∈ Y and the map Φ : (Y , d) → (Q X , D) : y → f y |X is an isometric embedding. Proof. (a) → (b): Let Y be a T0 -quasi-metric tight extension of X. By Proposition 6 the map Q Y → Q X defined by f → f |X defines a bijective isometric map between Q Y and Q X . Hence the extension Y of X, as a subspace of Q Y , fulfils condition (b) of Proposition 7, since the extension Q X of X satisfies it by [8, Remark 7]. (b) → (c): For any x1 , x2 ∈ X and y1 ∈ Y we have that d(x1 , x2 ) ≤ d(x1 , y1 ) + d(y1 , x2 ). Therefore for any x1 , x2 ∈ X and y1 , y2 ∈ Y we see that d(x1 , x2 ) − d(x1 , y1 ) − d(y2 , x2 ) ≤ d(y1 , x2 ) − d(y2 , x2 ). Consequently for any y1 , y2 ∈ Y we have by (b) that d(y1 , y2 ) = sup{(d(x1 , x2 ) − d(x1 , y1 ) − d(y2 , x2 )) ∨ 0 : x1 , x2 ∈ X } ≤ Similarly

˙ d(y2 , x2 )) : x2 ∈ X } ≤ d(y1 , y2 ). sup{(d(y1 , x2 )−

d(x1 , x2 ) ≤ d(x1 , y2 ) + d(y2 , x2 )

whenever x1 , x2 ∈ X and y2 ∈ Y . It follows that for each x1 , x2 ∈ X and y1 , y2 ∈ Y we have that d(x1 , x2 ) − d(y2 , x2 ) − d(x1 , y1 ) ≤ d(x1 , y2 ) − d(x1 , y1 ). Thus for any y1 , y2 ∈ Y we see by (b) that d(y1 , y2 ) ≤ sup{(d(x1 , x2 ) − d(y2 , x2 ) − d(x1 , y1 )) ∨ 0 : x1 , x2 ∈ X } ≤ ˙ d(x1 , y1 )) : x1 ∈ X } ≤ d(y1 , y2 ). sup{(d(x1 , y2 )−

Hence we conclude that d(y1 , y2 ) = D(f y1 |X , f y2 |X ) whenever y1 , y2 ∈ Y .

Tight Extensions of T0 -Quasi-Metric Spaces

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As we have just shown, for any y1 , y2 ∈ Y we have that

and

˙ d(x1 , y1 )) : x1 ∈ X } d(y1 , y2 ) = sup{(d(x1 , y2 )− ˙ d(y2 , x2 )) : x2 ∈ X }. d(y1 , y2 ) = sup{(d(y1 , x2 )−

Substituting x2 ∈ X for y2 and x1 ∈ X for y1 , respectively, we obtain the two equations ˙ d(x1 , y1 )) : x1 ∈ X } (f y1 )1 (x2 ) = d(y1 , x2 ) = sup{(d(x1 , x2 )−

whenever y1 ∈ Y and x2 ∈ X, and ˙ d(y2 , x2 )) : x2 ∈ X } (f y2 )2 (x1 ) = d(x1 , y2 ) = sup{(d(x1 , x2 )−

whenever y2 ∈ Y and x1 ∈ X. By [11, Remark 2] the restriction f y |X is minimal on X whenever y ∈ Y . (c)→ (a): Let q : Y × Y → [0, ∞) be a quasi-pseudometric on Y such that q ≤ d and q|X×X = d|X×X . According to (c) and since f y |X is minimal whenever y ∈ X we have d(y1 , y2 ) = ˙ d(y2 , x)) = supx∈X (d(x, y2 )− ˙ d(x, y1 )) whenD(f y1 |X , f y2 |X ) = supx∈X (d(y1 , x)− ever y1 , y2 ∈ Y by [8, Lemma 7]. Then substituting ˙ d(y2 , x2 )) : x2 ∈ X } d(x1 , y2 ) = sup{(d(x1 , x2 )−

into the formula ˙ d(x1 , y1 )) : x1 ∈ X }, d(y1 , y2 ) = sup{(d(x1 , y2 )−

we obtain d(y1 , y2 ) = sup sup {(d(x1 , x2 ) − d(x1 , y1 ) − d(y2 , x2 )) ∨ 0} x1 ∈X x2 ∈X

≤

sup {(q(x1 , x2 ) − q(x1 , y1 ) − q(y2 , x2 )) ∨ 0} ≤ q(y1 , y2 )

x1 ,x2 ∈X

whenever y1 , y2 ∈ Y by our assumption. Consequently q = d. Therefore condition (a) is satisfied. Remark 5. (compare [4, Section 1.14]) Let (Y , d) be a T0 -quasi-metric of X. Elaborating further on Proposition 7 we see that there is only one isometric embedding ϕ : Y → Q X satisfying ϕ(x) = f x whenever x ∈ X, since for such an embedding

20 | C. A. Agyingi, P. Haihambo, and H.-P.A. Künzi ϕ : Y → Q X , y ∈ Y and x ∈ X, we have (f y )2 |X (x) = d(x, y) = D(ϕ(x), ϕ(y)) = D(f x , ϕ(y)) = (ϕ(y))2 (x); therefore (f y )2 |X = (ϕ(y))2 . In a similar way, one can show that (f y )1 |X = (ϕ(y))1 whenever y ∈ Y . In particular we see that the tight extension Y of X can be understood as a subspace of the extension Q X of X. Hence Q X is maximal among the T0 -quasimetric tight extensions of X. We finally discuss an asymmetric version of a result due to Herrlich. To this end, given a quasi-pseudometric space (X, d X ), for each x ∈ X and ϵ > 0, by C d X (x, ϵ) = {y ∈ X : d X (x, y) ≤ ϵ} we denote the so-called closed ball of radius ϵ at x. (Note that it is closed with respect to the topology τ(d−1 X ) .) In X we say that the (double) family C = (C d X (x i , r i ), C d−1 (y i , s i ))i∈I of such X balls meets potentially provided that there exists a T0 -quasi-metric extension  (Y , d Y ) of (X, d X ) such that i∈I (C d Y (x i , r i ) ∩ C d−1 (y i , s i )) = ∅. Y

Proposition 8. (compare [6, Proposition]) If C = (C d X (x i , r i ), C d−1 (x i , s i ))i∈I is a X (double) family of balls in a T0 -quasi-metric space (X, d), then the following conditions are equivalent: (1) C meets potentially in X. (2) For any i, j ∈ I, C d X (x i , r i ) meets with any C d−1 (x j , s j ) potentially in X. X (3) d X (x i , x j ) ≤ r i + s j whenever i, j ∈ I. (4) There exists a minimal (ample) function pair t = (t1 , t2 ) on X with t2 (x i ) ≤ r i and t1 (x i ) ≤ s i whenever i ∈ I.

Proof. (1) → (2) → (3) are obvious. (3) → (4) is obvious for I = ∅. Otherwise define a function pair f = (f1 , f2 ) on Y = {x i : i ∈ I } by setting for each y ∈ Y , f1 (y) = inf{s i : x i = y} and f2 (y) = inf{r i : x i = y}. Choose y0 ∈ Y . Set g1 (x) = f1 (x) if x ∈ Y , and g1 (x) = f1 (y0 ) + d X (y0 , x) if x ∈ X \ Y . Furthermore set g2 (x) = f2 (x) if x ∈ Y , and g2 (x) = d X (x, y0 ) + f2 (y0 ) if x ∈ X \ Y. Then g1 (x i ) ≤ s i and g2 (x i ) ≤ r i whenever i ∈ I. Furthermore d X (x, x ) ≤ g2 (x) + g1 (x ) whenever x ∈ X and x ∈ X. Thus g = (g1 , g2 ) is an ample function pair on X and by Zorn’s Lemma there is a minimal ample pair t = (t1 , t2 ) on X such that t ≤ g. (4) → (1) : Let t = (t1 , t2 ) be a minimal ample pair on X with t1 (x i ) ≤ s i and  t2 (x i ) ≤ r i whenever i ∈ I. If t = f x for some x ∈ X, then x ∈ i∈I (C d X (x i , r i ) ∩ C d−1 (x i , s i )). Thus the family C meets in X. X Otherwise extend X to a space Y by adding one point y0 to X and by defining a T0 -quasi-metric d Y on Y extending d X and satisfying d Y (x, y0 ) = t2 (x) and

Tight Extensions of T0 -Quasi-Metric Spaces

| 21

d Y (y0 , x) = t1 (x) whenever x ∈ X. One readily checks that d Y is a T0 -quasi-metric on Y (compare [8, end of proof of Theorem 1]).  Then y0 ∈ i∈I (C d Y (x i , r i ) ∩ C d−1 (x i , s i )) and we are done. Y

The following example answers a question asked by one of the referees.

Remark 6. (see [8, Example 7]) Let X be the set {0, 1} equipped with the discrete metric. Then Q X is isometric to the product set [0, 1]×[0, 1] of two copies of real unit intervals [0, 1] equipped with the T0 -quasi-metric D((α, β), (α , β )) = u(α, α ) ∨ u(β, β ) whenever (α, β), (α , β ) ∈ [0, 1] × [0, 1], while T X can be identified with the subspace {(α, 1 − α) : α ∈ [0, 1]} in this product, that is, T X is isometric to the unit interval [0, 1] of the reals equipped with the restriction of the metric u s . Observe that, intuitively, Q X consists of all the points z determined by the equations 1 = D(f0 , z) + D(z, f1 ) and 1 = D(f1 , z) + D(z, f0 ). We recall from [8, Proposition 5] that for any metric space (X, m), there exists a canonical isometric embedding of T X into Q X . Acknowledgement: We would like to thank an anonymous referee for his useful comments.

Bibliography E. Colebunders, S. De Wachter and B. Lowen, Intrinsic approach spaces on domains, Topology Appl. 158: 2343–2355, 2011. [2] A. Collins Amburo, P. Haihambo and H.-P.A. Künzi, Endpoints in T0 -quasi-metric spaces, in preparation. [3] M. de Brecht, Quasi-Polish spaces, Ann. Pure Appl. Logic 164: 356–381, 2013. [4] A.W.M. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. Math. 53: 321–402, 1984. [5] P. Fletcher and W.F. Lindgren, Quasi-uniform Spaces, Dekker, New York, 1982. [6] H. Herrlich, Hyperconvex hulls of metric spaces, Topology Appl. 44: 181–187, 1992. [7] J.R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39: 65–76, 1964. [8] E. Kemajou, H.-P.A. Künzi and O.O. Otafudu, The Isbell-hull of a di-space, Topology Appl. 159: 2463–2475, 2012. [9] M.A. Khamsi and W.A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, John Wiley, New York, 2001. [10] H.-P.A. Künzi, An introduction to quasi-uniform spaces, Contemp. Math. 486: 239–304, 2009. [11] H.-P.A. Künzi and M. Sanchis, The Katětov construction modified for a T0 -quasi-metric space, Topology Appl. 159: 711–720, 2012. [1]

22 | C. A. Agyingi, P. Haihambo, and H.-P.A. Künzi [12] H.-P.A. Künzi and M. Sanchis, Addendum to “The Katětov construction modified for a T0 -quasi-metric space”, Math. Struct. in Comp. Science (to appear). [13] U. Lang, Injective hulls of certain discrete metric spaces and groups, arXiv:1107.5971v2 [math.GR] 28 June 2012. [14] J. Marín, S. Romaguera and P. Tirado, Weakly contractive multivalued maps and wdistances on complete quasi-metric spaces, Fixed Point Theory Appl. 2011:2, 9pp, 2011. [15] V. Selivanov, On the difference hierarchy in countably based T0 -spaces, Electronic Notes in Theoretical Computer Science 221: 257–269, 2008.

Klaus Ambos-Spies

On the Strongly Bounded Turing Degrees of Simple Sets¹ Abstract: We study the r-degrees of simple sets under the strongly bounded Turing reducibilities r = cl (computable Lipschitz reducibility) and r = ibT (identity bounded Turing reducibility) which are defined in terms of Turing functionals where the use function is bounded by the identity function up to an additive constant and the identity function, respectively. We call a c.e. r-degree a simple if it contains a simple set and we call a nonsimple otherwise. As we show, the ibTdegree of a c.e. set A is simple if and only if the cl-degree of A is simple, and there are nonsimple c.e. r-degrees > 0. Moreover, we analyze the distribution of the simple and nonsimple r-degrees in the partial ordering of the c.e. r-degrees. Among the results we obtain are the following. (i) For any c.e. r-degree a > 0, there are simple r-degrees which are below a, above a and incomparable with a. (ii) For any c.e. r-degree a > 0, there are nonzero nonsimple c.e. r-degrees which are below a and incomparable with a; and there is a nonsimple c.e. r-degree above a if and only if a is not contained in the complete wtt-degree. (iii) There are infinite intervals of c.e. r-degrees entirely consisting of nonsimple c.e. r-degrees respectively simple rdegrees. (iv) Any c.e. r-degree is the join of two nonsimple c.e. r-degrees whereas the class of the nonzero c.e. r-degrees is not generated by the simple r-degrees under join though any simple r-degree is the join of two lesser simple r-degrees. Moreover, neither the class of the nonsimple c.e. r-degrees nor the class of the simple r-degrees generates the class of c.e. r-degrees under meet. Keywords: Computable Lipschitz reducibility, identity bounded Turing reducibility, strong reducibilities, computably enumerable degrees, computably enumerable sets, simple sets, Post’s Program Mathematics Subject Classification 2010: 03D25, 03D30 || Klaus Ambos-Spies: Department of Mathematics and Computer Science, University of Heidelberg, Germany

1 This research was partially done whilst the author was a visiting fellow at the Isaac Newton Institute for the Mathematical Sciences in the programme ‘Semantics & Syntax’ in the spring of 2012.

24 | Klaus Ambos-Spies

1 Introduction The notion of a simple set is among the most well known concepts in computability theory. Simple sets were introduced in 1944 by Post in his seminal paper [15] where he raised the question of whether there are more than two computably enumerable (c.e.) Turing degrees which became known as Post’s Problem. This paper may be viewed as the origin of degree theory and specifically of the theory of the c.e. (Turing) degrees which became one of the most productive areas in computability theory (see e.g. Odifreddi [14] and Soare [16]). Post’s approach to solve Post’s Problem - now known as Post’s Program - was a structural one. He tried to find a nonvacuous property P of c.e. sets such that any c.e. set with this property is neither computable nor (Turing) complete. In order to get such a property, Post looked at c.e. sets with “thin” (but infinite) complements. In particular, Post called a c.e. set A simple if its complement is infinite but does not contain any infinite c.e. set as a subset. He showed that simple sets exist and that they are not complete under many-one reducibility (in fact under bounded truth-table reducibility) but that they may be complete under truth-table reducibility. So, in order to guarantee Turing incompleteness, Post suggested some refinements of simplicity, namely hyper-simplicity and hyper-hyper-simplicity. As it turned out, however, neither of these properties guarantees Turing-incompleteness and - in a strict sense - Post’s Program fails. The solution of Post’s Problem was given by Friedberg and Muchnik in 1956, by introducing the priority method. Though this constructive approach proved to be very powerful and most of the results on the structure of the c.e. degrees are proven by priority arguments, Post’s structural approach and variations of Post’s Program still attracted researchers in the past. So, by replacing thinness properties by other structural properties, several solutions to Post’s Problem were given where, probably, the solution found by Harrington and Soare [11] is the most satisfying one. Moreover, Post’s structural approach proved to be very useful for the study of the c.e. degrees under the strong reducibilities as many-one (m), bounded truth-table (btt) and truth-table (tt) reducibilities (see e.g. Chapter III in Odifreddi [14]). One may consider a generalization of Post’s Program in a different direction: instead of replacing thinness properties by more general structural properties one may look at degrees in general and not only at the complete degree. For this sake, given a property P of c.e. sets implying noncomputability, let Pρ (deg ρ (A)) ⇔ there is a set P with property P such that P =ρ A be the corresponding degree property (for a given reducibility ≤ρ ). Then, in order to solve Post’s Problem for ρ-reducibility, i.e., to show that there are at least

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two nonzero c.e. ρ-degrees, it suffices to show that there are noncomputable c.e. sets A and B such that the ρ-degree of A contains a set with property P (i.e., Pρ (deg ρ (A)) holds) and the ρ-degree of B does not contain a c.e. set with property P (i.e., Pρ (deg ρ (B)) fails). This Generalized Post’s Program is more technical but also more powerful than Post’s original program. Namely, the simple set property S does not suffice to solve Post’s Problem for tt-reducibility in the sense of Post’s Program (since, as mentioned before, Post has shown that there are tt-complete simple sets), but it suffices in the sense of Generalized Post’s Program since - by Post - there is a simple set A (hence Stt (degtt (A)) holds) while - as shown by Jockusch [12] there is a noncomputable c.e. set B which is not tt-equivalent to any simple set (hence Stt (degtt (B)) fails). But, even in the sense of Generalized Post’s Program, simplicity fails to solve Post’s Problems for Turing reducibility, since Dekker has shown in 1954 that any nonzero c.e. Turing degree contains a (hyper-)simple set. The stronger property HHS of hyper-hyper-simplicity, however, solves the original Post’s Problem for Turing reducibility in the sense of Generalized Post’s Program since Martin has shown that a c.e. Turing degree contains an hh-simple set if and only if the degree is high. So there are noncomputable c.e. sets A and B such that HHST (degT (A)) holds while HHST (degT (B)) fails. (For definitions and references see Chapter III in Odifreddi [14].) The above described role played by simple sets in the study of the degrees of c.e. sets led us to look at the degrees of simple sets under the strongly bounded Turing reducibilities. These reducibilities - computable Lipschitz reducibility (clreducibility for short) and identity bounded Turing reducibility (ibT-reducibility for short) are defined in terms of Turing functionals where the use function is bounded by the identity function up to an additive constant and the identity function, respectively. They were introduced only fairly recently by Downey, Hirschfeldt and Laforte [8] and Soare [17], respectively. Computable Lipschitz reducibility is of particular interest in the theory of algorithmic randomness since it does not only capture relative computability but also measures the relative degree of randomness in the sense of Kolmogorov complexity. We refer the reader to the recent monograph of Downey and Hirschfeldt [9] and to Ambos-Spies et al. [3] for more motivation and details. The strongly bounded Turing reducibilities have some interesting structural properties which distinguish them from the other computable reducibilities. So there are no complete sets under these reducibilities (Downey, Hirschfeldt and LaForte [8]); in fact, there are so-called r-maximal pairs of c.e. sets, i.e., c.e. sets A and B such that there is no c.e. set C such that A ≤r C and B ≤r C for r = ibT, cl (Barmpalias [5] and Fan and Lu [10]; see [3] for more on maximal pairs). (By the

26 | Klaus Ambos-Spies former, Post’s Program does not make sense in the context of these reducibilities, but one may consider Generalized Post’s Program here.) We start our investigation in the degrees of simple sets under the strongly bounded Turing reducibilities by showing that a set A is ibT-equivalent to a simple set if and only if it is cl-equivalent to a simple set, and we show that there are noncomputable c.e. sets which are not r-equivalent to any simple set (where r = ibT, cl throughout this paper). We then analyze the distribution of the classes Sr and Sr of the c.e. r-degrees containing respectively not containing simple sets in the partial ordering (Rr , ≤) of the c.e. r-degrees. While the class Sm of the simple many-one degrees is a proper ideal in the partial ordering (Rm , ≤) of the c.e. m-degrees, it turns out that the class Sr of the simple r-degrees is quite scattered in the partial ordering (Rr , ≤). For instance, for any nonzero c.e. r-degree a, there are simple and nonzero nonsimple r-degrees below a, there are simple and nonsimple r-degrees incomparable with a, and - unless a is the r-degree of a wtt-complete set - there are simple and nonsimple r-degrees above a. For wtt-complete A we show that deg r (A) is simple. So, in contrast to the situation for the m-degrees where the class of the simple degrees is an ideal, here the simple degrees are not closed downwards and a simple degree can be found above any c.e. degree. Moreover, we show that there are infinite intervals in (Rr , ≤) consisting only of simple degrees respectively nonsimple degrees and we discuss the question whether the classes Sr and Sr are closed under join and meet. Finally, we indicate that - in some sense - there are more nonsimple degrees than simple degrees. Namely while the class Sr of the nonsimple r-degrees generates the class Rr of all c.e. r-degrees under join (but not under meet), the class Sr of the simple r-degrees generates Rr neither under join nor under meet. The outline of the paper is as follows. In Sections 2 and 3 we introduce the facts on the strongly bounded Turing reducibilities and simple sets, respectively, to be used in the following. Then, after proving the coincidence theorem for simple ibTand cl-degrees (Section 4), we present our results on nonsimple r-degrees where we start with the basic construction of a nonzero nonsimple r-degree (Section 5) and then extend this construction in order to get the stronger positive results on Sr (Sections 6 and 7). We then turn to existence results for simple sets. First we show that there are simple sets above and below any noncomputable c.e. set (Section 8). By refining these results we obtain more insight in the distribution of the simple rdegrees among the c.e. r-degrees and can show that any noncomputable c.e. set is equivalent to a simple set under linearly bounded Turing reducibility (Section 9). We then show that wtt-complete sets have simple r-degrees (Section 10) and look at splittings into simple r-degrees (Section 11). Finally, in Section 12, we pose some open problems. There we also propose a further generalization of Post’s Program which allows a solution of Post’s Problem for wtt-reducibility in terms of simple

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sets. Whether the simple sets also give a solution of the original Post’s Problem for Turing reducibility along these lines is left open.

2 The strongly bounded Turing reducibilities We start with some basic definitions and notation on the reducibilities we consider here. For more details see e.g. Ambos-Spies et al. [3] or Ambos-Spies [2]. Let {Φ e }e≥0 be a standard enumeration of the Turing functionals obtained by goedelization of the oracle Turing machines, and let φ e be the use function² of Φ e . Then a set A is Turing reducible to a set B, A ≤T B, if A = Φ Be for some e ≥ 0. For any function f , we say that A is f -bounded Turing reducible to B (A ≤f -T B) if A is Turing reducible to B via a reduction where the use function is bounded by f , i.e., if there is a number e such that A = Φ Be and φ Be (x) ≤ f (x) for all x ≥ 0. This leads to the following reducibility notions. 1. A is identity-bounded-Turing (ibT-) reducible to B (A ≤ibT B) if A is f -bounded Turing reducible to B for the identity function f (x) = x. 2. A is computable Lipschitz (cl-) reducible to B (A ≤cl B) if there is a number k ≥ 0 such that A is f -bounded Turing reducible to B for f (x) = x + k. Moreover, for k ≥ 0, A is (i + k)bT-reducible to B (A ≤(i+k)bT B) if A is f -bounded Turing reducible to B for f (x) = x + k. (So A is ibT-reducible to B iff A is (i + 0)bTreducible to B, and A is cl-reducible to B iff A is (i + k)bT-reducible to B for some k ≥ 0.) 3. A is linearly bounded Turing (lbT-) reducible to B (A ≤lbT B) if A is f -bounded Turing reducible to B for a linearly bounded function f , i.e., for a function f satisfying f (x) ≤ k0 x + k1 for some numbers k0 , k1 ≥ 0 and all numbers x ≥ 0. 4. A is weak-truth-table (wtt-) reducible or bounded-Turing (bT-) reducible to B (A ≤wtt B) if A is f -bounded Turing reducible to B for some computable function f .

2 Warning: Here we work with a slightly nonstandard definition of the use. While most commonly the use function gives strict upper bounds on the oracle queries, here we consider nonstrict bounds. I.e., we let φ Xe (x) ↑ if Φ Xe (x) is undefined and we let φ Xe (x) be the greatest oracle query in the computation of Φ Xe (x) if Φ Xe (x) is defined (where we let φ Xe (x) = 0 if no queries are asked).

28 | Klaus Ambos-Spies In the following we refer to ibT and cl as the strongly bounded Turing reducibilities. Moreover we use the following convention. Convention. Since we will obtain many results for both of the strongly bounded reducibilities, ibT-reducibility and cl-reducibility, simultaneously, in order to simplify notation, r-reducibility will stand for both ibT-reducibility and cl-reducibility in the following. Similarly, if we simultaneously refer to ibT, cl and wtt then we shortly write r , and if we refer to ibT, cl, wtt and T then we write r . ρ will stand for any reducibility. We call a Turing functional Φ e an f -bounded Turing functional if φ Xe (x) ≤ f (x) for all oracle sets X and inputs x such that Φ Xe (x) is defined. ibT-, cl-, lbT- and wttfunctionals are defined correspondingly. Note that, for any computable function f , A ≤f -T B if and only if there is an f -bounded Turing functional Φ such that A ≤T B via Φ, and similarly for ibT, cl, lbT and wtt. (Namely, if A ≤f -T B via Φ e then A is f -T-reducible to B via the f -bounded Turing functional Ψ e where Ψ eX (x) = Φ Xe (x) if in the latter computation no oracle query exceeds f (x) and Ψ eX (x) is undefined otherwise.) Moreover, there are computable enumerations of the corresponding functionals (where in case of the wtt-functionals we have to include the Turing functionals bounded by partial computable functions). In the following we let ˆ e }e≥0 and {Φ ˜ e }e≥0 be computable enumerations of the ibT- and cl-functionals, {Φ ˆ e and φ ˜ e be the corresponding use functions, where the clrespectively, and let φ ˜ ˜ Xe (x) is defined. Finally we let ˜ Xe (x) ≤ x + e if Φ functionals Φ e are chosen so that φ {φ e }e≥0 and {W e }e≥0 be standard enumerations of the unary partial computable functions and c.e. sets, respectively, where W e = dom(φ e ). We use computable approximations of the functionals and the corresponding use functions which are denoted by an additional index s giving the stage of the approximation. So, for instance, Φ Xe,s (x) is the result of computing Φ Xe (x) for s steps and φ Xe,s (x) is the corresponding use. We assume that whenever Φ Xe,s (x) is defined then e, x, φ Xe,s (x) < s. This convention applies to ibT- and cl-functionals, partial computable functions and c.e. sets correspondingly. The ρ-degree of a set A is denoted by deg ρ (A), and (Rρ , ≤) denotes the partial ordering of the c.e. ρ-degrees, i.e., the ρ-degrees of c.e. sets. Lower case boldface letters denote c.e. ρ-degrees. 0 denotes the ρ-degree of the computable sets, and R+ ρ = Rρ \ {0} denotes the class of the c.e. ρ-degrees of noncomputable sets. For c.e. ρ-degrees a and b, a ∨ b denotes the join (least upper bound) of a and b in Rρ (if it exists) and a ∧ b denotes the meet (greatest lower bound) of a and b in Rρ (if it exists). In the following all sets and degrees are computably enumerable (unless explicitely stated otherwise). Note that, for any sets A and B, A ≤ibT B ⇒ A ≤cl B ⇒ A ≤lbT B ⇒ A ≤wtt B ⇒ A ≤T B

(1)

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hence degibT (A) ⊆ degcl (A) ⊆ deglbT (A) ⊆ degwtt (A) ⊆ degT (A).

(2)

The structures of the c.e. wtt- and T-degrees have been extensively studied in the literature (see e.g. Soare [16]) while the strongly bounded Turing degrees have been studied only more recently (see e.g. the recent monograph by Downey and Hirschfeldt [9] or the recent papers [2] and [3]). While the partial orderings of the c.e. wtt- and T-degrees are upper semi-lattices (but no lower semi-lattices) with least and greatest elements (namely the degrees of the computable sets and the complete sets, respectively), the partial orderings of the c.e. ibT- and cldegrees are neither upper semi-lattices nor lower semi-lattices (Downey and Hirschfeldt [9]) and do not possess greatest - in fact no maximal - elements (Downey, Hirschfeldt and LaForte [8], Barmpalias [5]). Also, while the partial orderings of the c.e. wtt- and T-degrees are dense, the partial orderings of the c.e. ibT- and cl-degrees are not dense (Barmpalias and Lewis [6], Day [7]). It seems that the partial ordering of the c.e. lbT-degrees has not been studied in the literature. One can easily show, however, that it shares the most basic properties with the partial ordering of the c.e. wtt-degrees: It is a dense and distributive upper semi-lattice with least and greatest elements but not a lower semi-lattice. We close with some simple but useful concepts and facts which will be applied in the following quite frequently. They are taken from [2] and [3]. A (computable) shift f is a strictly increasing (computable) function f : ω → ω. A shift f is nontrivial if f (x) > x for some (hence for almost all) x, and f is unbounded if, for any number k, there is a number x such that f (x) − x > k. For a set A and a computable shift f , the f -shift of A is defined by A f = {f (x) : x ∈ A}. Lemma 1 (Computable-Shift Lemma ([2], [3])). Let f be a computable shift and let A be a noncomputable c.e. set. (i) A f =wtt A. (ii) A f ≤ibT A. Moreover, if f is unbounded then A ≤cl A f (whence A f 0 for the strongly bounded Turing reducibilities r = ibT, cl, we first show that, for any c.e. set A, degibT (A) is simple if and only if degcl (A) is simple. Theorem 8 (Coincidence Theorem). For any c.e. set A, degibT (A) is simple if and only if degcl (A) is simple. For the proof of the nontrivial implication we will use the following lemma. Lemma 9 (Representation Lemma). Let A and B be noncomputable c.e. sets such that A ≤(i+k)bT B and B ≤(i+k)bT A, and let {A s }s≥0 and {B s }s≥0 be computable enumerations of A and B, respectively. There are one-to-one computable functions

32 | Klaus Ambos-Spies ˆ = range(a) and B ˆ = range(b) and for s ≥ 0, the following a and b such that, for A hold. ˆ⊆A&B ˆ⊆B A (3) ˆ =ibT A & B ˆ =ibT B A

(4)

a(s) ∈ A \ A s & b(s) ∈ B \ B s

(5)

|b(s) − a(s)| ≤ k

(6)

Lemma 9 is a symmetric version of a similar representation lemma in [2] where it is shown that any (i+ k)bT-reduction between c.e. sets is witnessed by a k-permitting argument. Since the proofs are quite similar we only sketch the proof and refer to [2] for more details. Proof (sketch).. Fix (i + k)bT-functionals Φ and Ψ such that B = Φ A and A = Ψ B . Define the length (of agreement) function l by l(s) = max{x ≤ s : ∀ y < x (B s (y) = Φ As s (y) & A s (y) = Ψ sB s (y))}. Then lims→ω l(s) = ω whence there are infinitely many expansionary stages s, i.e., stages s such that l(t) < l(s) for all t < s. Call an expansionary stage s critical if there is a number x < l(s) such that x ∈ A \ A s , and say that criticalness of s is witnessed by a stage t > s if A s  l(s) = A t  l(s). Note that criticalness of any critical stage s is witnessed by some t > s, and, for any witness t, all stages t ≥ t are witnesses too. Moreover, by noncomputability of A, there are infinitely many critical expansionary stages. So, since the set of all pairs (s, t) such that s is critical and criticalness of s is witnessed by t is computable, we can define a computable ascending sequence of expansionary stages s 0 < s1 < s2 < . . . such that s n is critical and s n+1 witnesses criticalness of s n . Now let a(n) = µ x (x ∈ A s n+1 \ A s n ) and b(n) = µ x (x ∈ B s n+1 \ B s n ). Note that, by choice of the stages s n and s n+1 , a(n) exists and a(n) < l(s n ) < l(s n+1 ). So B

Bs

n+1 A s n (a(n)) = Ψ s nsn (a(n)) = 0 and A s n+1 (a(n)) = Ψ s n+1 (a(n)) = 1.

Since Ψ is an (i + k)bT-functional it follows that B s n  a(n) + k + 1 = B s n+1  a(n) + k + 1.

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33

So b(n) exists too and b(n) ≤ a(n) + k. Moreover, the dual inequality a(n) ≤ b(n) + k follows for b(n) < l(s n ) by a symmetric argument and is for b(n) ≥ l(s n ) obvious. The above observations easily imply that the functions a and b are well defined, computable and one-to-one, and that the conditions (3) - (6) are satisfied. Proof of Theorem 8. By (2) it suffices to show that, for any c.e. set A such that degcl (A) is simple, degibT (A) is simple too. So fix c.e. sets A and B such that A =cl B and B is simple. It suffices to give a simple set C such that A =ibT C. By A =cl B we may fix k ≥ 0 such that A ≤(i+k)bT B ≤(i+k)bT A. So, given any computable enumerations {A s }s≥0 and {B s }s≥0 of A and B, respectively, by the Representation Lemma we may fix computable one-to-one functions a and b ˆ = range(a) and B ˆ = range(b), (3) to (6) hold. such that, for A Then C is given by the following computable enumeration. s = 0. C0 = {a(0)}. s > 0. C s = C s−1 ∪ {a(s)} ∪ {x : a(s) < x & [x − k, x + k] ⊆ B s }. ˆ The latter is done For a proof of C =ibT A, by (4) it suffices to show that C =ibT A. by showing that, for s > 0,

a(s) = µx(x ∈ C s \ C s−1 ).

(7)

ˆ s = {a(0), . . . , a(s)}, this implies ˆs \ A ˆ s−1 ) for A Note that, by a(s) = µx(x ∈ A ˆ ˆ that C ≤ibT A and A ≤ibT C by permitting. Now, for a proof of (7), observe that, by definition of C, a(s) ∈ C s and no new number < a(s) enters C at stage s. So it suffices to show that, for s > 0, a(s) ∈/ C s−1 . For a contradiction, pick s > 0 minimal such that a(s) ∈ C s−1 and fix t < s minimal such that a(s) ∈ C t . Then, by definition of C t , a(s) is an element of the third part of C t , hence [a(s)− k, a(s)+ k] ⊆ B t . It follows with (6) that b(s) ∈ B t . But, by t < s, this contradicts (5). Finally, for a proof that C is simple, note that, by C =ibT A and by noncomputability of A, C is noncomputable hence co-infinite. So, given an infinite c.e. set V, it suffices to show that C ∩ V = ∅. In order to do so, note that by the Simple Set Lemma there are infinitely many numbers x ∈ V such that the interval [x − k, x + k] is contained in the simple set B. So, given any computable enumeration {V s }s≥0 of V we can effectively enumerate numbers x0 < x1 < x2 < . . . and corresponding stages s0 < s1 < s2 < . . . such that x n ∈ V s n and [x n − k, x n + k] ⊆ B s n . By nonˆ this implies that there are some s and n such that s > s n and computability of A a(s) < x n . So, by definition of C s , x n ∈ C s hence C ∩ V = ∅.

34 | Klaus Ambos-Spies

5 A nonzero nonsimple cl-degree We now show that, for r = ibT, cl, there is a nonsimple c.e. r-degree a > 0. By the Coincidence Theorem, it suffices to show this for r = ibT. Theorem 10. There is a noncomputable c.e. set A such that degibT (A) is not simple. Proof. By a finite-injury argument we enumerate a noncomputable c.e. set A which is not ibT-equivalent to any simple set. As usual, we let A s denote the finite part of A enumerated by the end of stage s (A0 = ∅). In order to make A noncomputable it suffices to meet the requirements 2e : A = φ e (e ≥ 0)

and, in order to ensure that A is not ibT-equivalent to any simple set, it suffices to meet the requirements ˆ W e0 & W e = Φ ˆ Ae ] ⇒ W e is not simple 2e+1 : [A = Φ e1 0 0 2

for e = e0 , e1 , e2  ≥ 0. Before we give the formal construction of the set A, we describe the strategies for meeting the requirements n . Fix uniformly computable infinite and pairwise disjoint sets R n (n ≥ 0), say R n = ω[n] = {n, x : x ≥ 0}, and reserve R n for the strategy for meeting requirement n . The strategy for meeting a noncomputability requirement 2e is standard. We wait for a stage s > 2e such that there is a number x ∈ R2e satisfying φ e,s (x) = 0. Then, for the least such stage s and the least corresponding number x, x is put into A at stage s + 1 thereby ensuring A(x) = φ e (x). Note that if there are no such stage s and number x then φ e (x) = 0 for all x ∈ R2e , and no number from R2e is put into A. So 2e is met in this case too. The strategy for meeting a nonsimplicity requirement 2e+1 (e = e0 , e1 , e2 ) is a diagonalization strategy aiming at destroying the premise of the requirement. The strategy will fail to achieve this goal only if R2e+1 ∩ W e0 is finite hence (by Proposition 6) W e0 is not simple. So requirement 2e+1 is met no matter whether or not the strategy becomes active. The strategy is as follows. Wait for a stage s > 2e + 1 such that for some number x ∈ R2e+1 the following hold. (S1 ) x ∈ W e0 ,s (S2 ) x ∈/ A s ˆ W e0 ,s (x) (S3 ) A s (x) = Φ e1 ,s ˆ As  x (S4 ) W e0 ,s  x = Φ e2 ,s

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35

Then enumerate x into A at stage s + 1 and restrain all lesser numbers from A thereby guaranteeing that A  x = A s  x. To show that this action guarantees that the premise of requirement 2e+1 ˆ W e0 ,s (x) = Φ ˆ W e0 (x). Then, by (S2 ) and (S3 ), fails, first assume that Φ e1 ,s e1 W

W

ˆ e0 ,s (x) = Φ ˆ e0 ( x ) , A(x) = A s+1 (x) = A s (x) = Φ e1 ,s e1 W

W

W

ˆ e0 ,s (x) = Φ ˆ e0 . On the other hand, if Φ ˆ e0 (x) (where, by (S3 ), hence A = Φ e1 e1 ,s e1 ˆ W e0 ,s (x) ↓) then - since Φ ˆ e is an ibT-functional and since, by (S1 ), x has entered Φ e1 ,s 1 W e0 by stage s already - a number x < x has to enter W e0 after stage s, while, by the restraint imposed on A at stage s + 1, no number < x enters A after stage s. It follows with (S4 ) that ˆ Ae s ,s (x ) = Φ ˆ Ae (x ), W e0 (x ) = W e0 ,s (x ) = Φ 2 2 ˆ Ae . So in either case the premise of 2e+1 is not correct. hence W e0 = Φ 2 For a proof that the above strategy guarantees that requirement 2e+1 is met, w.l.o.g. assume that the premise of 2e+1 holds and the strategy for meeting 2e+1 does not become active, hence R2e+1 ∩ A = ∅. Then, for any x ∈ R2e+1 , the conditions (S2 ), (S3 ) and (S4 ) hold for all sufficiently large stages s. So, for any x ∈ R2e+1 , condition (S1 ) must fail at all sufficiently large stages s, hence x ∈/ W e0 . It follows that W e0 ∩ R2e+1 = ∅, hence W e0 is not simple. So requirement 2e+1 is met. Note that the above strategies are finitary. (More precisely, the 2e -strategy is positive and acts at most once while the 2e+1 -strategy is positive and negative and acts at most once unless its restraint is injured.) So the strategies can be easily combined in a standard finite injury fashion in the actual construction of A as follows where we attach the restraint function r(n, s) to requirement n denoting the restraint imposed by n at the end of stage s. Since the noncomputability requirements are purely positive, r(2e, s) = 0 for e, s ≥ 0; and r(2e + 1, 0) = 0 for e ≥ 0.

Stage s + 1. Requirement 2e requires attention at stage s + 1 if s + 1 > 2e, A s ∩ R2e = ∅, and there is a number x ∈ R2e such that x > max{r(n ) : n < 2e} and φ e,s (x) = 0; and, for the least such x (if any), 2e requires attention via x. Requirement 2e+1 (e = e0 , e1 , e2 ) requires attention at stage s + 1 if s + 1 > 2e + 1, r(2e + 1, s) = 0 and there is a number x ∈ R2e+1 such that (S1 ), (S2 ), (S3 ), (S4 ), and (S5 )

x > max{r(n , s) : n < 2e + 1}

hold; and, for the least such x (if any), 2e+1 requires attention via x.

36 | Klaus Ambos-Spies Fix n minimal such that n requires attention and say that n becomes active at stage s + 1. (If no requirement requires attention let r(n, s + 1) = r(n, s) for all n ≥ 0 and proceed to stage s + 2.) Let n = 2e + i, i ≤ 1. If n = 2e and 2e requires attention via x then enumerate x into A and let r(n , s + 1) = r(n , s) for n < 2e and r(n , s + 1) = 0 for n ≥ 2e. If n = 2e + 1 and 2e+1 requires attention via x then enumerate x into A and let r(n , s + 1) = r(n , s) for n < 2e + 1, r(2e + 1, s + 1) = x + 1, and r(n , s + 1) = 0 for n > 2e + 1. To show that the construction is correct, it suffices to show that all requirements n require attention at most finitely often and are met. The proof is by induction on n. So fix n and the unique e ≥ 0 and i ≤ 1 such that n = 2e + i. By inductive hypothesis, there is a stage s0 > n such that no requirement n with n < n requires attention after stage s0 . So, in particular, r(n , s) = r(n , s0 ) for n < n and s ≥ s0 , and we may fix x0 ∈ R n minimal such that r(n , s) < x0 for n < n and s ≥ s0 . Moreover, n becomes active whenever it requires attention after stage s0 . Now first assume that n = 2e. Then 2e becomes active at most once since if 2e becomes active at stage t + 1 then a number x ∈ R2e is put into A at stage t + 1 whence the clause A s ∩ R2e = ∅ in the definition of 2e requiring attention at stage s + 1 fails for all s > t. So 2e requires attention at most once after stage s0 . It remains to show that 2e is met. For a contradiction assume that 2e is not met. Then, obviously, 2e never becomes active, hence A ∩ R2e = ∅. Moreover, since 2e is not met, A(x) = φ e (x) = 0 for all x ∈ R2e . It follows that, for all sufficiently large s, 2e requires attention via x0 at stage s + 1. But this contradicts the fact that 2e requires attention only finitely often. Finally assume that n = 2e + 1. If 2e+1 becomes active at a stage s + 1 > s0 then r(2e + 1, t) = r(2e + 1, s + 1) > 0 for all t > s, hence 2e+1 does not require attention after stage s + 1. So 2e+1 requires attention at most once after stage s0 . It remains to show that 2e+1 is met. Since 2e+1 requires attention only finitely often, we may fix x1 ≥ x0 minimal such that there is no x ∈ R2e+1 ∩ A with x ≥ x1 . Now, for a contradiction, assume that 2e+1 is not met, i.e., that the premise of 2e+1 holds and W e0 is simple. Note that, by the observations on the 2e+1 -strategy preceding the construction, the premise of 2e+1 fails if there is a stage s + 1 at which 2e+1 becomes active provided that 2e+1 is not injured later. So, by assumption and by choice of s0 , r(2e + 1, s) = 0 for s ≥ s0 and 2e+1 does not require attention after stage s0 . But the latter can be refuted as follows. It suffices to show that there is a number x > x0 in R2e+1 and a stage s ≥ s0 such that (S1 ) - (S4 ) hold. By simplicity of W e0 , fix x ∈ W e0 ∩ (R2e+1 ∩ {x : x ≥ x1 }) and

On the Strongly Bounded Turing Degrees of Simple Sets |

37

s1 ≥ s0 such that x ∈ W e0 ,s1 . Then (S1 ) and (S2 ) hold at all stages s ≥ s1 ; and (S3 ) and (S4 ) hold at all sufficiently large stages s by the assumption that the premise of 2e+1 is satisfied.

By the Coincidence Theorem, Theorem 10 immediately carries over to cl-reducibility. Corollary 11. There is a noncomputable c.e. set A such that degcl (A) is not simple.

For an extension of Corollary 11, which we will prove in the next section, we have to use a direct argument for constructing a nonsimple cl-degree and cannot refer to the Coincidence Theorem. So, in the remainder of this section, we show how the proof of Theorem 10 can be adjusted in order to obtain a direct proof of Corollary 11. Direct proof of Corollary 11. In order to make degcl (A) nonsimple we have to replace the nonsimplicity requirements 2e+1 in the proof of Theorem 10 by the stronger requirements ˜ W e0 & W e = Φ ˜ Ae ] ⇒ W e is not simple 2e+1 : [A = Φ e1 0 0 2

(for e = e0 , e1 , e2  ≥ 0) where the ibT-functionals are replaced by cl-functionals (while the noncomputability requirements 2e are unchanged). Here we assume that the coding function is chosen so that e0 , e1 , e2 ≤ e0 , e1 , e2 . So, by choice ˜ W e0 (x) and of the enumeration of the cl-functionals, the use of the cl-functionals Φ e1 ˜ Ae (x) is bounded by x + e. Φ 2 The strategy for meeting the thus modified nonsimplicity requirements for clfunctionals is somewhat more involved than the previously described strategy for meeting the original requirements for ibT-functionals. Fix the computable partition of ω into finite intervals I n = [y n , y n+1 ) (y0 = 0) where, for e, m ≥ 0, the interval I2e,m has length 1 while the interval I2e+1,m has length 2e + 1, i.e., y2e+1,m+1 = y2e+1,m + 2e + 1, and let x2e+1,m = y2e+1,m + e be the middle element of the interval I2e+1,m . Then the uniformly computable infinite and pairwise disjoint sets R n reserved for the requirements  n , n ≥ 0, are defined by R n = m≥0 In,m . The revised strategy for meeting 2e+1 is as follows. Wait for a stage s > 2e + 1 such that for some m ≥ 0 the following hold. (S1 ) I2e+1,m ⊆ W e0 ,s (S2 ) x2e+1,m ∈/ A s W

˜ e0 ,s (x (S3 ) A s (x2e+1,m ) = Φ e1 ,s 2e+1,m )  ˜ As  y (S4 ) W e0 ,s  y2e+1,m = Φ e2 ,s 2e+1,m

38 | Klaus Ambos-Spies Then, for the least such s and for the least corresponding m, put x2e+1,m into A s+1 and restrain all numbers < x2e+1,m from A. To show that this action guarantees that the premise of requirement 2e+1  ˜ W e0 ,s (x ˜ W e0 fails, first assume that Φ e1 ,s 2e+1,m ) = Φ e1 ( x 2e+1,m ). Then, by (S2 ) and  (S3 ), W

W

˜ e0 ,s (x ˜ e0 A(x2e+1,m ) = A s (x2e+1,m ) = Φ e1 ,s 2e+1,m ) = Φ e1 ( x 2e+1,m ) , W

˜ e0 . So, for the remainder of the argument, we may assume that hence A = Φ e1 W  ˜ e0 ,s (x ˜ W e0 ,s ˜ W e0 Φ e1 ,s 2e+1,m ) = Φ e1 ( x 2e+1,m ) where, by (S3 ), Φ e1 ,s ( x 2e+1,m ) ↓. Then there must be a number z ∈ W e0 \ W e0 ,s such that W

˜ e1e,s0 ,s (x2e+1,m ) ≤ x2e+1,m + e < y 2e+1,m+1 . z≤φ

˜ A s (z) ↓. But, In fact, by (S1 ), z < y2e+1,m . So, by (S4 ), W e0 (z) = W e0 ,s (z) = Φ e2 ,s since ˜ Ae2s ,s (z) ≤ z + e < y2e+1,m + e = x2e+1,m , φ

˜ Ae (z) = Φ ˜ A s (z). So W e (z) = it follows with A  x2e+1,m = A s  x2e+1,m that Φ e2 ,s 0 2 A A ˜ ˜ Φ e2 (z), hence W e0 = Φ e2 . For a proof that the above strategy guarantees that requirement 2e+1 is met, w.l.o.g. assume that the premise of 2e+1 holds and the strategy for meeting 2e+1 does not become active, hence R2e+1 ∩ A = ∅. Then, for any m ≥ 0, the conditions (S2 ), (S3 ) and (S4 ) hold for all sufficiently large stages s. So, for any m ≥ 0, condition (S1 ) must fail at all sufficiently large stages s, hence I2e+1,m ⊆ W e0 , i.e., |I2e+1,m ∩ W e0 | < 2e + 1. (In the actual construction, due to the finite injuries we can only argue that this is true for all sufficiently large m.) Now fix k maximal such that |I2e+1,m ∩ W e0 | = k for infinitely many m, fix m0 minimal such that |I2e+1,m ∩ W e0 | ≤ k for all m > m0 , and effectively enumerate numbers m p and stages s p (p ≥ 1) such that m0 < m1 < m2 < m3 < . . . and |I2e+1,m p  ∩ W e0 ,s p | = k. Then, for z p = µz(z ∈ I2e+1,m p  \ W e0 ,s p ),

z p ∈/ W e0 . So W e0 does not intersect the infinite computable set {z p : p ≥ 1} hence is not simple. So 2e+1 is met. In the actual construction of A it suffices to adjust the definition of 2e+1 requiring attention and the action in case that 2e+1 becomes active as follows. Now requirement 2e+1 (e = e0 , e1 , e2 ) requires attention at stage s +1 if s +1 > 2e +1, r(2e + 1, s) = 0 and there is a number m ≥ 0 such that (S1 ), (S2 ), (S3 ), (S4 ), and (S5 )

x2e+1,m > max{r(n , s) : n < 2e + 1}

hold; and, for the least such m (if any), 2e+1 requires attention via m. If 2e+1 becomes active at stage s + 1 and 2e+1 requires attention via m then x2e+1,m is

On the Strongly Bounded Turing Degrees of Simple Sets |

39

enumerated into A at stage s +1 and the restraint of 2e+1 is set to r(2e +1, s +1) = x2e+1,m + 1. The correctness of the construction is established as in the proof of Theorem 10 using the above observations on the strategy for the modified nonsimplicity requirements.

6 An interval of nonsimple degrees The proof of the existence of nonzero nonsimple c.e. r-degrees (r = ibT, cl) can be easily extended to show that there are nontrivial - in fact infinite - intervals of c.e. r-degrees containing only nonsimple r-degrees. Theorem 12. There are c.e. sets A and B such that

and hold.

A ≤ibT B,

(8)

B ≤T A,

(9)

∀ C c.e.[A ≤cl C ≤cl B ⇒ C is not simple]

(10)

Proof. The proof is obtained by some rather straightforward modifications of the direct proof of Corollary 11 outlined at the end of the preceding section. So we only give the basic ideas and leave the details to the reader. By a finite-injury argument, we construct c.e. sets A and B with the required properties, and we let A s and B s denote the finite parts of A and B, respectively, enumerated by the end of stage s (A0 = B0 = ∅). In order to satisfy (9) and (10) it suffices to meet the requirements 2e : B = Φ Ae

(for e ≥ 0) and

˜ W e0 & W e = Φ ˜ Be ] ⇒ W e is not simple 2e+1 : [A = Φ e1 0 0 2

(for e = e0 , e1 , e2 , e ≥ 0), respectively. As in the direct proof of Corollary 11, fix the computable partition of ω into the finite intervals I n = [y n , y n+1 ) (y0 = 0) where interval I2e,m has length 1 and interval I2e+1,m has length 2e + 1, let x2e+1,m = y2e+1,m + e, and reserve

40 | Klaus Ambos-Spies  the infinite computable set R n = m≥0 In,m for requirement n . Moreover, let   R even = e≥0 R2e and R odd = e≥0 R2e+1 . Then (8) is guaranteed by ensuring A ∩ I even = ∅ & A ∩ I odd = B ∩ I odd .

(11)

The strategy for meeting requirement 2e is the standard Friedberg-Muchnik strategy. We pick the least unused number x from the reserved intervals I2e,m , m ≥ 0, which is not restrained by any higher priority requirement and wait for a stage s such that Φ Ae,ss (x) = 0 (and such that no higher priority requirement wants to act). Then, at stage s + 1, we put x into B and restrain all numbers ≤ φ Ae,ss (x) from A. Note that this strategy is finitary (i.e., if injured only finitely often, then 2e acts only finitely often and the restraint r(2e, s) imposed by 2e is bounded). The strategy for meeting requirement 2e+1 is essentially the strategy for meeting the corresponding requirement in the direct proof of Corollary 11. Since the role played by A there is now split between A and B, however, clause (S4 ) has to be replaced here by (S4 )

˜ Be s ,s  x W e0 ,s  x2e+1,m = Φ 2e+1,m 2

while clauses (S1 ) - (S3 ) remain unchanged. So now the positive part is played by A while the negative part is played by B. Accordingly, if the strategy acts at stage s +1 via m, now x2e+1,m is put into A and all numbers < x2e+1,m are restrained from B. In order to satisfy (11), however, we have to treat A and B equally on I odd . So, actually, we put x2e+1,m into A and B and we restrain the numbers < x2e+1,m from A and B. Note that this modification does not interfere with the basic strategy, and we can argue as before that the strategy succeeds. Theorem 12 immediately implies that, for r = ibT, cl, there are c.e. r-degrees a and b such that a < b and the interval [a, b] = {c ∈ Rr : a ≤ c ≤ b} contains only nonsimple r-degrees, i.e., [a, b] ⊆ Sr . In fact, by the following observation we may argue that the interval [a, b] is infinite. Lemma 13. (a) Let D, E, F be c.e. sets such that D ≤wtt E ≤wtt F and D ≤r F ˆ such that D ≤ibT E ˆ ≤r F and E ˆ =wtt E. (r = ibT, cl). There is a c.e. set E (b) Let D and F be c.e. sets such that D s). So it suffices to show that 2e is met. For a contradiction assume that this is not the case. Then C = φ e . Moreover, obviously, 2e is met if it acts at some stage. So 2e never acts, hence C ∩ R2e = ∅ and 2e does not require attention after stage s0 . It follows from C ∩ R2e = ∅ and C = φ e that, for any x ∈ R2e such that x > r2e , there is a stage s x > s0 such that clauses (i), (ii) and (iv) in the definition of 2e requiring attention hold for all s ≥ s x , and the least such stage s x can be computed from x. So, since 2e does not require attention after stage s0 , (iii) fails for s ≥ s x . Hence B s x  x + 1 = B  x + 1 for x ∈ R2e with x > r2e . So B is computable contrary to choice of B. This completes the proof for n = 2e. So, finally, assume that n = 2e + 1 (e = e0 , e1 , e2 ). We start with some observations on the restraint r(2e + 1, s) imposed by 2e+1 . Note that, by choice

46 | Klaus Ambos-Spies of s0 , r(2e + 1, s) is nondecreasing in s for s ≥ s0 , i.e., ∀ s, s (s0 ≤ s ≤ s ⇒ r(2e + 1 , s) ≤ r(2e + 1 , s )).

(21)

So sups→∞ r(2e + 1, s) = ω if and only if lims→∞ r(2e + 1, s) = ω. Moreover, for s ≥ s0 , r(2e + 1, s + 1) = r(2e + 1, s) if and only if 2e+1 becomes active at stage s + 1. So, again by (21), sups→∞ r(2e + 1, s) = ω if and only if 2e+1 becomes active infinitely often if and only if 2e+1 requires attention infinitely often. Finally, by construction, if l(2e + 1, s) is unbounded then 2e+1 requires attention infinitely often, and if l(2e +1, s) is bounded then r(2e +1, s) is bounded too. So, summarizing, we obtain the following equivalences. sups→∞ r(2e + 1 , s) = ω ⇔

lims→∞ r(2e + 1 , s) = ω

⇔

sups→∞ l(2e + 1 , s) = ω

⇔

2e+1 becomes active infinitely often

⇔

2e+1 requires attention infinitely often

(22)

Then, by (20) and (22), in order to show that 2e+1 is met and requires attention only finitely often it suffices to show sup l(2e + 1 , s) < ω.

s→∞

(23)

For a contradiction, assume that (23) fails. Then, by (20) and (22), lims→∞ l(2e + 1 , s) = sups→∞ l(2e + 1 , s) = ω

and

lims→∞ r(2e + 1 , s) = sups→∞ r(2e + 1 , s) = ω.

(24)

Now, for m ≥ 0, let t(m) be the least stage ≥ s0 such that

and

l(2e + 1, t(m)) ≥ m,

(25)

r(2e + 1, t(m)) ≥ x2e+1,m ,

(26)

A t(m)  x2e+1,m + 1 = A  x2e+1,m + 1

(27)

hold. Note that t(m) exists by (24) and t(m) is nondecreasing in m. Since B ≤wtt A, it follows that there are infinitely many numbers m such that B(m) = B t(m) (m).

On the Strongly Bounded Turing Degrees of Simple Sets | 47

(Namely otherwise B(m) were computable from A  f (m) for the computable function f (m) = x2e+1,m + 1.) So, for the remainder of the proof, we may fix m minimal such that x2e+1,m > r2e+1 and B(m) = B t(m) (m), and fix s ≥ t(m) with m ∈ B s+1 \ B s . In order to get the desired contradiction we will show that 2e+1 becomes active via (a) and m at stage s + 1 and that this action destroys the premise of 2e+1 . First observe that, by s0 ≤ t(m), no requirement n with n < 2e + 1 enumerates any numbers into C after stage t(m) and, by (26) and (21), no requirement n with n > 2e + 1 enumerates a number x ≤ x2e+1,m into C after stage t (m). Moreover, by minimality of m and by t(m ) ≤ t(m) for m < m, 2e+1 does not enumerate any number x2e+1,m with m < m into C after stage t(m). So (C ∩ R)  x2e+1,m = (C t(m) ∩ R)  x2e+1,m .

Since, by (18) and (27), (C ∩ R)  x2e+1,m = ( C t(m) ∩ R)  x2e+1,m

too, it follows that and

C  x2e+1,m = C t(m)  x2e+1,m

(28)

C s  x2e+1,m + 1 = C t(m)  x2e+1,m + 1

(29)

where the latter follows from the fact that, by m ∈ B s+1 \ B s , x2e+1,m ∈/ C s . Next observe that W e0  x2e+1,m + 1 = W e0 ,t(m)  x2e+1,m + 1. (30) Since, by (25), x2e+1,m ∈ W e0 ,t(m) , for a proof of (30) it suffices to show that W e0  x2e+1,m = W e0 ,t(m)  x2e+1,m . So, for a contradiction, assume that this is not the case. Since, by (25) and (28), C

ˆ t(m)  x2e+1,m = Φ ˆ Ce  x2e+1,m , W e0 ,t(m)  x2e+1,m = Φ 2 e ,t(m) 2

Φ Ce2 .

But, by (20), this contradicts (24). So (30) holds. it follows that W e0 = Now, by (25), (29) and (30), l(2e + 1, s) ≥ m. So 2e+1 requires and receives attention via (a) and m at stage s + 1 hence x2e+1,m ∈ C s+1 \ C s . Since, ˆ W e0 ,t(m) (x2e+1,m ) = ˆ W e0 ,t(m) (x2e+1,m ) and, by (30), Φ by (25), C t(m) (x2e+1,m ) = Φ e ,t(m) e ,t(m) W

1

W

1

ˆ e0 (x2e+1,m ), it follows that C(x2e+1,m ) = Φ ˆ e0 (x2e+1,m ). But, by (20), this Φ e1 e1 contradicts (24). This completes the proof of the claim and the proof of Lemma 15.

48 | Klaus Ambos-Spies We now turn to applications of Lemma 15. Most of these applications can be obtained from the following theorem. Theorem 16. Let A and B be c.e. sets such that A ≤r B and B ≤wtt A (r = ibT, cl). There is a c.e. set C such that degcl (C) is nonsimple and A ≤r C ≤r B. In fact, C can be chosen so that A < r C < r B and A 0, it holds that c|a or c|b. So Theorem 26 strengthens Theorem 24. ˆ such that a ˆ and b ˆ ≥ a Also note that, for any maximal pair (a, b) and for a ˆ ≥ b, the pair (a ˆ ) is maximal again; and, similarly, for any minimal pair ˆ, b and b ˆ such that 0 < a ˆ ≤ b, the pair (a ˆ ) is minimal ˆ and b ˆ ≤ a and 0 < b ˆ, b (a, b) and for a again. So, Theorem 26 is immediate by Theorem 16 and the following lemma. ˆ 0, A ˆ 1 such that the followLemma 27. There are noncomputable c.e. sets A0 , A1 , A ing hold (for i = 0, 1 and r = ibT, cl). ˆi (i) A i ≤ibT A ˆ (ii) A i ≤wtt A i (iii) (deg r (A0 ), deg r (A1 )) is a maximal pair ˆ 0 ), deg r (A ˆ 1 )) is a minimal pair (iv) (deg r (A

Proof. The proof requires a number of results on minimal and maximal pairs in the literature. We first note that it suffices to consider the case of r = ibT. This follows from the observation in [3] that, for a pair of c.e. sets A and B, (degibT (A), degibT (B)) is a maximal pair if and only if (degcl (A), degcl (B)) is a maximal pair, and, similarly, the pair (degibT (A), degibT (B)) is minimal if and only if the pair (degcl (A), degcl (B)) is minimal. Next we use the existence of pairs of c.e. ibT-degrees which are both minimal and maximal which has been established in [3] too. So we may fix noncomputable c.e. sets A0 and A1 such that the pair (degibT (A0 ), degibT (A1 )) is minimal and

52 | Klaus Ambos-Spies maximal (hence (iii) holds). Moreover, by Proposition 5, w.l.o.g. we may assume that, for i = 0, 1, there is an infinite computable set R i ⊆ A i . A further result from [3] we use is the fact that minimal pairs in the c.e. ibTdegrees and in the c.e. wtt-degrees coincide, i.e., that for a pair of c.e. sets A and B, (degibT (A), degibT (B)) is a minimal pair if and only if (degwtt (A), degwtt (B)) is a minimal pair. (So, in particular, the pair (degwtt (A0 ), degwtt (A1 )) is minimal.) This observation allows us to exploit the fact that the join of any minimal pair of c.e. wtt-degrees is the half of a minimal pair of c.e. wtt-degrees which has been proven in [1]. So, since (degwtt (A0 ), degwtt (A1 )) is a minimal pair,

(35)

we may fix c.e. sets B and C such that

and

(degwtt (A0 ⊕ A1 ), degwtt (B)) is a minimal pair

(36)

(degwtt (A0 ⊕ A1 ⊕ B), degwtt (C)) is a minimal pair.

(37)

(degwtt (A0 ⊕ B), degwtt (A1 ⊕ C)) is a minimal pair.

(38)

Moreover, by Proposition 3, w.l.o.g. we may assume that B ⊆ R0 and C ⊆ R1 . The final ingredient from the literature we use is the fact that the upper semilattice of the c.e. wtt-degrees is distributive (due to Lachlan; see [18]). By (35) - (37) this implies that

ˆ 0 = A0 ∪ B and A ˆ 1 = A1 ∪ C. Since A0 ∩ B = ∅ and A1 ∩ C = ∅, it Now let A ˆ 0 =wtt A0 ⊕ B and A ˆ 1 =wtt follows by the Splitting Lemma that (i) holds and A A1 ⊕ C. The latter implies (iv) and (ii) as follows. (iv) is immediate by (38) and by the coincidence of the wtt-minimal pairs with the ibT-minimal pairs. For a proof of (ii) it suffices to note that, by (36) and (37), B ≤wtt A0 and C ≤wtt A1 whereas ˆ 0 and C ≤wtt A1 ⊕ C =wtt A ˆ 1. B ≤wtt A0 ⊕ B =wtt A

8 Simple degrees below and above given degrees After having shown a variety of existence results for nonsimple r-degrees we now turn to existence results for simple r-degrees (r = ibT, cl). The existence of simple sets was shown by Post [15]. Using a stage construction the definition of a simple set A in the style of Post can be described as follows. The set A has to be c.e. and coinfinite and has to meet the requirements e : If W e is infinite then W e ∩ A = ∅.

On the Strongly Bounded Turing Degrees of Simple Sets |

53

for e ≥ 0. By effectivity of the enumeration {A s }s≥0 of A this is ensured by the following strategy for meeting e : If there is a stage s ≥ e such that W e,s ∩ A s = ∅ and there is a number x ∈ W e,s such that x ≥ 2e then put the least such x into A s+1 . Note that the condition x ≥ 2e ensures that |A ∩{0, . . . , 2e −1}| ≤ e - hence that A is coinfinite. Obviously, in order to guarantee the latter, the condition x ≥ 2e may be replaced by x ≥ f (e) for any strictly increasing computable function f such that lim supx→∞ f (x) − x = ∞. In the following we refer to this variant of Post’s construction as Post’s simple set construction with lower bound f (e). It is well known that Post’s simple set construction can be combined with the permitting technique. Since permitting yields an ibT-reduction this implies that for any noncomputable c.e. set B there is a simple set A such that A ≤ibT B. So the class Sr of the simple r-degrees is downward dense in the partial ordering (R+ r , ≤) of the nonzero r-degrees (r = ibT, cl). Theorem 28. Let B be a noncomputable c.e. set. There is a simple set A such that A ≤ibT B. Proof (sketch).. Given a computable enumeration {B s }s≥0 of B, it suffices to modify Post’s simple set construction as follows. The strategy for meeting requirement e is now combined with permitting: If there is a stage s ≥ e such that W e,s ∩ A s = ∅ and there is a number x ∈ W e,s such that x ≥ 2e and B s+1  x + 1 = B s  x + 1 then put the least such x into A s+1 . Then one can argue that, by noncomputability of B, all requirements are met. Namely, for a contradiction, assume that e is not met. Then W e is infinite and W e ∩ A = ∅. So, by the former, there are strictly increasing computable sequences {x n }n≥0 and {s n }n≥0 such that x n ≥ 2e and x n ∈ W e,s n ; and, by the latter, x n is not enumerated into A. Hence, by construction, B s n  x n + 1 = B  x n + 1. But this implies that B is computable contrary to choice of B. The dual of Theorem 28 holds too. Theorem 29. Let B be a c.e. set. There is a simple set A such that B ≤ibT A.

54 | Klaus Ambos-Spies Proof. Without loss of generality we may assume that 0 ∈/ B (by closure of =ibT under finite variations) and that B is noncomputable (since otherwise we may let A be any simple set) but not simple (since otherwise we may let A = B). So, in particular, we may assume that B is infinite and that the complement B of B contains an infinite computable set R as a subset where 0 ∈ R. Let b be a computable one-toone function which enumerates B and let r be the computable one-to-one enumeration of R in order of magnitude. Finally let B0 = ∅ and B s = {b(0), . . . , b(s − 1)} for s > 0. We will enumerate a set A with the required properties in stages where A s will denote the finite part of A enumerated by the end of stage s (A0 = ∅). We ensure B ≤ibT A by making A permit B. To be more precise, we use the odd stages of the construction for coding B into A by putting some new number y s ≤ b(s) into A at stage 2s + 1 thereby ensuring that ∀ s ∃ y s (y s ≤ b(s) & y s ∈ A2s+1 \ A2s )

(39)

holds. Obviously this implies B ≤ibT A. Since, by noncomputability of B, B ≤ibT A ensures that A is noncomputable (hence coinfinite), in order to make A simple, it suffices to meet the standard simplicity requirements for e ≥ 0. e : If W e is infinite then W e ∩ A = ∅.

For meeting e we use the standard strategy of Post with lower bound r(e) + 1 where action for meeting the simplicity requirements is limited to the even stages. Choosing the witness for e (if any) to be greater than r(e) ∈/ B will allow us to argue that the enumeration of the witnesses for the simplicity requirements will not interfere with coding B into A according to (39). The construction of A is as follows. Stage 2s > 0. For any e < s such that W e,s ∩ A2s−1 = ∅ and there is a number x > r(e) such that x ∈ W e,s , put the least such number x into A2s . Stage 2s + 1. If there is a number y ≤ b(s) such that y ∈/ A2s then let y s be the greatest such number y and put y s into A2s+1 . Correctness of the construction is established as follows. The proof that the simplicity requirements are met is straightforward. So it only remains to verify equation (39). By construction, it suffices to show that, for all s, the number y s specified at stage 2s + 1 of the construction exists, i.e., that ∀ s ∃ y (y ≤ b(s) & y ∈ / A 2s )

(40)

On the Strongly Bounded Turing Degrees of Simple Sets | 55

holds. Let A0 and A1 be the parts of A enumerated at the even and odd stages, respectively, and set A is = A i ∩ A s for s ≥ 0. (Note that A is the disjoint union of the c.e. sets A0 and A1 , that A00 = A10 = A0 = ∅, and that A02s = A02s+1 and A12s+1 = A12s+2 .) Then for a proof of (40) it suffices to show that ∀ s ∀ z [|( A0 ∪ A12s )  z + 1| ≤ |(R ∪ B s )  z + 1|]

(41)

holds. To show that (41) implies (40) fix s. Then |A2s  b(s) + 1|

≤ ≤ ≤

|(A0 ∪ A12s )  b(s) + 1| |(R ∪ B s )  b(s) + 1| b(s)

(since A2s ⊆ A0 ∪ A12s ) (by (41) applied to z = b(s))

where the last inequality follows from the facts that there are b(s) + 1 numbers ≤ b(s) and that b(s) ∈ / R ∪ B s (since R and B are disjoint and since b(s) enters B only at stage s + 1). So there is a number y ≤ b(s) with y ∈/ A2s . The proof of (41) is by induction on s. First note that |A0  r(e) + 1| ≤ e since a number x ≤ r(e) can enter A0 only for the sake of a requirement e with e < e, and any of these e requirements causes at most one number to be enumerated into A0 . Since |R  r(e) + 1| = e + 1 it follows that, for any number z, |A0  z + 1| ≤ |R  z + 1|

holds. Since A10 = B0 = ∅, this immediately implies (41) for s = 0. For the inductive step assume that (41) holds for s, and fix z in order to show |(A0 ∪ A12(s+1) )  z + 1| ≤ |(R ∪ B s+1 )  z + 1|].

(42)

W.l.o.g. we may assume that (A0 ∪ A12(s+1) )  z + 1 = (A0 ∪ A12s )  z + 1

since otherwise the claim is immediate by inductive hypothesis. Since A0 is fixed and since A12(s+1) = A12s+1 , it follows that at stage 2s + 1 of the construction the required y s exists, y s ≤ z, and A12(s+1) = A12s+1 = A12s ∪ {y s } whence (by A 0 ∩ A 1 = ∅) |(A0 ∪ A12(s+1) )  z + 1| = |(A0 ∪ A12s )  z + 1| + 1.

Now, if b(s) ≤ z then, by R ∩ B = ∅, |(R ∪ B s+1 )  z + 1| = |(R ∪ B s )  z + 1| + 1

(43)

56 | Klaus Ambos-Spies and the claim follows by inductive hypothesis. So for the remainder of the argument we can assume that b(s) > z. Now, for a contradiction, assume that (42) fails, i.e., |(A0 ∪ A12(s+1) )  z + 1| > |(R ∪ B s+1 )  z + 1|.

Since, by b(s) > z, |(R ∪ B s )  z + 1| = |(R ∪ B s+1 )  z + 1|, it follows, by inductive hypothesis and by (43), that |(A0 ∪ A12s )  z + 1| = |(R ∪ B s )  z + 1| = |(R ∪ B s+1 )  z + 1|

(44)

holds. Moreover, by definition of y s and by y s ≤ z < b(s), the interval (z, b(s)] is contained in A2s ⊆ A0 ∪ A12s . So |(A0 ∪ A12s )  b(s) + 1| = |(A0 ∪ A12s )  z + 1| + b(s) − z

(45)

while, by inductive hypothesis, |(A0 ∪ A12s )  b(s) + 1| ≤ |(R ∪ B s )  b(s) + 1|.

(46)

By replacing the left hand side of (46) according to (45) and (44) it follows that |(R ∪ B s )  z + 1| + b(s) − z ≤ |(R ∪ B s )  b(s) + 1|

hence

b(s) − z ≤ |(R ∪ B s ) ∩ (z, b(s)]|.

(47)

But, since b(s) ∈/ R ∪ B s ,

|(R ∪ B s ) ∩ (z, b(s)]| = |(R ∪ B s ) ∩ (z, b(s))| ≤ |(z, b(s))| = ( b(s) − z) − 1

contrary to (47). So (42) must hold, which completes the proof of (41) and the proof of the theorem.

9 Simple degrees below and above given degrees: stronger results and applications In the preceding section we have shown that, for any given noncomputable c.e. set B, there is a simple set A such that A ≤r B and a simple set A such that B ≤r A (r = ibT, cl). On the other hand, by Theorem 10, there are noncomputable c.e. sets B which are not r-equivalent to any simple set. So the above results cannot be combined. This makes it natural to ask how close a simple set A ≤r B and a

On the Strongly Bounded Turing Degrees of Simple Sets | 57

simple set A ≥r B can be to B. In the former case we will show that A can be chosen to be lbT-equivalent to B, i.e., that there is a simple set A such that B ≤lbT A ≤r B. So, in particular, all nonzero c.e. lbT-degrees are simple. In the latter case we can only show that A can be chosen to be wtt-equivalent to B, i.e., that there is a simple set A such that B ≤r A ≤wtt B. At the end of the section we apply the latter observation in order to prove the analog of Theorem 26 for simple degrees thereby showing that, for any nonzero c.e. r-degree a, there is a simple r-degree which is incomparable with a. Theorem 30. Let B be a noncomputable c.e. set. There is a simple set A such that A ≤ibT B and B ≤lbT A. Proof. Fix a computable 1-1 function b(n) enumerating B and let {B s }s≥0 be the computable enumeration of B defined by B0 = ∅ and B s+1 = {b(0), . . . , b(s)}. We effectively enumerate a c.e. set A with the required properties in stages where A s denotes the finite part of A enumerated by the end of stage s (A s = ∅). The strategy for meeting the simplicity requirements e : If W e is infinite then W e ∩ A = ∅.

is Post’s simple set strategy combined with permitting as introduced in the proof of Theorem 28. Here, however, we use a larger lower bound thereby leaving enough room for coding B into A in order to ensure B ≤lbT A. As in the proof of Theorem 29 we handle the simplicity requirements at the even stages and the coding of B into A at the odd stages of the construction. Stage 2s > 0. For any e < s such that W e,s ∩ A2s−1 = ∅ and there is a number x > max{8e, b(s)} such that x ∈ W e,s , put the least such number x into A2s . Stage 2s + 1. Put z s into A2s+1 where z s is the least number z such that 4b(s) < z and z ∈/ A2s . In order to show that the construction is correct, let A0 and A1 be the parts of A enumerated at the even and odd stages respectively. (So A is the disjoint union of the c.e. sets A0 and A1 , and, for A is = A i ∩ A s , A00 = A10 = A0 = ∅, A02s = A02s+1 and A12s+1 = A12s+2 .) As in the proof of Theorem 28 we may argue that, by noncomputability of B, the simplicity requirements e are met. Moreover, by construction, |A0  8e + 1| ≤ e + 1 and |A1  4e + 1| ≤ e + 1

whence A is coinfinite. So A is simple.

(48)

58 | Klaus Ambos-Spies The reduction A ≤ibT B holds by permitting since any number y entering A at stage 2s or 2s + 1 is greater than b(s). Finally, for a proof of B ≤lbT A, note that, by (48), |A  8y + 1| ≤ 3y + 2 for all y ≥ 0, hence [4y + 1, 8y] ⊆ A for y ≥ 3. So, for b(s) ≥ 3, z s ≤ 8b(s). Obviously this implies B ≤f -T A for the linearly bounded function f (n) = 8n. Theorem 30 immediately implies that any noncomputable c.e. set is lbT-equivalent to a simple set. Corollary 31. Any nonzero c.e. lbT-degree is simple. In the remainder of this section we prove a weak dual version of Theorem 30 where lbT is replaced by wtt. Theorem 32. Let B be a noncomputable c.e. set. There is a simple set A such that B ≤ibT A and A ≤wtt B. Proof. As in the proof of Theorem 29 we may assume that 0 ∈/ B and that B is not simple. So we may fix an infinite computable set R ⊆ B such that 0 ∈ R . From R we obtain a very sparse computable subset R of R by inductively defining the function r(s) enumerating R in order as follows: let r(0) = 0 and, given r(s), let r(s + 1) = (µ y ∈ R )(|R ∩ (r(s), y)| > r(s) + s + 2). Note that the definitions of R and r guarantee that 0 ∈ R, R ∩ B = ∅ and ∀ s ≥ 0 [|B ∩ (r(s), r(s + 1))| > r (s) + s + 2].

(49)

Now the desired set A is essentially constructed as in the proof of Theorem 29 based on the sets B and R and the enumeration r of R given above. The only difference is that now the strategy for making A simple is combined with B-permitting. So the even stages of the construction are now as follows Stage 2s > 0. For any e < s such that W e,s ∩ A2s−1 = ∅ and there is a number x > max(r(e), b(s)) such that x ∈ W e,s , put the least such number x into A2s . while the odd stages are unchanged: Stage 2s + 1. If there is a number y ≤ b(s) such that y ∈/ A2s then let y s be the greatest such number y and put y s into A2s+1 .

On the Strongly Bounded Turing Degrees of Simple Sets |

59

Then - as in the proof of Theorem 28 - we may argue that A is simple; and - as in the proof of Theorem 29 - we may argue that B ≤ibT A. In particular, as there, we can show that (41) holds and that the number y s specified at stage 2s + 1 exists. It remains to show that A ≤wtt B. Using the notation introduced in the proof of Theorem 29, it is immediate that the part A0 of A enumerated at the even stages is ibT-reducible (hence wtt-reducible) to B by permitting. So it suffices to show that the part A1 of A enumerated at the odd stages is wtt-reducible to B too. Since y s is the only number which enters A1 at stage 2s + 1, this can be established by giving a nondecreasing computable function f such that ∀∞ s ≥ 0 [b(s) ≤ f (y s )].

(50)

Namely, (50) implies that, for almost all numbers s and x, B s  f (x) + 1 = B  f (x) + 1 ⇒ A1s  x + 1 = A1  x + 1 whence A1 (x) can be computed from B  f (x) + 1. Now the desired function f is defined by ∀ x, s ≥ 0 [x ∈ [r(s), r(s + 1)) ⇒ f (x) = r(s + 2)].

(51)

Obviously, f is nondecreasing and computable. In order to show that f satisfies (50), fix s such that w.l.o.g. b(s) > r(1). Then, given x ≥ 1 such that b(s) ∈ (r(x), r(x + 1)) it suffices to argue that y s ≥ r(x − 1). Since, by definition, y s is the greatest number y ≤ b(s) such that y ∈/ A2s , it suffices to show that the interval [r(x − 1), r(x)] is not completely contained in A. This is done as follows. By |R  r (x) + 1| = x + 1 and by (41), |A  r (x) + 1| ≤ x + 1 + |B  r (x) + 1|

while, by (49), hence

(52)

|B ∩ (r(x − 1), r(x))| > r (x − 1) + x + 1 ,

|B  r (x) + 1| < r (x) + 1 − (r(x − 1) + x + 1) = (r(x) − r(x − 1)) − x.

It follows that |A ∩ [r(x − 1), r(x)]|

≤ ≤


0, c|a or c|b, Theorem 33 implies that any c.e. r-degree c > 0 is incomparable to some simple r-degree. Corollary 34. For any c.e. r-degree c > 0 there is a simple c.e. r-degree d such that c and d are incomparable (r = ibT, cl). Moreover, since the degree 0 is nonsimple, Theorem 33 immediately implies that the class of simple r-degrees is not closed under meet. Corollary 35. The class Sr of the simple c.e. r-degrees is not closed under meet (r = ibT, cl).

On the Strongly Bounded Turing Degrees of Simple Sets | 61

10 wtt-Complete sets have simple degree In Section 8 we have shown that Post’s simple set construction can be combined with permitting and coding in order to show that, for any given noncomputable c.e. set B, there is a simple set A such that A ≤ibT B and a simple set A such that B ≤ibT A, respectively. On the other hand, by the existence of nonzero nonsimple c.e. ibT-degrees (Theorem 10), in general permitting and coding cannot be combined in order to obtain a simple set A which is ibT-equivalent to B. The problem which arises in combining these techniques is as follows. If a number x (permitted by B) is put into A at a stage s in order to meet a simplicity requirement then it may happen that this number x is enumerated into B at a later stage t. Then, in order to record this change, a number < x has to be put into A (since x has been enumerated before) but this smaller number may not be permitted by B. (So, in general, we only obtain the weaker results presented in the preceding section.) This problem can be avoided, however, if the set B is wtt-complete. Then, by choosing a sufficiently large lower bound for the simple set construction, we may argue that if a number x used for meeting a simplicity requirement is enumerated into B later then a number x  < x can be forced into B at a still later stage so that the change of B at x has not to be recorded by the coding procedure. So any wtt-complete set is ibT-equivalent to a simple set. Theorem 36. Let B be wtt-complete. There is a simple set A such that A =ibT B. Proof. We start with some notation. Let {Ψ n }n≥0 and {ψ n }n≥0 be computable enumerations of the (partial) wtt-functionals together with corresponding partially computable bounds. (To be more precise, Ψ n and ψ n are chosen so that, whenever Ψ nX (x) is defined, then ψ n (x) is defined and the use of Ψ nX (x) is bounded by ψ n (x); and, for any wtt-functional Ψ and for any corresponding (w.l.o.g. strictly increasing) computable bound f , there is a number n such that Ψ = Ψ n and f = ψ n .) Moreover, let {Ψ n,s }s≥0 and {ψ n,s }s≥0 be uniformly computable enumerations of Ψ n and ψ n as usual. W.l.o.g. we may assume that, if ψ n (x) is defined then ψ n (y) is defined for all y < x and ψ n (y) < ψ n (x) (and similarly for ψ n,s ). Finally, let b be a 1-1 computable function enumerating B and let B0 = ∅ and B s+1 = {b(0), . . . , b(s)}. We construct a sequence {A n }n≥0 of (uniformly) c.e. sets together with an auxiliary c.e. set C such that, for n ≥ 0, C = Ψ nB ⇒ A n =ibT B & A n is simple

(54)

holds. Then, by wtt-completeness of B, there is a number n such that C = Ψ nB , and the set A = A n has the required properties.

62 | Klaus Ambos-Spies Call n correct if C = Ψ nB holds, let C s and A n,s be the finite parts of C and A n , respectively, enumerated by the end of stage s of the construction below, and let Bs l(n, s) = µx(C s (x) = Ψ n,s (x)).

Then, for correct n, lims→∞ l(n, s) = ∞. Moreover, for x < l(n, s) such that x ∈/ C s , we can force that a number y ≤ ψ n (x) enters B after stage s by putting x into C at stage s + 1. Now in order to make sure that, for correct n, the set A = A n has the required properties we ensure that for such n the conditions

and

x ∈ A n,s+1 \ A n,s ⇒ ∃ t ≥ s (b(t) ≤ x)

(55)

A n  b(s) + 1 = A n,s  b(s) + 1

(56)

are satisfied - thereby guaranteeing that A n ≤ibT B and B ≤ibT A n , respectively and the requirements ne : If W e is infinite then A ne ∩ W e = ∅

are met (for e ≥ 0) - thereby guaranteeing that A n is simple. For meeting the simplicity requirements ne we use Post’s strategy with lower bounds ψ n (n, e) + 1 combined with permitting. The latter guarantees (55). The chosen lower bound allows us to satisfy (56) as follows. If we enumerate y > ψ n (n, e) into A n in order to meet ne at stage s + 1 and y = b(t) for some t ≥ s + 1 then we can force a number < y into B after stage t + 1 by putting n, e into C at the least stage t + 1 ≥ t + 1 such that l(n, t ) > n, e. The construction of the sets C and A n is as follows (where C0 = A n,0 = ∅). Stage s + 1. The stage consists of 3 steps. Step 1 (Making A n simple). For any n, e ≤ s such that (i) W e,s ∩ A n,s = ∅, (ii) ψ n,s (n, e) ↓, and (iii) there is a number y such that y ∈ W e,s and max( b(s), ψ n (n, e) < y

hold, put the least such number y into A n and declare that y is an n-enumber. Step 2 (Partial coding of B into A n ). For any n, e ≤ s such that b(s) ∈/ A n,s put b(s) into A n .

On the Strongly Bounded Turing Degrees of Simple Sets | 63

Step 3 (Completing the coding of B into A n ). For any n, e ≤ s such that there is a stage t ≤ s such that b(t) ∈ A n,t , b(t) has been declared to be an n-e-number at a stage ≤ t, n, e < l(n, s), and n, e ∈/ C s , put n, e into C. In order to show that the construction achieves its goal, fix n such that n is correct. Since, obviously, (55) is satisfied (for all n), it suffices to show that the nrequirements, i.e., the requirements ne for e ≥ 0, are met and (56) holds. Note that, by correctness of n, ψ n is total and lims→∞ l(n, s) = ∞. By totality of ψ n , as in the proof of Theorem 28 we can argue that ne is met. Namely, for a contradiction, assume that ne is not met. Then W e is infinite and W e ∩ A n = ∅. So there are strictly increasing computable sequences {x n }n≥0 and {s n }n≥0 such that ψ n,s0 (n, e) ↓, x n > ψ n (n, e), x n ∈ W e,s n , and B s n  x n + 1 = B  x n + 1. Hence B is computable contrary to choice of B. Finally, the proof of (56) is indirect too. For a contradiction fix b(s) minimal such that A n  b(s) + 1 = A n,s  b(s) + 1. (57) Then there is a number e and a stage t such that n, e ≤ t < s and requirement ne enumerates b(s) into A n at stage t + 1. So ψ n,t (n, e) ↓, ψ n (n, e) < b(s), and b(s) becomes an n-e-number at stage t + 1. Since, by construction, there is at most one n-e-number (for any given n and e) and since lims→∞ l(n, s) = ∞, it follows that there is a stage s ≥ s such that l(n, s ) > n, e and n, e ∈/ C s and that, for the least such s , n, e is enumerated into C at stage s + 1, hence B C(n, e) = Ψ n,ss (n, e) ↓. By correctness of n this implies that a number z ≤ ψ n (n, e) has to enter B after stage s . I.e., there is a stage s ≥ s ≥ s such that b(s ) ≤ ψ n (n, e) < b(s). It follows, by minimality of b(s), that A n  b(s ) + 1 = A n,s  b(s ) + 1. But, by b(s ) < b(s) and by s ≤ s this contradicts (57). This completes the proof. Note that Theorem 36 shows that in Theorem 16 the assumption that A is not wttcomplete is necessary. Conversely, Corollary 19 implies that in Theorem 36 we cannot replace wtt-completeness by T-completeness. As a direct consequence of Theorem 36 we obtain the existence of simple upper cones in the c.e. ibT- and cl-degrees, which in turn directly implies that the class Sr of the nonsimple c.e. r-degrees does not generate the class Rr of all c.e. r-degrees under meet. Corollary 37. Let r = ibT, cl. There is a c.e. r-degree a such that ∀ b [a ≤ b ⇒ b simple].

(58)

64 | Klaus Ambos-Spies Proof. Since, for r = ibT, cl, r-reducibility is stronger than wtt-reducibility, for any c.e. sets A and B such that A is wtt-complete and A ≤r B, B is wtt-complete too. So the claim is immediate by Theorem 36. Corollary 38. The class Sr of the nonsimple c.e. r-degrees does not generate the class Rr of all c.e. r-degrees under meet (r = ibT, cl). Proof. This is immediate by Corollary 37. Another consequence of Theorem 36 is that there are infinite intervals of c.e. rdegrees containing only simple degrees. Corollary 39. Let r = ibT, cl. There are c.e. r-degrees a and b such that a < b, [a, b] is infinite, and c is simple for all c ∈ [a, b]. Proof. Given any wtt-complete set A and any unbounded computable shift f , the interval [deg r (A f ), deg r (A)] is contained in the class of r-degrees of wtt-complete sets, hence, by Theorem 36, contains only simple degrees. So it suffices to show that the interval [deg r (A f ), deg r (A)] is infinite. Since, by the Computable-Shift Lemma, B f < r B for any computable unbounded shift f and any noncomputable c.e. set B, this follows from the observation that for any unbounded computable shift f there are unbounded computable shifts g and h such that f (n) = g (h(n)) hence B f = (B h )g . Recall that in Section 6 we proved the analog of Corollary 39 for nonsimple degrees (Corollary 14). In fact, there we proved that there is a closed interval [deg r (A), deg r (B)] of nonsimple r-degrees such that not only A < r B but also A  Potential diagonalization witnesses are assigned to the strategies α (|α| = e) working on requirement e as follows. Fix a uniformly computable partition of ω into infinitely many mutually disjoint finite intervals I nα (α ∈ T, n ≥ 0) such that I nα = [x αn , y αn ] = [x αn , x αn + |α| + 1] has length |α| + 2 and x αn ≥ n. Then the intervals I nα , n ≥ 0, are reserved for the strategy α where the least element x αn ∈ I nα is the diagonalization candidate (follower) while y αn is the corresponding B-trace to be put into B if x αn is enumerated into A. ˆ e is relNow, for α and e < |α|, the guess α(e) = 0 expresses that requirement R evant, i.e., that its premise holds. In order to approximate this guess in the course ˆ e is approximated of the construction, the second condition in the premise of R with the help of the length function l(e, s) defined by ˆ Ae s ,s (x)). l(e, s) = µx(W e0 ,s (x) = Φ 1 ˆ e is a bounded functional, Then, since Φ 1 ˆ Ae ⇔ lim l(e, s) = ∞ ⇔ lim sup l(e, s) = ∞. W e0 = Φ 1 s→∞

s→∞

(67)

ˆ Ae if and only if there are infinitely many e-expansionary stages s, So W e0 = Φ 1 i.e., stages s such that l(e, s) > l(e, t) for all t < s (where we may consider only ˆ e, stages s and t from some given infinite set). The first part of the premise of R 0 namely that W e0 is simple, is too complex (namely, Π3 -complete; see e.g. Soare [16]), however, in order to be approximated in a similar fashion. So we replace the

On the Strongly Bounded Turing Degrees of Simple Sets | 69

test on simplicity of W e0 by a test on the following infinitary property used in the construction, namely the property that, for any β ∈ T and a corresponding infinite computable set D β (to be specified below), there are infinitely many n ∈ D β such that I nβ ⊆ W e0 . This leads to the following definitions where we say that a number e is α-relevant if e < |α| and α(e ) = 0. For n ≥ 0, call the node α n-saturated at stage s if s ≥ n and, for any node β β β β with |β| ≤ n there are at least n β-intervals I m = [x m , y m ] such that, for any αβ β relevant e , I m ⊆ W e0 ,s and l(e , s) > y m ; and call α saturated if for every number n there is a stage s n such that α is n-saturated at stage s n . Note that α is saturated if and only if ∀ e < |α| (α(e ) = 0 ⇒ lim l(e , s) = ω) (68) and

s→∞

β

∀ β ∃∞ m ( I m ⊆



{e :e 0, s is a λ-stage and, if s is an α-stage and |α| < s, then s is an α0-stage if, for the greatest t < s such that t is an α0-stage, α0 is (t + 1)-saturated at stage s, and s is an α1-stage otherwise. Let δ s be the unique string of length s such that s is a δ s -stage. (So δ s codes the guess at the end of stage of the construction about ˆ 0, . . . ,  ˆ s−1 . Also note that s is an α-stage relevance of the first s requirements  if and only if α  δ s .) Finally, the true path f of the construction is defined by f = lim inf s→∞ δ s . I.e., f  n is the leftmost string α of length n such that α  δ s for infinitely many stages s. The following True Path Lemma shows that the guesses on the true path are ˆ e is relevant then we guess so. (sufficiently) correct. Namely, if a requirement  ˆ And, if we erroneously guess that a requirement e is relevant, this will not do any harm to the construction since the assumptions actually used in the construction will be true. True Path Lemma. (a) Let α be a string such that α  f . Then α is saturated. ˆ Ae . (b) Assume that e = e0 , e1  ≥ 0 is relevant, i.e., W e0 is simple and W e0 = Φ 1 Then f (e) = 0. Proof. Part (a) is immediate by the definition of f . For a proof of part (b) let α = f  e. Then we have to show that there are infinitely many α0-stages. In order to do so, it suffices to show that α0 is saturated. Then, by (68) and (69) (for α0 in place of α) , for any t ≥ 0 and α  α0, α is (t +1)-saturated at all sufficiently large stages s which, by definition, implies that there are infinitely many α0-stages. Now, since α  f , it follows from part (a) that (68) and (69) hold. It suffices to establish these conditions for α0 in place of α. The former is immediate since, by

70 | Klaus Ambos-Spies ˆ Ae and by (67), lims→∞ l(e, s) = ω. For a proof of the latter fix β. Then, by W e0 = Φ 1 β (69) (for α), there is an infinite computable set D such that I m ⊆ W e0 for all e < e  such that e is α-relevant or, equivalently, α0-relevant. So it suffices to show that β there are infinitely many m ∈ D such that I m ⊆ W e0 . But since W e0 is simple this is immediate by the Simple Set Lemma. This completes the proof of the True Path Lemma. We are now ready to give the construction of A and B. Let A0 = B0 = ∅ and initialize all strategies α at stage 0. Moreover, say that a strategy α is satisfied at stage s if α has been active at a stage t ≤ s and α has not been initialized at any stage t such that t ≤ t ≤ s.

Stage s + 1 > 0. Strategy α requires attention at stage s + 1 if the following hold. (i) α  δ s (ii) α is not satisfied at stage s. (iii) There is a number m ≥ 0 such that (iii)a x αm is greater than the greatest stage t ≤ s such that α is initialized at stage t, (iii)b x αm ∈ A s and y αm ∈ B s , ˜ B s (x αm ) = 0 where e = |α|, and (iii)c Φ e α (iii)d for all α-relevant e < |α|, I m ⊆ W e0 ,s and l(e , s) > y αm . Moreover, if α requires attention then, for the least m satisfying (iii)a - (iii)d , α requires attention via m. If some strategy requires attention then fix α minimal such that α requires attention and fix m such that α requires attention via m. Declare α to be active, put x αm into A and y αm into B, and initialize all strategies β such that α < β. If no requirement requires attention then initialize all strategies β such that δ s ≤ β. This completes the construction. It remains to show that the construction is correct. We do this by proving the following claims. Claim 1. B ≤ibT A (i.e., (65) holds). Proof. Note that if a number y is enumerated into B at a stage s + 1 then, for some α and m, the α -follower x αm is enumerated into A at stage s + 1 and y = y αm > x αm . So B ≤ibT A by permitting. Claim 2. Let e ≥ 0 and let α = f  e. Then α requires attention only finitely often and e is met. Proof. The proof is by induction on e. Fix e and α = f  e. Then, by definition of f , α < δ s for all sufficiently large stages s. So, by inductive hypothesis, we may fix a stage s0 such that α < δ s for s ≥ s0 and such that no strategy α with α < α

On the Strongly Bounded Turing Degrees of Simple Sets | 71

requires attention after stage s0 . Then α is not initialized after stage s0 and α becomes active at any stage s + 1 > s0 at which it requires attention. It follows that α requires attention at most once after stage s0 . Namely, if α requires attention at stage s1 +1 > s0 then α becomes active at stage s1 +1. Since α is not initialized later, it follows that α is satisfied at all stages s ≥ s1 + 1. So condition (ii) in the definition of requiring attention fails for s ≥ s1 + 1, hence α does not require attention after stage s1 + 1. It remains to show that requirement e is met. First observe that if α becomes active at some stage s +1 and is not initialized later then e is met. Namely then, for some m ≥ 0, x αm is enumerated into A at ˜ B s (x αm ) = 0. Moreover, the only number which enters B at stage s + 1 where Φ e,s ˜ B s (x αm ) which is stage s + 1, namely y αm = x αm + e + 1, is greater than the use of Φ e,s α bounded by x m + e. Hence B

˜ Be,ss (x αm ) = Φ ˜ s+1 (x αm ). A(x αm ) = 1 = 0 = Φ e,s+1

(70)

Moreover, α does not act later (since α is not initialized after stage s hence permanently satisfied) and no strategy β with β < α acts later (since α is not initialized after stage s + 1). So only strategies β with α < β may act after stage s + 1. But theses strategies β are initialized at stage s + 1 hence (by clause (iii)a in the definition of requiring attention and by x βm < y βm ) enumerate only numbers > s + 1 into (A and) B later. So B  s + 2 = B s+1  s + 2. Since, by our convention on use ˜ B s+1 (x αm ) = 0 is bounded by s + 1, it follows that functions, the use of Φ e,s+1 B

˜ Be (x αm ) = Φ ˜ s+1 (x αm ) = 0. Φ e,s+1 ˜ Be (x αm ) by (70), hence e is met. So A(x αm ) = Φ ˜ Be . Fix m0 Now, for a contradiction, assume that e is not met, i.e., that A = Φ such that, for m ≥ m0 , α α ˜ Be (x αm ) = 0. x αm > s0 & I m ∩ A = Im ∩B=∅&Φ

˜ Be since α reNote that such a number m0 exists by the assumption that A = Φ α quires attention only finitely often, hence I m ∩ (A ∪ B) = ∅ for at most finitely many m. Since α is not initialized after stage s0 , it follows that for m ≥ m0 , the conditions (iii)a , (iii)b and (iii)c in the definition of requiring attention hold at all sufficiently large stages s. On the other hand, since α is saturated by the True Path Lemma, there are infinitely many m such that (iii)d holds for all sufficiently large s. So (iii) holds for all sufficiently large s. Moreover, by the preceding observation, if α becomes active at some stage s + 1 then - by the assumption that e is not met - α is initialized later. Since

72 | Klaus Ambos-Spies α is not initialized after stage s0 , this implies that α is not satisfied at any stage s ≥ s0 , hence (ii) in the definition of requiring attention holds for s ≥ s0 .

So, summarizing, (ii) and (iii) hold for all sufficiently large s. Since, by α  f , there are infinitely many α-stages, it follows that α requires attention infinitely often which gives the desired contradiction. ˆ e is met. Claim 3.  ˆ e is relevant, i.e., that Proof. Fix e = e0 , e1 . W.l.o.g. we may assume that  A ˆ W e0 is simple and W e0 = Φ e1 . We have to show that W e0 is ibT-reducible to B. Let α = f  e. Then, by the True Path Lemma, α0  f . So, in particular, there are infinitely many α0-stages. Moreover, by Claim 2, we may fix a stage s0 such that no strategy β with β ≤ α0 becomes active after stage s0 . Now given x, in order to compute W e0  x +1 from B  x +1, proceed as follows. Pick α0-stages s1 and s2 such that s2 > s1 > s0 , s1 > x and B s2  x + 1 = B  x + 1. We claim that W e0 ,s2  x + 1 = W e0  x + 1. For a contradiction assume that a number x ≤ x enters W e0 after stage s2 . Fix the least such x and let s +1 > s2 be the stage at which x is enumerated into W e0 . Note that at stage s1 + 1 > x all strategies β with α0 < L β are initialized. So, by choice of s0 and by s1 < s2 , a number x ≤ x can enter A at a stage s + 1 ≥ s2 + 1 only if s is an α0-stage and x is the follower of a strategy β where α0  β. Moreover, since s2 > s1 > x and s1 is an α0-stage, it follows that, for any α0-stage s ≥ s2 , α0 is (s1 + 1)-saturated at stage s, hence, by α0-relevance of e, l(e, s) > s1 > x. Now in order to get the desired contradiction distinguish the following two cases. First assume that A s +1  x + 1 = A s2  x + 1. Then, by l(e, s2 ) > x ≥ x , A

A

 ˆ s2 ( x  ) = Φ ˆ s +1 W e0 ,s +1 (x ) = W e0 ,s2 (x ) = Φ e1 ,s2 e1 ,s +1 ( x ) 

hence l(e, s + 1) ≤ x ≤ x. Since, as observed above, at a stage s + 1 ≥ s2 + 1 which is not an α0-stage no number ≤ x can enter A and since l(e, s) > x at any α0-stage s ≥ s2 , it follows that there is no α0-stage s ≥ s + 1. A contradiction. So we may assume that A s +1  x + 1 = A s2  x + 1 and we may fix x ≤ x ≤ x and s with s2 ≤ s ≤ s such that x enters A at stage s +1. Then, as observed above, s is α0-expansionary, l(e, s ) > x ≥ x and x is a follower x βm of some strategy β β β β β where α0  β. So, by construction, I m = [x m , y m ] ⊆ W e0 ,s and y m enters B at β β stage s + 1. By the latter and by choice of s2 , y m > x. So x m = x ≤ x ≤ x < y βm , β i.e., x ∈ I m . It follows that x ∈ W e0 ,s contrary to the assumption that x enters W e0 only at stage s + 1 > s . This completes the proof of Claim 3 and the proof of the theorem. Corollary 42. The class Sr of the simple r-degrees does not generate the class R+ r of the nonzero c.e. r-degrees under join (r = ibT, cl).

On the Strongly Bounded Turing Degrees of Simple Sets |

73

Proof. Fix c.e. sets A and B as in Theorem 41 and let a = deg r (A) and b = deg r (B). Then b < a and Sr ∩ Rr (≤ a) ⊆ Rr (≤ b). So, for any finite sequence c0 , . . . , cn of simple degrees ≤ a such that c0 ∨· · ·∨ cn exists, c0 , . . . , cn ≤ b hence c0 ∨· · ·∨ cn ≤ b < a. So a cannot be presented as the join of finitely many simple degrees. We conclude our investigations of the simple r-degrees by proving the dual of Corollary 42 for the meet operator. Theorem 43. The class Sr of the simple r-degrees does not generate the class Rr of the c.e. r-degrees under meet (r = ibT, cl). I.e., there is a c.e. set A such that, for any n ≥ 0 and for any simple sets A0 , . . . , A n , either deg r (A0 ) ∧ · · · ∧ deg r (A n ) does not exist or deg r (A0 ) ∧ · · · ∧ deg r (A n ) = deg r (A). Note that, by Theorem 33, the degree 0 is the meet of two simple degrees. So, in fact, Theorem 43 shows that the class Sr of the simple r-degrees does not generate the class R+ r of the nonzero c.e. r-degrees under meet. Proof of Theorem 43.. In order to simplify notation we only prove the binary case n = 1 and leave the straightforward extension to the general case to the reader. It suffices to enumerate a c.e. set A and auxiliary c.e. sets V e , e ≥ 0, such that, for any relevant e = e0 , e1 , e2 , e3  ≥ 0, V e ≤ibT W e0 , W e1

(71)

holds and the requirements ˜ An (n ≥ 0) e,n : V e = Φ

are met, where e is relevant if ˜ W ei & W e is simple). ∀ i ≤ 1 (A = Φ e2+i i

(72)

The strategy for meeting requirement e,n (for relevant e), which is similar to the strategy for meeting the corresponding requirements e in the proof of Theorem 41, is as follows. Let e = e0 , e1 , e2 , e3  where we assume that e0 , e1 , e2 , e3 ≤ e. We reserve a computable set of infinitely many pairwise disjoint intervals I = ˜ An (x0 ) [x0 , . . . , x e+n+1 ] of length e + n +2 for e,n . Then we ensure that V e (x0 ) = Φ for the first element of one of these intervals I. For this sake, we follow the stan˜ As (x0 ) = 0 dard Friedberg-Muchnik strategy. I.e., we wait for a stage s such that Φ n,s (if there is no such stage then e,n is trivially met since in this case x0 is not put ˜ A s (x0 ) = 0 by restraining all numinto V e ). Then we put x0 into V e and preserve Φ n,s bers ≤ x0 + n = x n from A. In order to make this compatible with (71), however, we have to make sure that x0 entering V e is permitted by W e i , i.e., that there is a

74 | Klaus Ambos-Spies number y i ≤ x0 which enters W e i after stage s (for i = 0, 1). Now, assuming that e is relevant, this can be achieved as follows. We put x0 into V e at stage s + 1 only if I ⊆ W e0 ,s ∩ W e1 ,s (73) and if, for i = 0, 1,

We

˜ i  x e+n+1 + 1 . A s  x e+n+1 + 1 = Φ e2+i ,s ,s

(74)

Note that, by relevance of e, there must be an interval I and a stage s with these properties. Namely, since W e0 and W e1 are simple, it follows from the Simple Set Lemma that, for some (in fact, infinitely many) of the reserved intervals I, I ⊆ ˜ W ei , (74) holds for all I at all sufficiently large stages W e0 ∩ W e1 ; and, by A = Φ e2+i s. Now, if (73) and (74) hold and we put x0 into V e then we simultaneously put x n+1 into A. Note that the latter does not interfere with the restraint imposed on A by the strategy. But, by (74) and (the first part of) (72), it forces a number y i ≤ x n+1 + e2+i ≤ x n+1 + e = x n+e+1 to enter W e i after stage s. In fact, by (73), this number y i must be less than x0 . Hence V e ≤ibT W e i by permitting. The formal construction is as follows. Let {I km }m,k≥0 be a computable partition of ω into finite intervals where the e,n e,n e,n interval I k = [x k,0 , . . . , x k,e+n+1 ] has length e + n + 2, and reserve the intere,n

vals I k , k ≥ 0, for the strategy meeting requirement e,n . Then stage s + 1 of the construction is as follows where r(e, n, s) is the restraint imposed by e,n at stage s and where A0 = V e,0 = ∅ and r(e, n, 0) = 0.

Stage s + 1 > 0. Requirement e,n requires attention at stage s +1 if e, n < s and r(e, n, s) = 0 and there is a number k ≥ 0 such that the following hold. e,n e,n (i) I k ∩ V e,s = ∅ ∩ As = Ik e,n

(ii) I k

⊆ W e0 ,s ∩ W e1 ,s e,n ˜ W ei ,s  xe,n (iii) A s  x k,e+n+1 + 1 = Φ + 1 for i = 0 , 1 e2+i ,s k,e+n+1 e,n As ˜ (iv) Φ (x )=0 n,s

k,0

e,n

(v) max{r(m, s) : m < e, n} < x k,0 And if e,n requires attention then say that e,n requires attention via k for the least k such that (i)-(v) hold. Then, for the least e, n such that e,n requires attention, e,n becomes e,n

active and for the unique k such that e,n requires attention via k, x k,0 is e,n

put into V e , x k,n+1 is put into A, and the restraint of e,n is set to r(e, n, s + e,n

1) = x k,n . Moreover, for m < e, n, r(m, s +1) = r(m, s) while, for m > e, n, r(m, s + 1) = 0.

On the Strongly Bounded Turing Degrees of Simple Sets | 75

Finally, if no requirement requires attention then r(m, s + 1) = r(m, s) for all m ≥ 0. This completes the construction. By a straightforward induction on m, any requirement m requires attention at most finitely often, hence lims→∞ r(m, s) < ω exists. Using these facts it easily follows by the above discussion of the strategy for meeting the requirements that, for relevant e, the e-requirements e,n , n ≥ 0, are met and condition (71) is satisfied.

12 Conclusion We have analysed the degrees of the simple sets under the strongly bounded Turing reducibilities r = ibT, cl. First we have shown, that a c.e. set A is ibTequivalent to a simple set if and only if A is cl-equivalent to a simple set (Coincidence Theorem; Theorem 8) and that there is a noncomputable c.e. set A which is not ibT-equivalent (hence not cl-equivalent) to any simple set (Theorem 10). We have also shown that the latter observation does not extend to weaker reducibilities like the linearly bounded Turing reducibility. Namely, any noncomputable c.e. set A is lbT-equivalent to a simple set (Corollary 31). Then we had a closer look at the distribution of the simple r-degrees and the nonsimple c.e. r-degrees among all c.e. r-degrees. It turned out that, both, the partial ordering of the simple r-degrees, (Sr , ≤), and the complementary class of the nonsimple c.e. r-degrees, (Sr , ≤), are quite scattered in the partial ordering of the c.e. r-degrees, (Rr , ≤). So, for instance, for any c.e. r-degree a > 0 there are degrees b, c, d ∈ Sr such that b is below a (Theorem 28), c is above a (Theorem 29) and d is incomparable with a (Theorem 34). For the class of the nonsimple c.e. r-degrees the corresponding results hold with one notable exception: for any c.e. r-degree a > 0 there are degrees b, d ∈ Sr such that b = 0 and b is below a (Theorem 17), and d is incomparable with a (Theorem 24). Moreover, if a is not the degree of a wtt-complete set, then there is a degree c ∈ Sr such that c is above a (Theorem 18). Here, however, this additional hypothesis is necessary, since the r-degree of any wtt-complete set is simple (Theorem 36). Moreover, neither Sr nor Sr is dense in (Rr , ≤) - to be more precise, there are infinite intervals in (Rr , ≤) which are entirely contained in Sr (Corollary 39) and there are infinite intervals in (Rr , ≤) which are entirely contained in Sr (Corollary 14) - and the classes Sr and Sr lack the most basic closure properties. Namely, Sr is neither closed under join (Corollary 22) nor closed under meet (Theorem 25). In

76 | Klaus Ambos-Spies case of the simple degrees we have only shown that Sr is not closed under meet (Corollary 35), but Kräling recently closed this gap by showing that Sr is not closed under join too. Finally, we shed some light on the question how common the simple and nonsimple r-degrees are by answering the question whether these degree classes generate all c.e. r-degrees under join or meet. In case of the nonsimple degrees, Sr generates Rr under join (Corollary 21) but not under meet (Corollary 38) while, in case of the simple degrees, Sr generates R+ r neither under join (Corollary 42) nor under meet (Theorem 43). It is an interesting open problem whether the fact that an r-degree contains a simple set is reflected by its ordering properties, i.e., whether the class Sr of the simple r-degrees is definable in the partial ordering (Rr , ≤). Open Problem 44. Is there a first order formula φ(x) in the language of partial orderings such that Sr = {a : (Rr , ≤)  φ(a/x)}? Finally, we come back to what we started with, Post’s Problem and Post’s Program. For what reducibilities ρ do simple sets help to solve Post’s Problem, i.e., allow us to show that there are at least two nonzero c.e ρ-degrees? As pointed out in the introduction already, Post himself gave a positive answer for ρ = m and even ρ = btt along the lines of his program: He established the existence of simple sets and showed that no simple set is btt-complete. Post also showed, however, that, by his approach, simple sets do not solve Post’s Problem for tt-reducibility since there are tt-complete simple sets. For tt-reducibility, however, simple sets provide a solution to Post’s Problem in the sense of Generalized Post Program (as defined in the introduction), i.e., there are noncomputable c.e. sets A and B such that the tt-degree of A contains a simple set (i.e., Stt (degtt (A)) holds) while the tt-degree of B does not contain any simple set (i.e., Stt (degtt (B)) fails). In the same sense, we obtain a solution for Post’s Problem for the strongly bounded Turing reducibilities ρ = ibT and ρ = cl by Theorem 10 and Corollary 11, respectively, in this paper. Even by this generalized approach, however, simplicity does not help to solve Post’s Problem for ρ = lbT, hence for ρ = wtt or ρ = T. Namely, by Corollary 31, SlbT (a) holds for any nonzero c.e. lbT-degree a (hence Swtt (a) and ST (a) hold for any nonzero c.e. wtt- respectively T-degree a). By some of the other results in this paper, however, we can further generalize Post’s Program in order to get a solution of Post’s Problem for ρ = wtt in terms of simple sets. Namely, given a reducibility ρ and a stronger reducibility ρ , we replace the property that the ρ-degree a contains a simple set, i.e., Sρ (a), by the

On the Strongly Bounded Turing Degrees of Simple Sets | 77

stronger property that every c.e. ρ -degree inside a contains a simple sets, i.e., Sρρ (a) ⇔ ∀ c.e. A ∈ a [Sρ (deg ρ (A))]. 

Then, by the existence of nonzero c.e. ibT-degrees which do not contain any simple set (Theorem 10) it follows that there is a c.e. wtt-degree a > 0 such that SibT wtt (a) ibT  fails, whereas, by Theorem 36, Swtt (0wtt ) holds for the wtt-complete degree 0wtt . We do not know, however, whether we can get a solution to the original version of Post’s Problem, i.e., to Post’s Problem for Turing reducibility, in terms of simple sets in this way. A positive answer would be quite remarkable by showing that Post’s simple set property indeed helps to establish the existence of noncomputable Turing-incomplete sets, though the way this property has to be used had to be much more technical than proposed by Post’s Program. In order to show that this is indeed possible it suffices to give an affirmative answer to the following question. Open Problem 45. Is there a c.e. set A such that SibT T ( deg T ( A )) holds, i.e., such that any c.e. set B which is Turing equivalent to A is ibT-equivalent to a simple set? Note that, by Corollary 19, there is a Turing complete set which has nonsimple ibT-degree. So in contrast to the case of weak truth-table reducibility, complete sets do not provide a positive answer to Open Problem 45. In fact, by Theorem 16, for any set A such that SibT T ( deg T ( A )) holds (if any), deg T ( A ) must be contiguous, i.e., degT (A) can contain only one c.e. wtt-degree. (The existence of contiguous degrees > 0 is shown in [13].)

Bibliography [1] [2] [3] [4]

[5]

[6]

K. Ambos-Spies. Cupping and noncapping in the r.e. weak truth-table and Turing degrees. Arch. Math. Logik Grundlag., 25:109–126, 1985. K. Ambos-Spies. On the Strongly Bounded Turing Degrees of the Computably Enumerable Sets. To appear. K. Ambos-Spies, D. Ding, Y. Fan and W. Merkle. Maximal pairs of computably enumerable sets in the computably Lipschitz degrees. Theory of Computing Systems, 52:2–27, 2013. K. Ambos-Spies, C. G. Jockusch, R. A. Shore and R.I. Soare. An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees. Trans. Amer. Math. Soc., 281:109–128, 1984. G. Barmpalias. Computably enumerable sets in the Solovay and the strong weak truth table degrees. In: Computability in Europe, Amsterdam, 2005. Lecture Notes in Comput. Sci., vol. 3526, pp. 8–17. Springer, Berlin, 2005. G. Barmpalias and A.E.M. Lewis. The ibT degrees of computably enumerable sets are not dense. Ann. Pure Appl. Logic, 141:51–60, 2006.

78 | Klaus Ambos-Spies [7] [8] [9] [10] [11] [12]

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A.R. Day. The computable Lipschitz degrees of computably enumerable sets are not dense. Ann. Pure Appl. Logic, 161:1588–1602, 2010. R.G. Downey, D.R. Hirschfeldt, G. LaForte. Randomness and reducibility. J. Comput. System Sci., 68:96–114, 2004. R.G. Downey and D.R. Hirschfeldt. Algorithmic randomness and complexity. Theory and Applications of Computability. Springer, New York, 2010. Y. Fan and H. Lu. Some properties of sw-reducibility. Journal of Nanjing University (Mathematical Biquarterly), 22:244–252, 2005. L. Harrington and R.I. Soare. Post’s program and incomplete recursively enumerable sets. Proc. Nat. Acad. Sci. U.S.A., 88:10242–10246, 1991. C. G. Jockusch. Three easy constructions of recursively enumerable sets. In: Logic Year 1979– 80, Proc. Seminars and Conf. Math. Logic, Univ. Connecticut, Storrs, CT, 1979/80. Lecture Notes in Math., vol. 859, pp. 83–91. Springer, Berlin, 1981. R.E. Ladner and P. Sasso. The weak truth table degrees of recursively enumerable sets. Ann. Math. Logic, 8:429–448, 1975. P. Odifreddi. Classical recursion theory. The theory of functions and sets of natural numbers. With a foreword by G. E. Sacks. Studies in Logic and the Foundations of Mathematics, 125. North-Holland Publishing Co., Amsterdam, 1989. E. L. Post. Recursively enumerable sets of positive integers and their decision problems. Bull. Amer. Math. Soc., 50:284–316, 1944. R.I. Soare. Recursively enumerable sets and degrees. A study of computable functions and computably generated sets. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987. R.I. Soare. Computability theory and differential geometry. Bull. Symbolic Logic, 10:457– 486, 2004. M. Stob. wtt-degrees and T-degrees of r.e. sets. J. Symbolic Logic, 48:921–930, 1983.

Matthew de Brecht

Levels of Discontinuity, Limit-Computability, and Jump Operators¹ Abstract: We develop a general theory of jump operators, which is intended to provide an abstraction of the notion of “limit-computability” on represented spaces. Jump operators also provide a framework with a strong categorical flavor for investigating degrees of discontinuity of functions and hierarchies of sets on represented spaces. We will provide a thorough investigation within this framework of a hierarchy of ∆02 -measurable functions between arbitrary countably based T0 -spaces, which captures the notion of computing with ordinal mind-change bounds. Our abstract approach not only raises new questions but also sheds new light on previous results. For example, we introduce a notion of “higher order” descriptive set theoretical objects, we generalize a recent characterization of the computability theoretic notion of “lowness” in terms of adjoint functors, and we show that our framework encompasses ordinal quantifications of the non-constructiveness of Hilbert’s finite basis theorem. Keywords: Borel complexity, computable analysis, descriptive set theory, category theory Mathematics Subject Classification 2010: 54H05, 26A21, 03D30, 03F60, 03G30 || Matthew de Brecht: National Institute of Information and Communications Technology, Center for Neural Systems and Information Networks, Osaka, Japan

1 Introduction This paper is concerned with two relatively new developments in the field of descriptive set theory. The first development is the extension of the classical descriptive set theory for metrizable spaces to more general topological spaces and mathematical structures. Although it is not uncommon, particularly in measure theory, to define the

1 This paper is dedicated to Victor Selivanov in celebration of his 60th birthday and his valuable contributions to the descriptive set theory of general topological spaces. The author thanks Arno Pauly, Luca Motto Ros, Vasco Brattka, and Takayuki Kihara for valuable discussions and comments on earlier drafts of this paper.

80 | Matthew de Brecht Borel algebra for an arbitrary topological space, detailed analysis of the Borel hierarchy has been mainly restricted to the class of metrizable spaces, or possibly Hausdorff spaces on rare occasion. However, relatively recent work by V. Selivanov [37; 35; 36; 38; 39], D. Scott [34], A. Tang [45; 46], and the author [11; 12], have demonstrated that a significant portion of the descriptive set theory of metrizable spaces generalizes naturally to countably based T0 -spaces. This development opens up the possibility of finding new applications of descriptive set theory to mathematical fields heavily relying on non-Hausdorff topological spaces, such as theoretical computer science (e.g., ω-continuous domains) and modern algebraic geometry (e.g., the Zarisiki topology on the prime spectrum of a countable commutative ring). These generalizations can also shed new light on old results. For example, although the Gandy-Harrington space (a non-metrizable space that plays an important role in effective descriptive set theory) cannot be topologically embedded into any Polish space, it can be embedded as a co-analytic set into a quasi-Polish space [11]. The second development is a shift from a focus on the complexity of subsets of a space to a focus on the complexity of functions between spaces. Certainly Baire’s hierarchy of discontinuous functions has a long history, but it is fair to say that Borel’s hierarchy of sets has played a more prominent role in the development of the theory. However, recently there has been growing interest within the field of computable analysis concerning the relationship between hierarchies of discontinuous functions, Turing degrees, and limit-computability, in particular by researchers such as V. Brattka, P. Hertling, A. Pauly, M. Ziegler, T. Kihara, and the author [6; 10; 20; 7; 48; 25; 12]. Furthermore, recent extensions of the Wadge game by researchers such as A. Andretta, L. Motto Ros, and B. Semmes [2; 29; 27; 28; 40] have provided new classifications of discontinuous functions and new methods to generalize classical results like the Jayne-Rogers theorem [22]. V. Selivanov has made contributions in this area as well, for example by generalizing the HausdorffKuratowski theorem for the difference hierarchy to a hierarchy of ∆02 -measurable functions into finite discrete spaces [38; 39]. These two developments should not be considered independent. For example, if we simply add to our framework the two-point non-trivial non-metrizable space S, known as the Sierpinski space, then we can obtain an elegant bijective correspondence between the family of Σ 0α -subsets of a space X and the family of Σ 0α -measurable functions from X to S. This is a natural generalization of the known bijection between open subsets of a space and continuous functions into the Sierpinski space, and is also similar to the role of the subobject classifier in a topos. Domain theory teaches us that the mathematical object Σ 0α (X ), now viewed as a family of functions into S, will certainly not be metrizable, even if we can hope for it to be a topological space at all.

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Continuing a little more our analogy with a topos, if we wish to work within a single category then we are faced with a dilemma if we have only one “sub-object” classifier S but want a hierarchy of classes of subobjects such as Σ10 (X ) ⊆ Σ20 (X ) ⊆ · · · . One natural solution is to abandon the idea of having a single subobject classifier, and instead have a sequence S1 , S2 , . . . of subobject classifiers that respectively classifiy the families Σ10 , Σ20 , . . .. Such a theory would be very unwieldy if the subobject classifiers were all unrelated, but we might have some hope for the theory if the subobject classifiers S1 , S2 , . . . are defined as the iterates of a single endofunctor F applied to a single subobject classifier S. We now only have to worry about which functors F to consider, and what the “base” subobject classifier S should be. This abstract view is closely related to recent work initiated by A. Pauly on synthetic descriptive set theory [31; 32]. Ultimately, an axiomatic approach in the same spirit as topos theory would be most desireable, as it might help expose connections between the descriptive set theory of general topological spaces and the descriptive complexity of finite structures [1]. However, it seems a little premature to attempt that now, and instead we develop these ideas within the category of represented spaces and continuously realizable functions [4]. In this context, we introduce (topological) “jump operators”, which modulate the representation of a space and in effect play the role of the endofunctors F described above. The concept of a jump operator that we present here has its roots in the work of M. Ziegler [48] and V. Brattka [6; 10], where numerous connections are made between levels of discontinuity, limit-computability, and the representation of a function’s output space. The hierarchy of discontinuity that jump operators characterize turns out to be a subset of the strong Weihrauch degrees [9; 8], but we believe that the categorical framework that jump operators provide has much to offer. This paper is organized into five major sections. After this Introduction, we will develop the general theory of (topological) jump operators. The third major section will investigate a lower portion of the jump operator hierarchy consisting of ∆02 -measurable functions. Our main contribution here is to extend some previous results concerning functions between metrizable spaces to functions between arbitrary countably based T0 -spaces. The results in this section are also important because they demonstrate that the jump operator framework is powerful enough to characterize functions as finely as P. Hertling’s hierarchy of discontinuity levels [21; 20]. The fourth major section presents several examples and applications, such as connections with the difference hierarchy, a quantification of the nonconstructiveness of Hilbert’s basis theorem in terms of the ordinal ω ω (essentially due to S. Simpson [41] and F. Stephan and Y. Ventson [44]), and show some applications to the Jayne-Rogers theorem. It is our attempt to find a common thread

82 | Matthew de Brecht between the results in this section that should be considered new, more so than the results themselves, so in several cases we omit proofs. We conclude in the fifth major section. We will expect that the reader is familiar with classical descriptive set theory [24] and domain theory [17]. The reader should also consult [37] and [11] for definitions and results concerning the descriptive set theory of arbitrary countably based T0 -spaces. Although we will not be concerned much with computability issues, the reader will benefit from an understanding of the Type Two Theory of Effectivity [47]. In particular, we will make much use of M. Schröder’s extended definition of an admissible representation [33], as well as the notion of realizability of functions between represented spaces (see [4]). Our notation will follow that of [11]. The following modification of the Borel hierarchy, due to V. Selivanov, is required in order to provide a meaningful classification of the Borel subsets of non-metrizable spaces. Definition 1. Let X be a topological space. For each ordinal α (1 ≤ α < ω1 ) we define Σ0α (X ) inductively as follows. 1. Σ10 (X ) is the set of all open subsets of X. 2. For α > 1, Σ0α (X ) is the set of all subsets A of X which can be expressed in the form  A= B i \ Bi , i∈ω

where for each i, B i and Bi are in Σ0β i (X ) for some β i < α. We define Π α0 (X ) = {X \ A | A ∈ Σ 0α (X )} and ∆0α (X ) = Σ 0α (X ) ∩ Π α0 (X ).

The above definition is equivalent to the classical definition of the Borel hierarchy for metrizable spaces, but it differs for more general spaces. A function f : X → Y is Σ0α -measurable if and only if f −1 (U ) ∈ Σ 0α (X ) for every open subset U of Y. We will also be interested in ∆02 -measurability, which requires the preimage of every open set to be a ∆02 -set. Later in the paper we will present some results specific to quasi-Polish spaces, which are defined as the countably based spaces that admit a Smyth-complete quasi-metric. Polish spaces and ω-continuous domains are examples of quasiPolish spaces. A space is quasi-Polish if and only if it is homeomorphic to a Π20 subset of P(ω), the power set of ω with the Scott-topology. The reader should consult [11] for additional results on quasi-Polish spaces.

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83

2 Jump operators A represented space is a pair X, ρ where X is a set and ρ : ⊆ ω ω → X is a surjective partial function. If X, ρ X  and Y , ρ Y  are represented spaces and f : ⊆ X → Y is a partial function, then a function F : ⊆ ω ω → ω ω realizes f , denoted F  f , if and only if f ◦ ρ X = ρ Y ◦ F. If there exists a continuous realizer for f then we say that f is continuously realizable and write  f . Note that if F  f and G  g, then G ◦ F  g ◦ f , assuming the composition g ◦ f makes sense. In some cases, a function f : X → Y between represented spaces may fail to be continuously realizable, but will become continuously realizable if we strengthen the information content of the representation of X or weaken the information content of the representation of Y. The notion of “limit computability” is a common example of weakening the output representation. The motivation for the following definition is to create an abstract framework to investigate in a uniform manner how modifications of the information content of a representation effects the realizability of functions. Definition 2. A (topological) jump operator is a partial surjective function j : ⊆ ω ω → ω ω such that for every partial continuous F : ⊆ ω ω → ω ω , there is partial continuous F  : ⊆ ω ω → ω ω such that F ◦ j = j ◦ F  . The identity function id : ω ω → ω ω is a trivial example of a jump operator. Let f : X → Y be a function between represented spaces. A j-realizer of f is a function F : ⊆ ω ω → ω ω such that j ◦ F  f . We use the notation F j f to denote that F is a j-realizer for f . If there exists a continuous j-realizer for f then we will say that f is j-realizable and write j f . The definition of “jump operator” given above is appropriate for the category of represented spaces and continuously realizable functions. Given a represented space X, ρ X  and a jump operator j, we can write j(X ) to denote the represented space X, ρ X ◦ j. For each function f : X → Y between represented spaces, we define j(f ) to be the same function as f but now interpretted as being between the represented spaces j(X ) and j(Y ). It is now clear that the definition of a jump operator is precisely what is needed to guarantee that j(·) is a well-defined endofunctor on the category of represented spaces. If working in the category of represented spaces and computably realizable functions, then the appropriate definition of a (computability theoretic) jump operator would be to require that for every computable F : ⊆ ω ω → ω ω , there is computable F  : ⊆ ω ω → ω ω such that F ◦ j = j ◦ F  . The definition of j-realizability would also be modified in a similar manner. These modifications are necessary

84 | Matthew de Brecht because, for example, the operator L introduced in [7] to characterize low computability is a computability theoretic jump operator but it is not a topological jump operator. In this paper, unless explicitly mentioned otherwise we will assume the topological jump operator definition given above and shall drop the term “topological”. However, much of the theory we develop will also apply to the computability theoretic jump operators as well. Examples 3, 4, and 5 below provide typical examples of jump operators. In the following, · · · n∈ω : (ω ω )ω → ω ω is some fixed (computable) encoding of countable sequences of elements of ω ω as single elements of ω ω . Example 3. Define j Σ20 : ⊆ ω ω → ω ω as: ξ n n∈ω ∈ dom(j Σ0 ) 2

j Σ20 (ξ n n∈ω )

⇔ =

ξ0 , ξ1 , . . . converges in ω ω lim ξ n

n∈ω

Example 4. Define j ∆ : ⊆ ω ω → ω ω as: ξ n n∈ω ∈ dom(j ∆ )

j ∆ (ξ n n∈ω )

⇔ =

(∃n)(∀m ≥ n)[ξ m = ξ n ] lim ξ n

n∈ω

Example 5. For each countable ordinal α, define j α : ⊆ ω ω → ω ω as: β n α  ξ n n∈ω ∈ dom(j α )

⇔

j α (β n α  ξ n n∈ω )

=

(∀n)(α > β n ≥ β n+1 ) and

(∀n)(ξ n = ξ n+1 ⇒ β n = β n+1 ) lim ξ n

n∈ω

where ·α : α → ω is some fixed encoding of ordinals less than α as natural numbers, and βα  ξ is the element of ω ω obtained by prepending the encoding of β to the beginning of ξ . The jump operators j Σ20 and j ∆ are also computability theoretic jump operators. If α < ω1CK then j α is a computability theoretic jump operator assuming that the encoding ·α is effective. Intuitively, j Σ20 -realizing a function only requires the realizer to output a sequence of “guesses” which is guaranteed to converge to the correct answer. Each guess is an infinite sequence in ω ω , and convergence means that each finite prefix of the guess can be modified only a finite number of times. The jump operator j Σ20 and its connections with limit computability have been extensively studied in the field of computable analysis, for example by V. Brattka [6; 10] and M. Ziegler [48].

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85

In the context of Wadge-like games and reducibilities, the jump operator j Σ20 essentially captures the notion of an “eraser” game (see [27; 28; 40] and the references therein). It can also be shown that j Σ20 is a Σ20 -admissible representation of ω ω in the sense of [12]. It follows from these results that if X and Y are countably based T0 -spaces, then f : X → Y is j Σ20 -realizable if and only if f is Σ20 -measurable. In general, for any ordinal α < ω1 , any Σ0α -admissible representation j Σ0α of ω ω will be a jump operator which precisely captures the Σ0α -measurable functions between countably based spaces. For finite n > 2 the inductive definition j Σ0 = j Σ0n ◦ j Σ20 n+1 suffices. The jump operator j ∆ defines a stricter notion of limit computability. In this case, the realizer is also allowed to output guesses that converge to the correct answer, but the realizer can only modify his guess a finite number of times. This jump operator has been investigated by M. Ziegler [48] in terms of finite revising computation, and was shown in [7] by V. Brattka, A. Pauly, and the author to correspond to closed choice on the natural numbers. In the context of Wadgelike games, j ∆ corresponds to the “backtrack” game (see [40]). A. Andretta [2] has shown that a total function on ω ω is j ∆ -realizable if and only if it is ∆02 -piecewise continuous. It follows from the Jayne-Rogers theorem ([22], see also [29; 23; 43; 25]) that a total function on ω ω is j ∆ -realizable if and only if it is ∆02 -measurable. However, it should be noted that this relationship between j ∆ -realizability and ∆02 measurability does not extend to functions between arbitrary spaces, and in fact we will show in the latter half of this paper that there is no jump operator that completely captures the notion of ∆02 -measurability. The family of jump operators j α for α < ω1 are further restrictions of j ∆ , where the realizer must output with each guess an ordinal bound on the number of times it will change its guess in the future. For example, when 1 ≤ n < ω, a j n -realizer can only make a maximum of n guesses. A j ω -realizer must output a bound n < ω along with its first guess, and then can only modify its guess a maximum of n times thereafter. This concept is closely related to the Ershov hierarchy [15], to the notion of ordinal mind change complexity in the field of inductive inference (see [16; 26; 13; 14]), and to the Hausdorff difference hierarchy [24]. We will show later in this paper that a function between countably based T0 -spaces is j α -realizable if and only if the discontinuity level of the function (in the sense of P. Hertling [21; 20]) does not exceed α.

2.1 Lattice structure Jump operators are (quasi-)ordered by j ≤ k if and only if j is k-realizable. We will say j and k are equivalent, written j ≡ k, if j ≤ k and k ≤ j. In the examples given

86 | Matthew de Brecht previously, it is clear that j α ≤ j β ≤ j ∆ ≤ j Σ20 when α ≤ β < ω1 . It is straightforward to prove that if j ≤ k, then j f implies k f for every function f between represented spaces. In this section we will prove that the topological jump operators form a lattice which is complete under countable (non-empty) meets and joins. The definitions and main points of the proofs in this section are mostly due to A. Pauly [30]. The author is indebted to A. Pauly for pointing out that the proofs in this section only apply to topological jump operators and may fail to hold for computability theoretic jump operators in general. In the following, given i ∈ ω and ξ ∈ ω ω , we will write i  ξ to denote the element of ω ω obtained by prepending i to the beginning of ξ . Definition 6. Let (j i )i∈ω be a countable sequence of jump operators. Define   j i : ⊆ ω ω → ω ω and j i : ⊆ ω ω → ω ω by  1. ( j i )(i  ξ ) = j i (ξ ), where  dom( j i ) = {i  ξ ∈ ω ω | ξ ∈ dom(j i )}. 2. (



j i )(ξ n n∈ω ) = j0 (ξ0 ), where  dom( j i ) = {ξ n n∈ω ∈ ω ω | ∀i, k : j i (ξ i ) = j k (ξ k )}.

The next two theorems show that the above definitions are in fact (topological) jump operators corresponding to the supremum and infimum of (j i )i∈ω . Theorem 7.



j i is a jump-operator and is the supremum of (j i )i∈ω .

Proof. Assume f : ⊆ ω ω → ω ω is continuous. For i ∈ ω there is continuous g i : ⊆ ω ω → ω ω such that f ◦ j i = j i ◦ g i . Define g : ⊆ ω ω → ω ω as g (i  ξ ) = i  g i (ξ ). Clearly g is continuous and  f (( j i )(i  ξ )) = f (j i (ξ )) =

= =

j i (g i (ξ ))  ( j i )(i  g i (ξ ))  ( j i )(g (i  ξ )),

   hence f ◦ j i = j i ◦ g. Therefore j i is a jump operator.  Next, for i ∈ ω define f i (ξ ) = i  ξ . Then f i is continuous and j i = ( j i ) ◦ f i .  Therefore j i ≤ j i for all i ∈ ω. Finally, assume p : ω ω → ω ω is such that j i = p ◦ q i for some continuous q i : ω ω → ω ω (for all i ∈ ω). Define q : ω ω → ω ω so that q(i  ξ ) = q i (ξ ). Then

Levels of Discontinuity and Jump Operators |

q is continuous and (



j i )(i  ξ )

=

j i (ξ )

=

p(q i (ξ ))

87

p(q(i  ξ ))  j i ≤ p. It follows that j i is the supremum of =

hence



( j i )i∈ω .

j i = p ◦ q. Therefore

Theorem 8.





j i is a jump-operator and is the infimum of (j i )i∈ω .

Proof. Assume f : ⊆ ω ω → ω ω is continuous. For i ∈ ω there is continuous g i : ⊆ ω ω → ω ω such that f ◦ j i = j i ◦ g i . Define g : ⊆ ω ω → ω ω as

g (ξ n n∈ω ) = g n (ξ n )n∈ω .  Clearly g is continuous and if ξ n n∈ω ∈ dom( j i ) then for all i, k ∈ ω, j i (ξ i ) = j k (ξ k ) hence j i ◦ g i (ξ i ) = f ◦ j i (ξ i ) = f ◦ j k (ξ k ) = j k ◦ g k (ξ k ), and it follows that   g n (ξ n )n∈ω ∈ dom( j i ). So for ξ n n∈ω ∈ dom( j i ) we have  f (( j i )(ξ n n∈ω )) = f (j0 (ξ0 )) =

=







j0 (g0 (ξ0 ))  ( j i )(g n (ξ n )n∈ω ),

hence f ◦ j i = j i ◦ g. Therefore j i is a jump operator. Next, define π i (ξ n n ∈ ω) = ξ i , which is clearly continuous. Then  ( j i )(ξ n n∈ω ) = j0 (ξ0 ) 

=

j i (ξ i )

=

j i (π i (ξ n n∈ω )),

hence j i ≤ j i for all i ∈ ω. Assume p : ω ω → ω ω is such that p = j i ◦ q i for some continuous q i : ω ω → ω ω (for all i ∈ ω). Define q : ω ω → ω ω so that q(ξ ) = q n (ξ )n∈ω . Clearly q is continuous. If ξ ∈ dom(q) then p(ξ ) = j i (q i (ξ )) for all i ∈ ω, so q n (ξ )n∈ω ∈  dom( j i ) and p(ξ )

=

= =

hence p ≤



j i . It follows that



j0 (q0 (ξ ))  ( j i )(q n (ξ )n∈ω )  ( j i )(q(ξ )),

j i is the infimum of (j i )i∈ω .

For example, let (j α i )i∈ω be a sequence of jump operators from Example 5. Then it   is straightforward to verify that j α i = j α i and j α i = j α i .

88 | Matthew de Brecht

2.2 The “jump” of a representation Let J be the lattice of jump operators. It is easy to show that if j and k are jump operators, then so is j ◦ k. Furthermore, if j1 ≤ j2 , then j1 ◦ k ≤ j2 ◦ k. Thus, every jump operator k defines a monotonic function on J, which we call the k-jump, that maps j to j ◦ k. This notion of iterating “jumps” can be found in [48] and [10] for the case of j Σ20 . A jump operator j is extensive if the identity function id : ω ω → ω ω is jrealizable. Currently the author is unaware of any topological jump operators that are not extensive, but the non-extensive computability theoretic jump operators have a non-trivial structure (for example, the inverse of the Turing jump, or integral, in [10] and [7] is non-extensive). A jump operator j is idempotent if j ◦ j ≡ j. The jump operator j ∆ is idempotent, but j Σ20 is not. We will say that j-realizability is closed under compositions if for every pair of j-realizable functions f : X → Y and g : Y → Z we have that g ◦ f is also j-realizable. Theorem 9. If j is an extensive jump operator, then j-realizability is closed under composition if and only if j is idempotent. Proof. Assume j is extensive and closed under compositions. Clearly, j is jrealizable, so j ◦ j must be j-realizable, hence j ◦ j ≤ j. On the other hand, since id ≤ j it follows by the monotonicity of the j-jump that j ≤ j ◦ j. Therefore, j ◦ j ≡ j. For the converse, assume j is extensive and idempotent, and assume F j f and G j g and the composition g ◦ f is possible. Then j ◦ F  f and j ◦ G  g, and composition gives j ◦ G ◦ j ◦ F  g ◦ f . Since j is a jump operator, there is continuous G such that j ◦ j ◦ G ◦ F  g ◦ f . Now using the idempotent property of j we obtain G  ◦ F j g ◦ f . Recall that a closure operator on a partially ordered set is a function which is monotonic, extensive, and idempotent. The above theorem can be reworded as follows.

Corollary 10. If j is an extensive jump operator, then j-realizability is closed under composition if and only if the j-jump is a closure operator on J. It is easy to see that if the j-jump is a closure operator on J, then j is the least fixed point of the j-jump above id. In particular, the j ∆ -jump of j α is equivalent to j ∆ for each α < ω1 . It turns out that j Σ0α is a fixed point of the j ∆ -jump for each α < ω1 because j Σ0α ◦ j ∆ is Σ0α -measurable.

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2.3 Strong Weihrauch Degrees In this section we will compare jump operators with the notion of strong Weihrauch reducibility (see [9; 8; 30; 7] for more on Weihrauch reducibility). We only consider the topological version of reducibility for the case of single valued functions. Definition 11. Let f : X → Y and g : W → Z be functions between represented spaces. Define f ≤sW g if and only if there are continuous functions K, H : ⊆ ω ω → ω ω satisfying K ◦ G ◦ H  f whenever G  g. Theorem 12. Let f : X → Y be a function between represented spaces, and let j be a jump operator. Then f ≤sW j if and only if f is j-realizable. Proof. Assume f ≤sW j and let K and H be the relevant continuous functions. Since ω ω is represented by the identity function, it follows that K ◦ j ◦ H  f . Using the fact that j is a jump operator, there is continuous K  : ⊆ ω ω → ω ω such that j ◦ K  ◦ H  f . Therefore, the continuous function K  ◦ H j-realizes f . For the converse, assume F : ω ω → ω ω is a continuous j-realizer of f . Then j ◦ F  f by definition. Again, since ω ω is represented by the identity function we have J  j if and only if J = j. Thus, taking K as the identity function and H = F demonstrates that f ≤sW j.

The above theorem shows that jump operators form a subset of the strong Weihrauch degrees. However, this inclusion is strict, in the sense that there are strong Weihrauch degrees that do not correspond to any jump operator. For example, a constant function on ω ω is not strong Weihrauch equivalent to any jump operator because jump operators are surjective.

2.4 Adjoints This section actually applies more to computability theoretic jump operators than topological jump operators, but the basic definitions and immediate results are the same in both cases. This section mainly consists of generalizations of results found in [10] and [7]. Let j and k be jump operators and let id : ω ω → ω ω be the identity function. We say that j is left adjoint to k or that k is right adjoint to j, and write j  k, if and only if k ◦ j ≤ id ≤ j ◦ k. This is equivalent to stating that the j-jump on J is left adjoint to the k-jump, and it also implies that the associated endofunctors are adjoint.

90 | Matthew de Brecht Example 13 (see [10] and [7]). Let (U n )n∈ω be a standard enumeration of the computably enumerable open subsets of ω ω . Define J : ω ω → ω ω by J (ξ )(n) = 1 if ξ ∈ U n and J (ξ )(n) = 0, otherwise. Then J −1 , the inverse of J, is a computability theoretic jump operator and J −1  j Σ20 . Note, however, that J −1 is not a topological jump operator [7]. Proposition 14. If j  k then the (j ◦ k)-jump is a closure operator. In particular, (j ◦ k)-realizable functions are closed under composition. Proof. This is a well known property of adjoints. Since k ◦ j ≤ id it follows that j ◦ k ◦ j ◦ k ≤ j ◦ id ◦ k ≡ j ◦ k. Furthermore, id ≤ j ◦ k implies j ◦ k ≡ id ◦ j ◦ k ≤ j ◦ k ◦ j ◦ k, and it follows that (j ◦ k) is idempotent. Therefore, the (j ◦ k)-jump is a closure operator. The low-jump-operator is defined as L = J −1 ◦ j Σ20 . It is shown in [7] that Lrealizability captures the notion of “lowness” from computability theory. It immediately follows from the above proposition that L-realizable functions are closed under composition. The general theory of adjoints provides much information about j and k when it is known that j  k. For example, the j-jump preserves joins on J and the k-jump preserves meets. Viewed as functors, j preserves colimits and k preserves limits. This means, in particular, that k(X )× k(Y ) will be isomorphic to k(X × Y ) for every pair of represented spaces X and Y. Although so far we have been investigating the effects of weakening the output representation, it is also interesting to investigate the effects of strengthening the input representation. Given jump operators j and k, represented spaces X, ρ X  and Y , ρ Y , and a function f : X → Y, we will say that a function F : ⊆ ω ω → ω ω j, k-realizes f if and only if f ◦ ρ X ◦ j = ρ Y ◦ k ◦ F. This simply means that F realizes f reinterpretted as a function between j(X ) and k(Y ). We will say that a function is j, k-realizable if and only if it has a continuous j, k-realizer. Clearly, j-realizability as defined earlier corresponds to id, j-realizability. The following theorem shows that if j  k, then strengthening the input representation by j is equivalent to weakening the output representation by k. Theorem 15. If j and k are jump operators and j  k, then j, id-realizability is equivalent to id, k-realizability. Proof. Assume j  k and that f : X → Y is j, id-realizable. Let F j be any continuous j, id-realizer for f . Since k is a jump operator there is a partial continuous F j that k, k-realizes F j , hence F j is a j ◦ k, k-realizer of f . If we let I be a contin-

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uous function reducing id to j ◦ k, then F j ◦ I is a continuous id, k-realizer for f . Therefore, f is id, k-realizable. Proving that id, k-realizability implies j, id-realizability is done similarly. Finally, the following proposition shows that it is easy to create new pairs of adjoint jump operators from a given pair of adjoint operators. We leave the proof as an easy exercise. Proposition 16. If j  k, then k ◦ k ◦ j ◦ j ≤ k ◦ j ≤ id ≤ j ◦ k ≤ j ◦ j ◦ k ◦ k. In particular, we have j ◦ j  k ◦ k.

2.5 Additional properties In our final section on the general theory of jump operators, we would like to emphasize how they can contribute to the development of a categorical framework for descriptive set theory. The observations in this section are closely related to recent work initiated by A. Pauly on synthetic descriptive set theory [31; 32]. Let S = {⊥, } be the Sierpinski space and let 2 = {0, 1} be the discrete two point space. It is well known that there is a bijection between the open (resp., clopen) subsets of a topological space X and the continuous functions from X to S (resp., 2). In the same manner, there is an obvious bijection between Σ20 (X ) and the set of j Σ20 -realizable functions χ : X → S. Furthermore, ∆02 (X ) is in bijective correspondence with the set of j Σ20 -realizable functions χ : X → 2. In general, given an arbitrary jump operator j and a represented space X, we can define Σ j (X ) to be the set of j-realizable functions from X into S, and define ∆ j (X ) to be the set of j-realizable functions from X into 2. Thus, each jump operator j determines a “j-decideable” class ∆ j (X ) of subsets of X and a “j-semi-decideable” class Σ j (X ).² It is well known that the category of represented spaces and continuously realizable (total) functions is cartesian closed (see [4], for example). Given a represented space Y and a jump operator j, recall that j(Y ) denotes the represented space obtained by composing the representation with j (this is the image of Y under the endofunctor determined by j). Then for any pair of represented spaces X and Y, the exponential object j(Y )X is the natural candidate for the represented space of j-realizable functions from X to Y. In particular, j(S)X corresponds to Σ j (X ) and j(2)X corresponds to ∆ j (X ). 2 Note that ∆ j and Σ j will completely coincide for some jump operators, such as j ∆ .

92 | Matthew de Brecht We can therefore define notions such as “Σ20 -set” on an arbitrary represented space X, and we can interpret the set of Σ20 -sets as a new represented space. This can be done even when it is impossible to interpret X as a topological space in any natural way. What kind of a space is Σ20 (Σ20 (X ))? Note that j Σ20 (S) is isomorphic to the Sierpinski space with the total representation ρ : ω ω → S sending ξ ∈ ω ω to  if and only if (∃n)(∀m)[ξ (n, m) = 1]. Thus, Σ20 (X ) represents in a sense the family of Σ20 -predicates on X, and Σ20 (Σ20 (X )) is the second-order object corresponding to the family of Σ20 -predicates on the Σ20 -predicates on X. This connection between Σ0n (X ) and Σ20 -predicates easily extends to n > 2. It is the topic of future research to determine what kind of general “topological” information can be extracted from spaces like Σ20 (Σ20 (X )).

3 Levels of discontinuity The next part of this paper will be dedicated to characterizing j ∆ - and j α -realizability (1 ≤ α < ω1 ) for functions between arbitrary countably based T0 -spaces. A characterization of j ∆ -realizability for functions on ω ω has already been given by A. Andretta [2]. In addition, L. Motto Ros [27] has independently investigated a notion related to j α -realizability on metric spaces. However, the extension of the theory to arbitrary countably based T0 -spaces that we provide here appears to be new. In the following sections, we will assume that all represented spaces are countably based T0 -topological spaces with admissible representations. Recall from [33; 47] that a representation ρ : ⊆ ω ω → X to a topological space X is admissible if ρ is continuous and for any continuous f : ⊆ ω ω → X there is continuous F : ω ω → ω ω such that f = ρ ◦ F. It is well known that a function f : X → Y between admissibly represented spaces is continuously realizable if and only if it is continuous.

3.1 Characterization of j ∆ -realizability A total function f : X → Y is ∆02 -piecewise continuous if and only if there is a family  {A i }i∈ω of sets in ∆ 02 (X ) such that X = i∈ω A i and f |A i : A i → Y, the restriction of f to A i , is continuous for all i ∈ ω. Let ω∞ be the one point compactification of the natural numbers, with ∞ the point at infinity. Recall that a function ξ : ω∞ → X is continuous if and only if the

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sequence (ξ (i))i∈ω converges to ξ (∞) in X. Given a continuous function ξ : ω∞ → X and S ⊆ X, we say that ξ is eventually in S if and only if ξ (∞) ∈ S and ξ (m) ∈ S for all but finitely many m ∈ ω. We will say that ξ is eventually equal to x for some x ∈ X if ξ is eventually in the singleton set {x}, and in this case we will also say that ξ is eventually constant. Assuming, as we do, that X and Y are countably based, a function f : X → Y is ∆02 -piecewise continuous if and only if there is a ∆02 -measurable function ι : X → ω such that for any continuous function ξ : ω∞ → X, if ι ◦ ξ is eventually constant then f ◦ ξ is continuous. Converting from the previous definition to this definition only requires the equivalence between continuity and sequential continuity for countably based spaces, and the generalized Σ20 -reduction principle which allows us to convert a Σ20 -covering into a Σ20 -partitioning. We will call the function ι : X → ω above a ∆02 -indexing function for f . For example, the function ι ∆ : dom(j ∆ ) → ω that maps each ξ n n∈ω ∈ dom(j ∆ ) to the least n ∈ ω satisfying (∀m ≥ n)[ξ m = ξ n ] is a ∆02 -indexing function for j ∆ . The next theorem generalizes a result by A. Andretta [2]. Theorem 17. Let f : X → Y be a function between admissibly represented countably based T0 -spaces. Then f is j ∆ -realizable if and only if f is ∆02 -piecewise continuous. Proof. Let ρ X be the admissible representation for X and ρ Y the admissible representation for Y. We can assume without loss of generality that ρ X is an open map and has Polish fibers (i.e. ρ−1 X ( x ) is Polish for each x ∈ X), and similarly for ρ Y . Assume F : ⊆ ω ω → ω ω j ∆ -realizes f . Then ι = ι ∆ ◦ F is a ∆02 -indexing function for f ◦ ρ X = ρ Y ◦ j ∆ ◦ F. Since ι is ∆02 -measurable, we can write ι−1 (n) =  n n i∈ω A i for suitably chosen closed sets A i . Let { B i }i∈ω be a countable basis for k k k k ω ω , and define U n,i = ρ X (B k ) and V n,i = ρ X (B k \ A ni ). Note that U n,i and V n,i are open subsets of X by our assumption that ρ X is an open map. k k We first show that each x ∈ X is in U n,i \ V n,i for some choice of k, n, i ∈ ω.  −1 n Since ρ X (x) ⊆ n,i∈ω A i , the Baire category theorem implies some A ni must have −1 non-empty interior in ρ−1 X ( x ). Thus there is some k ∈ ω such that B k ∩ ρ X ( x ) = ∅ −1 −1 n k k and B k ∩ ρ X (x) ⊆ A i ∩ ρ X (x). It follows that x ∈ U n,i \ V n,i . Let ·, ·, · : ω3 → ω be a bijection, and define ι : X → ω so that ι(x) = k, n, i, k k where k, n, i is the least number satisfying x ∈ U n,i \ V n,i . It is immediate that ι 0 is ∆2 -measurable. Let ξ : ω∞ → X be a continuous function such that ι ◦ ξ is eventually constant. The admissibility of ρ X implies there is continuous ξ  : ω∞ → ω ω such that ξ = ρ X ◦ ξ  . Assume (ι ◦ ξ )(∞) = k, n, i. Then ξ is eventually in ρ X (B k ) \ ρ X (B k \ A ni ), hence ξ  is eventually in B k ∩ A ni , and it follows that ι ◦ ξ  is eventually equal to

94 | Matthew de Brecht n. Since ι is a ∆02 -indexing function for f ◦ ρ X , it follows that f ◦ ξ = f ◦ ρ X ◦ ξ  is continuous. Therefore, ι is a ∆02 -indexing function for f , and we have proven that f is ∆02 -piecewise continuous. For the converse, let ι : X → ω be a ∆02 -indexing function for f . Then ι = ι ◦ ρ X  is a ∆02 -indexing function for f ◦ ρ X . We can write ι−1 (n) = i∈ω A ni for suitably chosen closed sets A ni , and we have that f ◦ ρ X restricted to A ni is continuous. By the admissibility of ρ Y , there is continuous F in : ⊆ ω ω → ω ω that realizes the restriction of f ◦ ρ X to A in . By relabeling, we can assume that {A i }i∈ω is a family of closed sets covering the domain of f ◦ ρ X , and F i is a continuous realizer for the restriction of f ◦ ρ X to A i . The most intuitive way to explain how to “glue” together the continuous realizers F i into a single j ∆ -realizer F, is to define an algorithm for a Type Two Turing Machine that computes F (possibly with access to some oracle). This description will also help clarify the connections between limit computing with finite mind changes and the j ∆ jump operators. The reader should consult [47] for more on Type Two Turing Machines, and [48] for an intuitive description of computing with finite mind changes. The realizer F first initializes a pointer p := 0, and begins reading in the input ξ ∈ ω ω . While reading in the input, F attempts to write to its output tape (an encoding of) an infinite sequence of copies of the output of F p (ξ ). In parallel, F will try to determine whether or not ξ really is in A p . If ξ is not in A p , then this will be observed after reading in some finite prefix of ξ because A p is a closed set. In such a case, F will increment the pointer p := p + 1, and then resume outputting copies of the output of F p (ξ ) and testing whether x ∈ A p for the updated value of the pointer p. When p is incremented, it is possible that F has already written some finite prefixes of a finite number of elements of ω ω to the output tape. After incrementing the pointer, F will consider these initial guesses to be invalid, and will complete the prefixes it has already written by extending them with infinitely many zeros. This guarantees that F will produce a valid encoding of an infinite sequence of elements of ω ω as output. Since {A i }i∈ω covers the domain of f ◦ ρ X , after a finite number of “mind changes” the pointer p will reach a value such that ξ ∈ A p , and the pointer will never be modified again afterwards. Since F p realizes the restriction of f ◦ ρ X to A p , we see that the output of F converges after a finite number of modifications to the desired output.

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3.2 Characterization of j α -realizability In this section we will characterize j α -realizability in terms of a hierarchy of discontinuity levels introduced by P. Hertling [21; 20]. Recall that a function f : X → Y is continuous at x ∈ X iff for any neighborhood V of f (x) there is an open neighborhood U of x such that f (U ) ⊆ V. If f is not continuous at x then f is discontinuous at x. Definition 18 (P. Hertling [21; 20]). Let cl(·) be the closure operator on X and let f : X → Y be a function. For each ordinal α, define Lα (f ) recursively as follows: 1. L0 (f ) = X 2. Lα+1 (f ) = cl({x ∈ Lα (f ) | f |Lα (f ) is discontinuous at x})  3. If α is a limit ordinal, then Lα (f ) = β δ j , δ j−1 , . . . , δ0 > 0.

126 | J. Duparc, O. Finkel, and J.-P. Ressayre Let k be the least integer such that ∀i ≤ j δ i < ω k+1 . Then there exist ωlanguages B, C ∈ BC(k) such that B ≡w Ω(α) and C ≡w Ω(α) . We recall that Ω(α) is defined by n

n

Ω(α) = (∅ • ω1j ) • δ j + (∅ • ω1j−1 ) • δ j−1 + . . . + (∅ • ω1n0 ) • δ0 . Proof. For every i ≤ j the set ∅ • ω1n i is ω-regular, so that for each integer i ≤ j we − − have machines Ai and Ai such that L(Ai ) ≡w ∅ • ω1n i and L(Ai ) ≡w (∅ • ω1n i ) hold for every i ≤ j. The case j = 0 was already proved in Proposition 24, so that we may assume that j > 0 holds. Now we also consider for each i ≤ j the “exact" k i -partially blind-counter − automata Aiδ i and Aiδ i that were designed in the proof of Proposition 24. (Notice that k i was defined as the least integer such that δ i < ω k i +1 .) We then form for each − i ≤ j, some k-partially blind-counter automata Aˆiδ i and Aˆiδ i which work exactly the way Aiδ i and Aiδ i do on the first k i -many counters, leaving untouched the last k − k i -many ones. For simplicity – and without loss of generality – we may assume that both − Σ h ∩ Σ i = ∅ and Σ i = Σ − i hold for every 0 ≤ h < i ≤ j, where Σ i , Σ i stand for the −

respective alphabets of Aˆiδ i and Aˆiδ i . Then, for each i ≤ j we form, some k-partially blind-counter automata Aˇiδ i and  − − Aˇiδ i which work on the alphabet Σ = i≤j Σ i . The machine Aˇiδ i (resp. Aˇiδ i ) works −

as Aˆiδ i (resp. Aˆiδ i ) on the alphabet Σ i , and both machines reject if they eventually read a letter not in Σ i . At last we use new letters t+ , t− for getting from one machine to another, and we build Aα , A− α as follows. −

ˇ0 1. As long as neither t+ nor t− is encountered, Aα (resp. A− α ) works as Aδ0 (resp. −

Aˇ0δ0 ).

2. If a letter among {t+ , t− } is encountered for the ith time for some i < j, then right after this letter is read and until another letter of the form t+ or t− is eventually read: ˇi – if this letter is t+ , Aα (resp. A− α ) now behaves as if it were Aδ i , and –

ˇi if this letter is t− , Aα (resp. A− α ) now behaves as if it were Aδ i . −

3. If a letter among {t+ , t− } is encountered for the jth , then both Aα and A− α reject right away.

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 The result is L(Aα ) ≡w Ω(α) and L(A− α ) ≡w Ω ( α ) . We leave the tedious but straightforward details to the reader.

6 Localisation of BC(k) This section is dedicated to proving that no other Wadge class than the ones described in Theorem 25 is generated by some non-self dual ω-language in BC(k). As a preliminary, we present a technical result about the Wadge hierarchy, a few others about ordinal combinatorics, and some notations. For any A ⊆ X ω and u ∈ X ∗ , we write u−1 A for the set {x ∈ X ω | ux ∈ A}, and we say that A is initializable if the second player has a w.s. in the Wadge game W (A, A) even though she is restricted to positions u ∈ X ∗ that satisfy u−1 A ≡w A. Lemma 26. For A ⊆ X ω any initializable set, B ⊆ Y ω , and δ, θ any countable ordinals,  −1   u B ≡w A • (θ + 1) ∗ A • (θ + 1) ≤w B ≤w A • δ =⇒ ∃u ∈ Y or   u−1 B ≡ (A • (θ + 1)) . w

Proof. The case θ + 1 = δ is obvious since the empty word works for u. So in the sequel we assume θ + 1 < δ. The proof goes by induction on δ. Assume δ is limit. 1.If B ≡w A • δ, then clearly the set {d0W u−1 B : u ∈ Y ∗ and u−1 B < w B}

is unbounded in d0W (A • δ) = d0W (A) · δ. Hence there exists some ordinal ξ < δ and some v ∈ Y ∗ that both satisfy A • (θ + 1) ≤w v−1 B ≤w (A • ξ ) < w A • δ Then by induction hypothesis one gets some u ∈ Y ∗ such that  −1 −1   u v B ≡w A • (θ + 1) or   u−1 v−1 B ≡ (A • (θ + 1)) w

Hence u = vu works. 2.If B < w A • δ, then d0W (B) < d0W (A • δ) = d0W (A) · δ. Hence, for some ξ < δ we have A • (θ + 1) ≤w B ≤w (A • ξ )

128 | J. Duparc, O. Finkel, and J.-P. Ressayre which gives the result using the induction hypothesis on ξ . Assume δ is successor. – Assume δ = ζ + 2. 1. Assume (A • (ζ + 1)) ≤w B ≤w A • (ζ + 2). We consider the following combination of Wadge games with 3 players : I, II and III: * I is in charge of ((A • ζ ) + A ) – which is Wadge equivalent to (A • (ζ + 1)) , * II is in charge of B, and * III is in charge of A • (ζ + 2). II applies a w.s. that reduces I and III applies a w.s. that reduces II. This means that if I plays x1 , II plays x2 and III plays x3 then II reduces I if x1 ∈ (A • ζ ) + A ⇐⇒ x2 ∈ B; and III reduces II if x3 ∈ A • (ζ + 2) ⇐⇒ x2 ∈ B. Assume now that player I remaining in the right tail A (i.e. without going into (A • ζ ) or (A • ζ ) ) applies a winning strategy in the Wadge game W (A , A) against Player III as long as III stays in the tail part A of A • (ζ + 1) + A. Necessarily after a finite number of moves player III exits the right most A and chooses (A • (ζ + 1)) – for the other choice A • (ζ + 1) would be a losing one. We let v be the position of player II at that point, so that we obtain: (A • (ζ + 1)) ≤w v−1 B ≤w (A • (ζ + 1)) ,

hence

v−1 B ≡w (A • (ζ + 1)) .

If θ + 1 = ζ + 1 we are done. Otherwise we have (A • (θ + 1)) ≤w v−1 B ≤w A • (ζ + 1)

By induction hypothesis there exists u extending v such that u−1 B ≡w A • (θ + 1) or u−1 B ≡w (A • (θ + 1)) which gives u−1 B ≡w A • (θ + 1) or u−1 B ≡w (A • (θ + 1)) . 2. Assume B ≤w A • (ζ + 1). Since ζ + 1 < δ holds, the result relies on the induction hypothesis for A • (θ + 1) ≤w B ≤w A • (ζ + 1).

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Assume δ = ζ + 1, ζ limit: If B ≤w A • ζ holds the result follows from the induction hypothesis. Thus we assume that the following holds: ( A • ζ ) ≤ w B ≤ w ( A • ζ ) + A

Now consider the following combination of Wadge games with 3 players where: * I is in charge of (A • θ) + A + A + A (this is Wadge equivalent to (A • (θ + 3)) ), * II is in charge of B, and * III is in charge of A • ζ + A. II applies a w.s. that reduces I and III applies a w.s. that reduces II. Player I applies a winning strategy in the Wadge game W (A , A) against Player III as long as Player III remains in the tail part A of (A • ζ ) + A. Necessarily after a finite number of moves player III exits the first A and chooses (A • ζ ) or (A • ζ ). Now notice that since A is non-self dual, the set {w ∈ X ∗ | w−1 A ≡w A} is a tree – it is closed under prefixes – that contains an infinite branch. We let x be such an infinite branch. * If x ∈ A, then player I chooses to go into (A • θ + A + A) if III chooses (A • ζ ) and into (A • θ + A + A) if III chooses (A • ζ ). If x ∈/ A, then player I chooses to go into (A • θ + A + A) if III chooses (A • ζ ) and into (A • θ + A + A) if III chooses (A • ζ ). Then I plays along x, so that III is forced to choose A • γ for some γ < ζ (by definition A • ζ = supγ 0 holds, at least one of the l i ’s, and one of the m i ’s are different from zero.

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1. If α = β = 1, then the result is immediate. 2. We assume α ≥ β, and we let (α, β) be the < lex -least pair such that there exists some ordinal γ together with f : γ −→ {0, 1} that satisfy – α = ot(f −1 [0]), – β = ot(f −1 [1]), and – γ > H(α, β). We consider the order types of the following two sets of ordinals: (a) α = ot({θ < H(α, β) | f (θ) = 0}), and (b) β = ot({θ < H(α, β) | f (θ) = 1}),

together with f  the restriction of f to H(α, β). Necessarily either α < α or β < β holds. Therefore we have (max{α , β }, min{ α , β }) < lex (max{α, β}, min{ α, β}).

Hence we get the ordinal γ = H(α, β), together with the mapping f  : H(α, β) −→ {0, 1} such that α = ot({θ < γ | f  (θ) = 0}) and β = ot({θ < γ | f  (θ) = 1}). By Lemma 29, we obtain γ = H(α, β) > H(α , β ) which contradicts the induction hypothesis. Corollary 31. Let k, n be nonzero integers, γ be any ordinal, 0 ≤ α0 , . . . , α k < ω n , and f : γ −→ {0, . . . , k}. If ∀i ≤ k α i = ot(f −1 [i]) holds, then γ < ω n . Proof. This is immediate from Lemma 30. Lemma 32. Let k be some nonzero integer, (Nk , ) be a well-ordering such that  ∀i < k a i ≤ b i    (a0 , . . . , a k−1 )  (b0 , . . . , b k−1 ) =⇒ or    ∃i, j < k a i < b i and a j > b j

holds for every k-tuples (a0 , . . . , a k−1 ), (b0 , . . . , b k−1 ) ∈ Nk . Then, the order type of (Nk , ) is at most ω k . Proof. The proof goes by induction on k ≥ 1. 1. The initial case k = 1 is immediate since (N1 , ) is nothing but the usual ordering on integers.

132 | J. Duparc, O. Finkel, and J.-P. Ressayre 2. We assume the result holds for k ≥ 1, and we show that it holds for k + 1. Claim 33. For any integer n, the order type of the following set (ordered by ) A n = {(a0 , a1 , . . . , a k ) ∈ Nk+1 | (a0 , a1 , . . . , a k ) < (n, n, . . . , n)} is strictly below ω(k+1) . Proof. Notice that if (a0 , a1 , . . . , a k ) < (n, n, . . . , n) holds then a i < n must hold for some i ≤ k + 1. For each i ≤ k + 1 and each j < n we consider A(i,j) = {(a0 , a1 , . . . , a k ) ∈ Nk+1 | a i = j}, and α(i,j) = ot(A(i,j) ) its order type (ordered by ). Notice that for (a0 , . . . , a i−1 , j, a i+1 , . . . , a k )  (b0 , . . . , b i−1 , j, b i+1 , . . . , b k )

      

=⇒ ∀l ∈ {0, . . . , i − 1, i + 1 , . . . , k} a l ≤ b l

or

∃l, m ∈ {0, . . . , i − 1, i + 1 , . . . , k} a l < b l and a m > b m

Therefore by induction hypothesis, α(i,j) < ω k+1 holds for all i ≤ k + 1 and j < n. It follows from Corollary 31 that ot(A n ) < ω k+1 holds. On the other hand for every integer n ≥ 0 it holds that (n, n, . . . , n) < (n + 1 , n + 1 , . . . , n + 1). Moreover if n = max{a0 , a1 , . . . , a k } + 1 then (a0 , a1 , . . . , a k ) < (n, n, . . . , n ). Therefore, the sequence (ot(A n ))n≥1 is cofinal in ot(Nk+1 ) and thus the order type of (Nk+1 , ) is at most ω k+1 . Lemma 34. We let k be any nonzero integer, B ∈ BC(k), A ⊆ X ω be any initializable set, and δ any countable ordinal. B ≤w A • δ =⇒ B ≤w A • α for some α < ω k+1 . Notice that an immediate consequence is that B ≡w A • δ holds only for ordinals δ < ω k+1 . Proof. First notice that for every B ⊆ X ω , and every u ∈ X ∗ , if B ∈ BC(k) holds, then u−1 B ∈ BC(k) holds too. Towards a contradiction, we assume that A • α < w B ≤w A • δ holds for all α < ω k+1 . We let B be a k-partially blind counter automaton that recognizes B.

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By Lemma 26, for each successor ordinal α < ω k+1 there exists some u α ∈ X ∗ −1  such that u−1 α B ≡w A • α or u α B ≡w ( A • α ) . For each such u α , we form (q α , c α,0 , c α,1 , . . . , c α,k−1 ) where q α denotes the control state that B is in after having read u α , and c α,i the height of its counter number i (any i < k). Now there exists necessarily some control state q such that the order type of the set S = {α < ω k+1 | α successor and q α = q} is ω k+1 . Now, by Lemma 32 there exist α, α ∈ S such that α < α holds together with c α,i ≤ cα,i holds for all i < k. (Without loss of generality, we may even assume that ω ≤ α < α holds.) Let us denote Bα the k-partially blind counter automaton B that starts in state (q α , c α ,0 , c α ,1 , . . . , c α ,k−1 ), and Bα the one that starts in state (q α , c α,0 , c α,1 , . . . , c α,k−1 ). Notice that since c α,i ≤ cα,i holds for all i < k, Bα performs exactly the same as Bα except when the latter crashes for it tries to decrease a counter that is already empty. But it is then not difficult to see that −1 given the above assumption – that ω ≤ α < α holds – u−1 α B ≤w u α B holds    which leads to either A • α ≤w A • α or (A • α) ≤w A • α . In both cases, it contradicts α < α. Notice that the set ∅ • ω1n is initializable, so we have in particular the following result. Lemma 35. For k, n any integers, A any non-self dual ω-language in BC(k), and any nonzero countable ordinal α, A or A ≡w (∅ • ω1n ) • α =⇒ α < ω k+1 . In a similar way, we can now state the following lemma. Lemma 36. We let k be any nonzero integer, B ∈ BC(k), A ⊆ X ω be any initializable set, δ be any countable ordinal, and C be any set of the form C = A • ω1n • ν n + . . . + A • ω1n−1 • ν n−1 + . . . + A • ω1 • ν1 for any nonzero integer n, and countable multiplicative coefficients ν n , ν n−1 , . . . , ν1 with at least one of them being nonzero. B ≤w C + A • δ =⇒ B ≤w C + A • α for some α < ω k+1 . Proof. The proof is very similar to the one of Lemma 34, so we leave it to the reader.

134 | J. Duparc, O. Finkel, and J.-P. Ressayre Theorem 37. Let k be any nonzero integer, B ⊆ X ω be non-self dual. If B ∈ BC(k), then either B or B is Wadge equivalent to some n

n

Ω(α) = (∅ • ω1j ) • δ j + (∅ • ω1j−1 ) • δ j−1 + . . . + (∅ • ω1n0 ) • δ0 . where j ∈ N, n j > n j−1 > . . . > n0 and ω k+1 > δ j , δ j−1 , . . . , δ0 > 0. Proof. This is an almost immediate consequence of Lemmas 34 and 36. This settles the case of the non-self dual ω-languages in BC(k). For the self-dual ones, it is enough to notice the easy following: 1. Given any A ⊆ X ω , if A ∈ BC(k) is self dual, then there exists two non-self dual sets B, C ⊆ X ω such that both B and C belong to BC(k), B ≡w C , and A ≡w X0 B ∪ X1 C, where {X0 , X1 } is any partition of X in two non-empty sets.

2. If A ⊆ X ω and B ⊆ X ω are non-self dual, satisfy A ≡w B , and both belong to BC(k), then, given any partition of X in two non-empty sets {X0 , X1 }, X0 A ∪ X1 B is self-dual, and also belongs to BC(k). As a consequence, we obtain the following general result if we come back to the original definition of the Wadge degree of a set (denoted d◦ ) – from which we slightly departed from to define d W – namely: Definition 38. For A ⊆ X ω , we set d◦ (A) = sup{d◦ (B) + 1 | B < W A}. (Notice that this definition implies d◦ (∅) = d◦ (∅ ) = 0.) Theorem 39. For any A ⊆ X ω , there exists an ω-language B ⊆ X ω recognized by some deterministic Petri net, such that A ≡w B if and only if d◦ A is of the form α = ω1n · δ n + . . . + ω01 · δ0 .

for some n ∈ N, and ω ω > δ n , . . . , δ0 ≥ 0.

From where we immediately obtain the following: Corollary 40. The height of the Wadge hierarchy of ω-languages recognized by 2 deterministic Petri nets is (ω ω )ω = ω ω .

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7 Conclusions We provided a description of the extension of the Wagner hierarchy from automata to deterministic Petri Nets with Muller acceptance conditions. Of course the results would be rigorously the same if we replace Muller acceptance conditions with parity acceptance conditions. But with Büchi acceptance conditions instead, it becomes even simpler since the ω-languages are no more boolean combinations of Σ20 -sets, but Π20 -sets. So, the whole hierarchy comes down to the following: Corollary 41. For any A ⊆ X ω , there exists an ω-language B ⊆ X ω recognized by some deterministic Petri net with Büchi acceptance conditions, such that A ≡w B if and only if either – d◦ A = ω1 , and A is Π20 -complete, or –

d◦ A < ω ω .

Deciding the degree of a given ω-language in BC(k), for k ≥ 2, recognized by some deterministic Petri net – either with Büchi or Muller acceptance conditions, remains an open question. Notice that for k = 1 this decision problem has been shown to be decidable in [13]. Another rather interesting open direction of research is to go from deterministic to non-deterministic Petri nets. It is clear that this step forward brings new Wadge classes – for instance there exist ω-languages recognized by nondeterministic Petri nets with Büchi acceptance conditions that are Σ30 -complete, hence not ∆03 , [21] – but the description of this whole hierarchy still requires more investigations.

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Willem L. Fouché

Diophantine Properties of Brownian Motion: Recursive Aspects¹ Abstract: We use recent results on the Fourier analysis of the zero sets of Brownian motion to explore the diophantine properties of an algorithmically random Brownian motion ( also known as a complex oscillation). We discuss the construction and definability of perfect sets which are linearly independent over the rationals directly from Martin-Löf random reals. Finally we explore the recent work of Tsirelson on countable dense sets to study the diophantine properties of local minimisers of Brownian motion. Keywords: Brownian motion, Fourier dimension, algorithmic randomness, additive structure in fractals Mathematics Subject Classification 2010: 68Q30, 03D30, 60D99 || Willem L. Fouché: Department of Decision Sciences, University of South Africa, Pretoria, South Africa

1 Introduction A Brownian motion on the unit interval is algorithmically random if it meets all effective (Martin-Löf) statistical tests, now expressed in terms of the statistical events associated with Brownian motion on the unit interval. The class of functions corresponds exactly, in the language of Weihrauch [24; 25], Gács [11] and specialised by Hoyrup and Rojas [13], in the context of algorithmic randomness, to the Martin-Löf random elements of the computable measure space R = (C0 [0, 1], d, B, W ), where C0 [0, 1] is the set of the continuous functions on the unit interval that vanish at the origin, d is the metric induced by the uniform norm, B is the countable set of piecewise linear functions vanishing at the origin with slopes and points of non-differentiability all rational numbers and where W is the Wiener measure. We shall also refer to such a Brownian motion as a complex oscillation. This termi-

1 Dedicated to Victor Selivanov on the occasion of his 60th birthday.

140 | Willem L. Fouché nology was suggested to the author by the following Kolmolgorov theoretic interpretation of this notion [1; 2]: One can characterise a Brownian motion which is generic (in the sense just stated) as an effective and uniform limit of a sequence (x n ) of “finite random walks”, where, moreover, each x n can be encoded by a finite binary string s n of length n, such that the (prefix-free) Kolmogorov complexity, K (s n ), of s n satisfies, for some constant d > 0, the inequality K (s n ) > n − d for all values of n. (See Definition 3, introduced by Asarin and Prokovskiy [2], in Section 3 below.) We shall study the images of certain ultra-thin sets (perfect sets of Hausdorff dimension zero) under a complex oscillation. We have shown in [6] that these images are perfect sets whose elements are linearly independent over the field of rational numbers. In this paper we discuss the definability of these sets, within the recursion-theoretic hierarchy, by exploiting the recursive isomorphism constructed in [5] between the Kolmogorov-Chaitin random reals and the class of suitably encoded versions of complex oscillations. We shall also utilise Tsirelson’s theory of countable dense random sets [23] to study the diophantine properties of the local minimisers of Brownian motion. The local minimizers of a complex oscillation is studied in [7]. We shall utilise recent results by Mukeru and the author [8] on the rate of decay of the Fourier transform of the delta function of a continuous version of Brownian motion to identify some diophantine properties of the zero set of a complex oscillation. For more on the Fourier and consequent Diophantine properties of the sample paths of Brownian motion the reader is referred to the paper by Łaba and Pramanik [15]. We shall also show that some of these phenomena can be expressed within the hyperaritmetical hierarchy and pose the problem as to whether this is essentially so. The author is very grateful to the Department of Mathematics at the Corvinus University, Budapest, for hosting my frequent visits to the department and for teaching and sharing with me so much of the subtleties of measure theory and stochastic processes. This research is being partially supported by the National Research Foundation (NRF) of South Africa as well as by a Marie Curie International Research Staff Exchange Scheme Fellowship (COMPUTAL PIRSES-GA-2011-294962) within the 7th European Community Framework Programme.

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2 Preliminaries from Brownian motion and geometric measure theory A random variable X with mean µ and variance σ2 is normal if it has a density function of the form 2 2 1 √ e−(t−µ) /2σ . 2π σ If (Ω, F, P) is a probability space and X is a real-valued random variable on Ω, the measure µ on F given by F → P(X −1 (F )), F ∈ F, is called the distribution of X. A Brownian motion on the unit interval is a real-valued function (ω, t) → X ω (t) on Ω × [0, 1], where Ω is the underlying space of some probability space, such that X ω (0) = 0 a.s. and for t1 < . . . < t n in the unit interval, the random variables X ω (t1 ), X ω (t2 )− X ω (t1 ), · · · , X ω (t n )− X ω (t n−1 ) are statistically independent and normally distributed with means all 0 and variances t1 , t2 − t1 , · · · , t n − t n−1 , respectively. It is a fundamental fact that any Brownian motion has a “continuous version”(see, for example [10]). This means the following: Write Σ for the σ-algebra of Borel sets of C[0, 1] where the latter is topologised by the uniform norm topology. There is a unique probability measure W on Σ such that for 0 ≤ t1 < . . . < t n ≤ 1 and for a Borel subset B of Rn , we have P({ω ∈ Ω : (X ω (t1 ), · · · , X ω (t n )) ∈ B}) = W (A), where

A = {x ∈ C[0, 1] : (x(t1 ), · · · , x(t n )) ∈ B}).

The measure W is known as the Wiener measure. We shall usually write X (t) instead of X ω (t). For a compact subset A of Euclidean space Rd and real numbers α, ϵ with 0 ≤ α < d and ϵ > 0, consider all coverings of A by balls B n of diameter ≤ ϵ and the corresponding sums  | B n |α , n

where |B| denotes the diameter of B. All the metric notions here are to be understood in terms of the standard 2 norms on Euclidean space. The infimum of the sums over all coverings of A by balls of diameter ≤ ϵ is denoted by H αϵ (A). When

142 | Willem L. Fouché ϵ decreases to 0, the corresponding H αϵ (A) increases to a limit (which may be infinite). The limit is denoted by H α (A) and is called the Hausdorff measure of A in dimension α. If 0 < α < β ≤ d, then, for any covering (B n ) of A,   |B n |β ≤ sup |B n |β−α | B n |α , n

n

n

from which it follows that

H βϵ (A) ≤ ϵ β−α H αϵ (A). Hence if H α (A) < ∞, then H β (A) = 0. Equivalently, H β (A) > 0 =⇒ H α (A) = ∞. Therefore,

sup{α : H α (A) = ∞} = inf{β : H β (A) = 0}.

This common value is called the Hausdorff dimension of A and denoted by dimh A. If α is such that 0 < H α (A) < ∞, then α = dimh A. However, if α = dimh A, we cannot say anything about the value of H α (A). It is easy to check that A → H α (A) defines an outer measure which is invariant under translations and rotations, and homogeneous of degree α with respect to dilations. If A is a Borel subset of Euclidean space, the set of non-zero Radon measures with support contained in A is denoted by M + (A). For a given µ ∈ M + (A), the energy integral of µ with respect to the kernel |x|−α is given by   dµ(x)dµ(y) I α (µ) = . | x − y |α Rd Rd

We say that µ has finite energy with respect to |x|−α when I α (µ) < ∞. If A carries positive measures of finite energy with respect to |x|−α we say that A has positive capacity with respect to |x|−α and we write Capα (A) > 0. If A carries no positive measure of finite energy with respect to |x|−α , we say that A has capacity zero with respect to this kernel and we write Capα (A) = 0. It follows from the Fourier analysis of temperate distributions that  dξ ˆ (ξ ))|2 |ξ |α I α (µ) = C(α, d) |µ , (1) | ξ |d Rd

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when 0 < α < d, where C(α, d) is a positive constant and where, moreover,  ˆ (ξ ) = µ e isξ dµ(s), Rd

is the Fourier transform of the measure µ. (For a proof see Chapter 12 of[17].) We shall make frequent use of the following very fundamental fact. Proposition 1. For a compact subset A of Rd and 0 < α < β < d, H β (A) > 0 ⇒ Capα (A) > 0 ⇒ H α (A) > 0. Hence or, equivalently,

sup{α : Capα (A) > 0} = dimh A, sup{α : I α (µ) < ∞} = dimh A.

A proof of this proposition can be found in Chapter 8 of [17], for example. The Fourier dimension of a compact set E is the supremum of positive real numbers α < 1 such that for some non-zero Borel measure µ supported by E , it is the case that 1 ˆ (ξ )|2 ≤ , |µ | ξ |α

for |ξ | sufficiently large. The Fourier dimension of E is denoted by dimf (E). Clearly, by (1), dimf (E) ≤ dimh (E), (2)

for all compact sets E. The set is called a Salem set if dimf (E) = dimh (E). The following question posed by Beurling was addressed and solved in the affirmative by Salem [21] in 1950.

Given a number α ∈ (0, 1), does there exist a compact set E on the line whose Hausdorff dimension is α that carries a Borel measure µ whose Fourier transform  e iux dµ(x)

ˆ (u) = µ

R

is dominated by |u|

−α/2

as |u| → ∞?

It follows from (1) and (2) that given a compact subset E of [0, 1] with Hausdorff dimension α ∈ (0, 1), the number α/2 is critical for this question to have an affirmative answer. Of course, such a set E will be a Salem set.

144 | Willem L. Fouché Salem proved this result by constructing for every α in the unit interval, a random measure µ (over a convenient probability space) whose support E has Hausdorff dimension α and which satisfies the Beurling-requirement with probability one. It was recently shown [9] by the author, in collaboration with George Davie and Safari Mukeru, that such sets can also be constructed by looking at Cantor type ternary sets E with computable ratios ξ and then to consider the image of E under a complex oscillation. The following theorem illustrates the rich diophantine structure of sets E of non-zero Fourier dimension. Even though the proof method is well-known in geometric measure theory, we give a full proof, for we need sharper estimates than what we could find in the literature. Theorem 2. (Folklore) Suppose E is a compact subset of reals such that, for every ϵ > 0, there is some µ ∈ M + (E) and 0 < α < 1, such that, for some constant C = C(ϵ), it is the case that ˆ (ξ )|2 ≤ C|ξ |−α+ϵ , |µ as |ξ | → ∞. Then, if k is a natural number such that kα > 1, it will follow, upon writing E k = E + · · · + E (k times), that

R=



n 0 such that k(α − ϵ) > 1 + ϵ, we have , for |ξ | large, ˆ (ξ )2 |k ≤ C k |ξ |−kα+kϵ ≤ C k |ξ |−1−ϵ . |ˆν(ξ )|2 = |µ

It follows that the function ˆν is in L2 (R). Since ν is a non-zero measure, it follows from Parseval’s theorem that ν is absolutely continuous with respect to Lebesgue measure. In particular, supp ν has non-zero Lebesgue measure. Since supp ν ⊂ E k ,

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we conclude that E k has non-zero Lebesgue measure. It follows from Steinhaus’s theorem [22] that E k − E k has zero as an interior point. This concludes the first part of the theorem. The second part follows from the following beautiful remark [15]: If F is any set of positive Lebesgue measure, then F will contain an affine copy of any finite set A of real numbers. This is, as noted by Łaba and Pramanik [15], a consequence of Lebesgue’s density theorem.

3 Complex oscillations The set of non-negative integers is denoted by ω and we write B for the Cantor space {0, 1}ω . The set of words over the alphabet {0, 1} is denoted by {0, 1}∗ . If a ∈ {0, 1}∗ , we write |a| for the length of a. If α = α0 α1 . . . is in B, we write α(n)  for the word j 0 such that K (c(x n )) ≥ n − d for all n. A function x ∈ C[0, 1] is a complex oscillation if there is a complex sequence (x n ) such that x − x n  converges effectively to 0 as n → ∞.

146 | Willem L. Fouché The class of complex oscillations is denoted by C. In [5] the author constructed a bijection Φ : KC → C which is effective in the following sense: If α ∈ KC and m < ω, one can effectively construct from the first m bits of α, a function p m , where p m is a finite linear combination of piecewise linear functions with rational coefficients, such that, for some absolute positive constant C, the complex oscillation Φ(α) is approximated by the sequence (p m ) as follows: √ supt∈[0,1] |Φ(α)(t) − p m (t)| ≤ C log m/ m (4)

for all m > M α , where M α is a constant that depends on α only. Conversely, if x ∈ C, then one can compute, relative to an infinite binary string which encodes the values of a complex oscillation x at the rational numbers in the unit interval, the KC-string α such that Φ(α) = x. In [5] the author proved that these results have the following implication:

Theorem 4. There is a uniform algorithm that, relative to any KC-string α, with input a rational number t in the unit interval and a natural number n, will output the first n bits of the the value of the complex oscillation Φ(α) at the value t. This result plays a crucial rôle in this paper, for it will enable us to show how the sample path properties of a complex oscillation Φ(α) (and hence of a typical Brownian motion) can be described within the arithmetical hierarchy relative to the associated KC-string α. In this way, as was stated in the introduction of this paper, one finds an explicit unfolding of the incredibly rich geometry that is enfolded in every KC-string α by merely regarding such an α as an encoding of a complex oscillation or, equivalently, of an (effectively) generic Brownian motion. The mapping Φ is also a measure-theoretic isomorphism in the following (standard) sense: Write λ for the Lebesgue measure on the space {0, 1}ω and write W for the Wiener measure on C[0, 1]. Then, for any Borel subset A of C[0, 1] with the uniform norm topology, we have λ(Φ−1 (A)) = W (A). In other words, W is the pushout of λ under Φ. We shall frequently denote Φ(α) by x α . We follow [4] to define an analogue of a Π20 subset of C[0, 1] which is of constructive measure 0. If F is a subset of C[0, 1], we denote by F its topological closure in C[0, 1] with the uniform norm topology. For ϵ > 0, we let O ϵ (F ) be the ϵ-ball {f ∈ C[0, 1] : ∃g∈F f − g  < ϵ} of f . (Here . denotes the supremum norm.) We write F 0 for the complement of F and F 1 for F.

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Definition 5. A sequence F0 = (F i : i < ω) in Σ is an effective generating sequence if 1. for F ∈ F0 , for ϵ > 0 and δ ∈ {0, 1}, we have, for G = O ϵ (F δ ) or for G = F δ , that W (G) = W (G), 2. there is an effective procedure that yields, for each sequence 0 ≤ i1 < . . . < i n < ω and k < ω a binary rational number β k such that | W ( F i 1 ∩ . . . ∩ F i n ) − β k | < 2− k ,

3. for n, i < ω, a strictly positive rational number ϵ and for x ∈ C n , both the relations x ∈ O ϵ (F i ) and x ∈ O ϵ (F 0i ) are recursive in x, ϵ, i and n, relative to an effective representation of the rationals. If F0 = (F i : i < ω) is an effective generating sequence and F is the Boolean algebra generated by F0 , then there is an enumeration (T i : i < ω) of the elements of F (with possible repetition) in such a way, for a given i, one can effectively describe T i as a finite union of sets of the form F = F iδ11 ∩ . . . ∩ F iδnn

where 0 ≤ i1 < . . . < i n and δ i ∈ {0, 1} for each i ≤ n. We call any such sequence (T i : i < ω) a recursive enumeration of F. We say in this case that F is effectively generated by F0 and refer to F as an effectively generated algebra of sets. Let (T i : i < ω) be a recursive enumeration of the algebra F which is effectively generated by the sequence F0 = (F i : i < ω) in Σ. It is shown in [4] that there is an effective procedure that yields, for i, k < ω, a binary rational β k such that | W ( T i ) − β k | < 2− k ,

in other words, the function i → W (T i ) is computable. A sequence (A n ) of sets in F is said to be F-semirecursive if it is of the form (T ϕ(n) ) for some total recursive function ϕ : ω → ω and some effective enumeration (T i ) of F. (Note that the sequence (A cn ), where A cn is the complement of A n , is also an F-semirecursive sequence.) In this case, we call the union ∪n A n a Σ10 (F) set. A set is a Π10 (F)-set if it is the complement of a Σ10 (F)-set. It is of the form ∩n A n for some F-semirecursive sequence (A n ). A sequence (B n ) in F is a uniform sequence of Σ10 (F)- sets if, for some total recursive function ϕ : ω2 → ω and some effective enumeration (T i ) of F, each B n is of the form  Bn = T ϕ(n,m) . m

In this case, we call the intersection ∩n B n a Π20 (F)-set. If, moreover, the Wienermeasure of B n converges effectively to 0 as n → ∞, we say that the set given by ∩n B n is a Π20 (F)-set of constructive measure 0.

148 | Willem L. Fouché The proof of the following theorem appears in [4]. Theorem 6. Let F be an effectively generated algebra of sets. If x is a complex oscillation, then x is in the complement of every Π20 (F)-set of constructive measure 0. This means, that every complex oscillation is, in an obvious sense, F-Martin-Löf random. Definition 7. An effectively generated algebra of sets F is universal if the class C of complex oscillations is definable by a single Σ20 (F)-set, the complement of which is a set of constructive measure 0. In other words, F is universal iff a continuous function x on the unit interval is a complex oscillation iff x is F-Martin-Löf random. We introduce two classes of effectively generated algebras G and M which are very useful for reflecting properties of one-dimensional Brownian motion into complex oscillations. Let G0 be a family of sets in Σ each having a description of the form: a1 X (t1 ) + · · · + a n X (t n ) ≤ L

(5)

or of the form (5) with ≤ replaced by 1 and n sufficiently large it is the case 1

|x α (t n ) − z| ≤ C|t n − t| 2 log

Since |t n − t| ≤

1 2n2

1 . |t n − t|

we conlude that for all n sufficiently large |x α (t n ) − z|
m such that |x α (t n ) − z| < 21n for some t n ∈ D n . The sequence (t(m)) has some convergent sequence with a limit τ say. Clearly τ ∈ E, and by the continuity of x α , we can conclude that x α (τ) = z. This concludes the proof of the Proposition. Note that |x α (t) − z|
2 and for k ≥ 1 set s k = (1 + m k ).

154 | Willem L. Fouché Theorem 16. The sequence (s k ) is linearly independent over Q. Proof: For standard measure spaces (Ω1 , P1 ) and (Ω2 , P2 ), let there be some P i measurable strongly random variable X i : Ω i → CS(0, 1) such that the induced probability distributions on CS(0, 1) are the same. We say in this case that the strongly random sets X1 and X2 are statistically similar relative to the probabilities P1 , P2 and we write X1 ∼ X2 . This means exactly that P1 (X1−1 (Σ )) = P2 (X2−1 (Σ )), for all Borel subsets Σ of CS(0, 1). Write λ∞ for the product measure on (0, 1)∞ which is the countable product of the Lebesgue measure λ on the unit interval and write Λ for the measure on CS(0, 1) which is the pushout of λ∞ under π. In other words, for a Borel subset Σ of CS(0, 1), Λ(Σ ) = λ∞ (π −1 Σ). Write U : (0, 1)∞ → CS(0, 1) for the strongly random set as defined by the following commutative diagram: Id

(0, 1)∞ =

(0, 1)∞ = π

U

CS(0, 1) = (0, 1)∞ = /S∞ In [23] Tsirelson proved the truly remarkable result that MIN ∼ U.

(9)

The theorem with the uniform sequence (u k ) replacing the local minimizers (m k ) is known to be true. (See pages 256-260 in Meyer [18].) Hence the theorem follows from the statistical similarity of MIN and U and the fact that the notion of linear independence over Q of a sequence of reals is preserved under the action of S∞ on the space of real sequences. Remark. In statistics U is known as a model of an unordered uniform infinite sample. Moreover, it follows from the Hewitt-Savage theorem that, for every Borel subset Σ of CS(0, 1), it is the case that Λ(Σ ) ∈ {0, 1}.

(10)

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155

Note that Λ is non-atomic. As has been noted before [23], this shows that the wellknown fact that CS(0, 1) is not a standard Borel space, is a direct consequence of a frequently used fact in statistics. Open problem. In [7] the author showed how the local minimizers of a complex oscillation Φ(α) can be computed from a KC-string α. This opens the possibility of finding analogues of Theorem 16 for complex oscillations. Let us call a continuous function x on the unit interval strongly random if it belongs to every Σ20 (G) set of Wiener measure one, for some gaussian algebra G. The set of strongly random functions is a subclass of the complex oscillations. By using the constructions in [7], it can be shown that the s k associated with a strongly random function will be linearly independent over the rationals. Whether this result can be extended to complex oscillations, is an open problem.

Bibliography [1] [2]

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

Asarin, E.A.: Individual random signals: an approach based on complexity, doctoral dissertation, Moscow State University, 1988 Asarin, E. A. and Prokovskiy, A. V.: Use of the Kolmogorov complexity in analysing control system dynamics, Automation Remote Control 47 (1986) 21-28. Translated from: Primeenenie kolmogorovskoi slozhnosti k anlizu dinamiki upravlemykh sistem, Automatika i Telemekhanika (Automation Remote Control) 1 (1986) 25-33. Chaitin, G. A.: Algorithmic information theory, Cambridge University Press, 1987. Fouché, W. L.: Arithmetical representations of Brownian motion I, J. Symb. Logic 65 (2000), 421-442. Fouché, W. L.: The descriptive complexity of Brownian motion, Advances in Mathematics 155, (2000), 317-343 Fouché, W. L.: Fractals generated by algorithmically random Brownian motion, K. AmbosSpies, B. Löwe, and W. Merkle (Eds.): CiE 2009, LNCS 5635 (2009), 208-217. Fouché, W. L.: Kolmogorov complexity and the geometry of Brownian motion. Accepted by Mathematical Structures in Computer Science. Fouché, W. L. and Mukeru S.: On the Fourier structure of the zero set of fractional Brownian motion, Statistics and Probability Letters 83 (2013), 459-466. Fouché, W. L., Mukeru, S. and Davie, G.: Fourier spectra of measures associated with algorithmically random Brownian motion. Submitted. Freedman, D.: Brownian motion and diffusion, (second edition) Springer-Verlag, New York, 1983. Gács, P.: Uniform test of algorithmic randomness over a general space, Theoretical Computer Science 341 (2005), 91-137. Hinman, P. G.: Recursion-theoretic hierarchies, Springer-Verlag, New York, 1978. Hoyrup, M. , Rojas, C.: Computability of probability measures and Martin-Löf randomness over metric spaces, Information and Computation 207 (2009), 830-847. Kahane, J. -P.: Some random series of functions (second edition), Cambridge University Press, 1993.

156 | Willem L. Fouché [15] Łaba I., and Pramanik, M.: Arithmetical progressions in sets of fractional dimension, Geometric and Functional Analysis, 19 (2009), 429-456. [16] Martin-Löf, P.: The definition of random sequences, Information and Control 9 (1966), 602-619. [17] Mattila, P.: Geometry of sets and measures in Euclidean spaces, Cambridge University Press, 1995. [18] Meyer, Y.: Algebraic numbers and harmonic analysis, North Holland, Amsterdam, 1972. [19] Nies, A.: Computability and randomness, Oxford Logic Guides 51, Clarendon Press, Oxford, 2008. [20] Rudin, W.: Fourier Analysis on Groups, Interscience Publishers, New York - London, 1960. [21] Salem, R.: On singular monotonic functions whose spectrum has a given Hausdorff dimension, Ark Mat 1 (1950), 353-365. [22] Steinhaus, H.: Sur les distances des distances des ensemble de mesure positive, Fund Math 1 (1920), 93-104. [23] Tsirelson, S.: Brownian local minima, random dense countable sets and random equivalence classes, Electronic Journal of Probability 11 (2006), 162-198. [24] Weihrauch, K.: Computability on the probability measures on the Borel sets of the unit interval, Theoretical Computer Science 219 (1999), 421-437. [25] Weihrauch, K.: Computable Analysis, Springer, Berlin, 2000.

Sy-David Friedman

The Completeness of Isomorphism¹ Abstract: This paper provides a survey of results concerning the complexity of the isomorphism relation when restricted to classes ranging from the class of computable structures to the class of arbitrary countable structures. The aim is to determine in which cases this relation is Σ11 -complete. Keywords: countable structure, computable structure, isomorphism, hierarchy, completeness. Mathematics Subject Classification 2010: 03C57, 03C75, 03D45, 03D60, 03E15, 03E45 || Sy-David Friedman: Kurt Gödel Research Center, University of Vienna, Austria

In classical descriptive set theory, analytic equivalence relations (i.e., Σ11 equivalence relations with parameters) are compared under the relation of Borel reducibility (for example, see [5]). An important subclass of the Σ11 equivalence relations are the isomorphism relations, i.e., the restrictions of the isomorphism relation on countable structures (viewed as an equivalence relation on reals coding such structures) to the models of a sentence of the infinitary logic L ω1 ω . Scott’s Theorem implies that the equivalence classes of any isomorphism relation are Borel, and therefore no isomorphism relation can be complete (under Borel reducibility) within the class of Σ11 equivalence relations as a whole, some of which contain non-Borel equivalence classes. (This is clarified below.) The picture is different in the computable setting. It is shown in [2] that isomorphism on computable structures (viewed as an equivalence relation on natural numbers coding such structures), indeed on computable trees, is complete for Σ11 equivalence relations under the natural analogue of Borel-reducibility for equivalence relations on numbers: E0 is reducible to E1 iff for some computable f : N → N, E0 (m, n) iff E1 (f (m), f (n)) for all m, n. In this article we survey the situation for classes of structures between the class of computable structures and the class of arbitrary countable structures. Our

1 The author would like to congratulate Professor Victor Selivanov on the occasion of his 60th birthday for his broad and significant contributions to the field of mathematical logic. He also wishes to thank the FWF (Austrian Science Fund) for its generous support of this research through Project P 22430-N13.

158 | Sy-David Friedman aim is to determine in which cases isomorphism is complete and in which cases it is not. Work has also been done by considering not arbitrary isomorphisms, but isomorphisms of a restricted type (such as computable or hyperarithmetic isomorphism). For this I refer the reader to [3].

1 Classes of structures To discuss classes of structures intermediate between the class of computable structures and the class of arbitrary countable structures we make use of the Lhierarchy. We fix a computable first-order language and consider structures for that language with universe ω. Assume V = L; thus every structure is definable over L α for some infinite countable ordinal α. For pairs (α, n) where α is an infinite countable ordinal, 0 < n ∈ ω, define: X (α, n) = all reals (subsets of ω) which are ∆ n definable over L α Also when α is a countable ordinal greater than ω we define: X (α, 0) = all reals (subsets of ω) which are elements of L α Now fix α, n as above and let E be an equivalence relation on reals which is Σ11 with parameter from X (α, n). We say that E is complete on X (α, n) iff whenever F is another such equivalence relation there exists a function f from reals to reals sending X (α, n) into X (α, n) such that for x, y ∈ X (α, n): F (x, y) iff E(f (x), f (y)), where f is Hyp (i.e. ∆11 ) in a parameter from X (α, n). Note that isomorphism (viewed as an equivalence relation on reals coding countable structures) is a parameter-free Σ11 equivalence relation. Main Question 1. For which α, n is isomorphism complete on X (α, n)?

2 When isomorphism is complete The basic positive result from [2] reads as follows. Theorem 2. ([2]) Isomorphism is complete on X (ω, 1), the set of computable reals. Roughly speaking, the proof goes as follows. Suppose that E(m, n) is a Σ11 equivalence relation on computable reals with a computable parameter; we can trans-

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| 159

late E into a Σ11 equivalence relation E on natural numbers without parameter. By Kleene’s Representation Theorem choose a computable sequence (T (m, n) | m, n ∈ ω) of computable trees such that E (m, n) iff T (m, n) is ill-founded. Using “rank-saturated” trees (see [1]) we can assume that the isomorphism type of T (m, n) depends only on the rank of T (m, n) (which is ∞ if T (m, n) is ill-founded). The main trick is to ensure that this rank depends only on the E -equivalence classes of m, n. Then by defining T ∗ (m) to be the “join” of the T (m, n), n ∈ ω, we obtain: E (m0 , m1 ) iff T ∗ (m0 ) is isomorphic to T ∗ (m1 ). For the details see [2]. Now using a Hyp function which takes a computable real to a Turing-index for it, we obtain the desired Hyp reduction of E to isomorphism on computable structures. Now Theorem 2 clearly relativises to a real parameter. Say that isomorphism is complete on the p-computable reals (where p is a real parameter) iff whenever E is a Σ11 equivalence relation with a p-computable parameter there is a Hyp function f with p-computable parameter sending p-computable reals to p-computable structures such that for p-computable x, y: E(x, y) iff f (x), f (y) are isomorphic. Corollary 3. For any parameter p, isomorphism is complete on the set of p-computable reals. This reduces the Main Question 1 to the cases where n = 0, using the following fine-structural fact (see [6] or [4]). Theorem 4. For any α, n, X (α, n) either equals X (α, 0) or equals the set of pcomputable reals for some real p. The reason for this is the following: If X (α, n) does not equal X (α, 0) then there is a real which is ∆ n over L α but does not belong to L α ; in fact there is a “canonical” such real called the “∆ n master code” for L α , which serves as the parameter p in the conclusion of the theorem. We can reduce our Main Question 1 even further. For example, consider X (ω + 1, 0), the set of arithmetical reals. There is a Hyp function which takes an arithmetical real to an arithmetical code for it and this reduces the completeness of isomorphism on X (ω + 1, 0) to its completeness on X (ω, 1), the content of Theorem 2. More generally, suppose that X (α, 0) is distinct from X (β, 0) for each β < α (an assumption we can make without loss of generality) and that for some real p in L α , α is less than the least p-admissible ordinal ω1p ; then there is a Hyp in p function which send the reals of X (α, 0) injectively into ω, thereby reducing the completeness of isomorphism on X (α, 0) to its completness on the p-computable

160 | Sy-David Friedman reals, Corollary 3. Thus we have the completeness of isomorphism on X (α, n) in all cases except when n = 0 and one of the following holds: 1. α is admissible but not the limit of admissibles. 2. α is a limit of admissibles. We now show that isomorphism is not complete on X (α, 0) in the second of these cases.

3 When isomorphism is not complete First we need to clarify why isomorphism on arbitrary countable structures is not complete for Σ11 equivalence relations on arbitrary reals. Proposition 5. There is a Σ11 equivalence relation E on reals with an equivalence class which is not Borel (i.e., not Hyp with a real parameter). Proof. Let X be a Σ11 set of reals which is not Borel. Define E by: E(x, y) iff x, y ∈ X or x = y. Then X is an equivalence class of E. Theorem 6. (Scott, see [5]) For any countable structure A, the set of (codes for) countable structures which are isomorphic to A is Borel. Proof. Let φ be the Scott sentence of A, i.e., the canonical sentence of L ω1 ω whose countable models are exactly those isomorphic to A. This set of models is Borel, as the set of countable models of any sentence of L ω1 ω is Borel. Corollary 7. Isomorphism on countable structures is not complete for Σ11 equivalence relations (under Borel, i.e. Hyp in a real parameter, reducibility). Proof. A Borel reduction from a Σ11 equivalence relation E to another such equivalence relation F takes non-Borel equivalence classes to non-Borel equivalence classes. Now suppose that we replace the set of all reals by some subset X (α, 0) of the reals; what do we need to know about α for the above argument to still work? Proposition 8. Suppose that α is a limit of admissibles. Let A be a countable structure with code in L α . Then the set of codes for countable structures isomorphic to A is Hyp with parameter in L α .

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Proof. The canonical Scott sentence φ for A belongs to the second admissible set containing x whenever x is a real coding A. As α is a limit of admissibles, φ is coded by a real in L α . It follows that the set of countable structures isomorphic to A, i.e., the set of countable models of φ, is Hyp with parameter in L α . Corollary 9. Isomorphism is not complete on X (α, 0) when α is a limit of admissibles. Proof. Let X be the set of reals which code linear orders having infinite descending chains. Then X is Σ11 and not Borel. Now X ∩ L α is the set of reals in L α which code linear orders which have infinite descending chains in L α , using the fact that α is a limit of admissibles. Thus X ∩ L α is Σ11 but not ∆11 in L α . And if B is a Hyp set of reals with parameter in L α then B ∩ L α is ∆11 in L α , so it follows that X and B disagree on the reals of L α . Now as before consider the equivalence relation E(x, y) iff x ∈ X or x = y; this equivalence relation is not reducible to isomorphism on X (α, 0) as its restriction to L α has an equivalence class which is not ∆11 in L α but the intersection with L α of the equivalence classes of isomorphism are each ∆11 in Lα .

Successor admissibles? We are left with cases of X (α, 0) when α is a successor admissible, i.e. an admissible ordinal which is not the limit of admissibles. Note the following. Proposition 10. Suppose that α is a successor admissible. Then either X (α, 0) equals X (β, 0) where β is a limit of admissibles or the reals of X (α, 0) are exactly those which are hyperarithmetic in p for some fixed real p. Proof. If L α thinks that ℵ1 exists then X (α, 0) equals L(β, 0) where β is the ℵ1 of L α . Otherwise we may choose a real p in L α which codes the supremum of the admissibles less than α and then the reals of L α are exactly those which are hyperarithmetic in p. Thus the only remaining cases are relativisations to a real parameter of the following. Open Question 11. Is isomorphism complete on X (ω1ck , 0), the set of hyperarithmetic reals?

162 | Sy-David Friedman Recall that this asks the following: Suppose that E is a Σ11 equivalence relation on reals. Is there a Hyp function f which takes reals to countable structures such that for hyperarithmetic x, y, E(x, y) iff f (x), f (y) are isomorphic? The proof methods of Theorem 2 and Corollary 9 do not appear to cover this case.

4 A variant There is a strengthening of Corollary 9 for the case of X (α, 0) when α is a limit of limits of admissibles. Let E1 be the equivalence relation E1 (x, y) iff for sufficiently large n, (x)n = (y)n , where (x)n is the n-th “column” of x via some computable pairing funcion ·, · on the natural numbers: (x)n (m) = x(m, n) for all m. Then E1 is a Hyp equivalence relation. Theorem 12. Let α be a limit of limits of admissibles. Then E1 is not reducible to isomorphism on structures with codes in X (α, 0), the set of reals in L α , via a Hyp function with parameter from X (α, 0). Proof. Suppose that there were such a reduction f with parameter p in L α and choose a limit of admissibles α0 < α so that p belongs to L α0 . Let M denote L α , M0 denote L α0 and let (z n | n ∈ ω) ∈ M be generic for the ω-product of Sacks forcing over M0 . Define x n so that (x n )k is the 0-real for k < n and is z k otherwise. The x n ’s are pairwise E1 -equivalent so the f (x n )’s are pairwise isomorphic. As α0 is a limit of admissibles, they are in fact pairwise isomorphic in M0 [x0 ]. Choose a permutation π of ω in M which is Cohen-generic over M0 [x0 ]. Let z be π · f (x0 ), the code for the structure obtained from f (x0 ) by applying π. Then the structure coded by z is isomorphic to the structures coded by the f (x n )’s. Also z is Cohen over each M0 [x n ]: Write z = π · f (x0 ) = π · π −1 n f ( x n ) where π n is an isomorphism between f (x0 ) and f (x n ) in M0 [x0 ] and note that π · π −1 is n Cohen over M0 [x n ]. Now choose a real y in M0 [z] so that f (y) is isomorphic to the structure coded by z; this is possible as α0 is a limit of z-admissibles. Then y is E1 equivalent to x0 and therefore some z n is a component of y. But then M0 [x n+1 ][y], a Cohen-generic extension of M0 [x n+1 ], contains a real which is Sacks-generic over M0 [x n+1 ], a contradiction.

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W. Calvert, J.Knight and J.Millar, Computable trees of Scott rank ω1ck and computable approximations, J. Symbolic Logic, 71 (2006), no.1, 283–298. E.Fokina, S.Friedman, V.Harizanov, J.Knight, C.McCoy and A.Montalban, Isomorphism relations on computable structures, Journal of Symbolic Logic, vol.77, no.1, pp. 122–132, March 2012. E.Fokina, S.Friedman and A.Nies, Equivalence Relations that are Σ3 Complete for Computable Reducibility - (extended abstract), Lecture Notes in Computer Science 7456, pp.26–33, 2012. S.Friedman, Negative Solutions to Post’s Problem II, Annals of Mathematics, Vol. 113, 1981, pp. 25-43. S.Gao, Invariant descriptive set theory, Pure and Applied Mathematics 293. Taylor and Francis Group, 2009. C.Jockusch and S.Simpson, A degree-theoretic definition of the ramified analytical hierarchy, Annals of Math. Logic 10 (1975), pp. 1-32.

Peter Hertling, Victor Selivanov

Complexity Issues for Preorders on Finite Labeled Forests¹ Abstract: We prove that three preorders on the finite k-labeled forests are polynomial time computable. Together with an earlier result of the first author, this implies polynomial-time computability for an important initial segment of the corresponding degrees of discontinuity of k-partitions on the Baire space. Furtermore, we show that on ω-labeled forests the first of these three preorders is polynomial time computable as well while the other two preorders are NP-complete. Keywords: Topological complexity, Weihrauch reducibility, labeled forest, polynomial-time computability, NP-completeness. Mathematics Subject Classification 2010: 03D15, 03D30, 06A06 || Peter Hertling: Institut für Theoretische Informatik, Mathematik und Operations Research Universität der Bundeswehr München, Germany Victor Selivanov: A.P. Ershov Institute of Informatics Systems, Siberian Division of the Russian Academy of Sciences and Novosibirsk State Pedagogical University, Russia

1 Introduction As is well-known, reducibilities serve as useful tools for understanding the complexity (or non-computability) of decision problems on discrete structures. In computable analysis, many problems of interest turn out to be non-computable, even discontinuous. Thus, there is a need for tools to measure their non-computability or discontinuity. Accordingly, also in this context of decision problems on continuous structures people employed some reducibility notions. Weihrauch [24; 25] (see also the thesis [23] supervised by Weihrauch) introduced a topological reducibility relation ≤2 for functions f between products of the Cantor space and a discrete space. Independently, Hirsch [11] gave a similar definition for functions between arbitrary topological spaces. Weihrauch [24; 25] also introduced a generalisation of this reducibility relation to sets of functions.

1 The results contained in this paper have been presented at the conference CiE 2011. A short version [10] of this paper has appeared in the proceedings volume of CiE 2011. Both authors were supported by DFG-RFBR (Grant 436 RUS 113/1002/01, 09-01-91334).

166 | Peter Hertling and Victor Selivanov A computability-theoretic version of this generalisation, transferred via representations to computational problems on arbitrary represented spaces, is now called Weihrauch reducibility; see [4]. These notions have turned out to be very useful for understanding the noncomputability and discontinuity of important computation problems in computable analysis [9; 3] and constructive mathematics [25; 4]. The first author has also considered slightly weaker topological reducibility relations ≤0 and ≤1 for functions between arbitrary topological spaces in [7; 8; 9]. The notions are nontrivial even for the case of discrete spaces Y = k = {0, . . . , k − 1} with k points, 1 < k < ω. We call functions f : X → k k-partitions of X because they are in a natural bijective correspondence with the partitions (A0 , . . . , A k−1 ) of X where A i = f −1 (i)). For k = 2 the relation ≤0 coincides with the classical Wadge reducibility [12]. In [7] (without proofs) and [8; 9] (with proofs) the first author gave a “combinatorial” characterization of the important initial segment of the degree structures under these reducibilities of k-partitions of the Baire space B = ω ω formed by the k-partitions of B the components of which are finite Boolean combinations of open sets. Namely, he introduced preorders ≤0 , ≤1 , ≤2 on the set Fk of finite k-labeled forests (precise definitions are given in the next section) such that, for each i ≤ 2, the structure of the topological ≤i -degrees of the specified initial segment is isomorphic to the quotient-poset of (Fk \ {∅}; ≤i ). In fact, he showed that with any k-partition f in the initial segment just described one can associate a non-empty finite k-labeled forest B(f ) such that, for any k-partitions f and g on the Baire space one has f ≤i g if, and only if, B(f ) ≤i B(g ). Furthermore, the mapping f → B(f ) to the set of finite non-empty k-forests is onto. This result provides a natural naming system for the specified initial segment of topological ≤i -degrees of k-partitions of the Baire space. The second author has extended this result for ≤0 to a much larger initial segment of k-partitions [22]. Note that the Baire space is important because it is commonly used in computable analysis [26] to represent many other spaces of interest. The structure (Fk ; ≤0 ) and its extension to the structure (Pk ; ≤0 ) of finite klabeled posets are also important due to their close relationship to the Boolean hierarchy of k-partitions that extends the classical Boolean (or difference) hierarchy of sets [14; 13; 21] and to some other fields of discrete mathematics like clones of functions on k [20; 19]. The mentioned results motivated the study of (Fk ; ≤0 ) and (Pk ; ≤0 ) in a series of publications. In [15] it was shown that for any k ≥ 3 the first-order theory of the quotient-poset of (Fk ; ≤0 ) is undecidable. It is even computably isomorphic to first-order arithmetic [16]. In [18] the same was shown for ≤1 and ≤2 . In [17] a complete definability theory for the quotient-poset of (Fk ; ≤0 ) was developed.

Complexity Issues for Preorders on Finite Labeled Forests | 167

According to [20], the quotient-poset of (Pk ; ≤0 ) is universal for each k ≥ 2, i.e., it contains any countable poset as a substructure. The naming systems (Fk ; ≤i ) and (Pk ; ≤i ) are obviously computable, and the natural next step in understanding their computational properties is to look at their complexity. In [19] it is shown that the relation ≤0 on Pk is NP-complete for each k ≥ 2. In this paper we answer some natural complexity questions about the relations ≤0 , ≤1 , ≤2 on Fk , and also on the set Fω of finite ω-labeled posets. Note that in [9] it is shown that the structure of ≤2 -degrees of functions f : B → ω such that all f −1 (k) are finite Boolean combinations of open sets is isomorphic to the quotient-poset of (Fω ; ≤2 ). Our main results are summarized in the following three theorems. Theorem 1. The relation ≤0 is computable in cubic time on Fk , for any k < ω, and on Fω . Theorem 2. The relation ≤1 is computable in cubic time on Fk , for any k < ω, and NP-complete on Fω . Theorem 3. The relation ≤2 is computable in time O(n k+3 ) on Fk , for any k < ω, and NP-complete on Fω . Furthermore, we show that minimization with respect to ≤0 is polynomial-time computable. In Section 2 we recall some relevant definitions and facts. In Sections 3, 5 and 6 we prove the results for ≤0 , ≤1 and ≤2 , respectively. In Section 4 we consider the minimization problem with respect to ≤0 . In Section 7 we mention some related open questions. The results contained in this paper have been presented at the conference CiE 2011. A short version [10] of this paper has appeared in the proceedings volume of CiE 2011. The statements concerning NP-completeness in Theorems 2 and 3 have been proved already in [10]. For completeness sake, we include the proofs here as well. For the other statements either only proof sketches or no proofs at all have been given in [10]. Here we give complete proofs of all assertions.

2 Preliminaries We use some standard notation and terminology on posets which may be found e.g. in [5]. Throughout this paper, k denotes an arbitrary integer, k ≥ 2, which is identified with the set {0, . . . , k − 1}. A poset is a partially ordered set (P, ≤),

168 | Peter Hertling and Victor Selivanov that is, a set P with a binary relation ≤ on it that is reflexive, transitive, and antisymmmetric. As usual, by x < y we mean (x ≤ y ∧ x = y). For an element x ∈ P the upper cone of x is the set ↑ x := {y ∈ P | x ≤ y}, and the lower cone of x is the set ↓ x := {y ∈ P | y ≤ x}. A chain is a poset (P, ≤) such that additionally x ≤ y or y ≤ x, for all x, y ∈ P. By a forest we mean a finite poset in which every upper cone ↑ x (endowed with the order obtained by restricting ≤ to ↑ x) is a chain. A tree is a forest that has a largest element, that is, an element x satisfying y ≤ x for all y ∈ P. When such an element exists then it is uniquely determined, and it is called the root of the tree. The cardinality of a forest F is denoted |F |. A k-labeled forest (or just a k-forest) is a triple (P; ≤, c) consisting of a forest (P; ≤) and a labeling c : P → k. We call a k-forest (P; ≤, c) repetition-free iff c(x) = c(y) whenever y is an immediate successor of x in P, i.e., whenever x < y and there does not exist a z ∈ P with x < z and z < y. We often simplify the notation of a k-forest to P. By default, we denote the labeling in a given k-forest by c. We are mainly interested in the set Fk of finite k-labeled forests, but also in the set Tk of finite k-labeled trees and in the set Ck of finite k-labeled chains. The finite klabeled chains are identified with the finite non-empty words over the alphabet k. Any such word u is a sequence u(0) · · · u(n − 1) of letters where n = |u| is the length of u. We will view this as a k-labeled chain of length n with u(0) being the label of the root of the chain. For any k-forests F, G ∈ Fk and i < k, let p i (F ) denote a k-tree obtained from F by adding a new greatest element with the label i. Let F  G be the disjoint union of F, G. Note that any tree is of the form p i (F ), and that any proper forest, i.e., a forest which is neither empty nor a tree, is of the form F  G where F and G are nonempty forests. The singleton k-tree with label i is identified with i. Lemma 4. Let k ≥ 2, and let (P; ≤P , c P ) and (Q; ≤Q , c Q ) be k-forests. For a monotone function φ : (P; ≤P , c P ) → (Q; ≤Q , c Q ) the following two conditions are equivalent. 1. There exists a function f : k → k with c P = f ◦ c Q ◦ φ. 2. For all x, y ∈ P, c P (x) = c P (y) → c Q (φ(x)) = c Q (φ(y)). The same holds true for ω-forests. Proof. We prove the assertion for k-forests. The proof of the assertion for ω-forests is essentially identical. Let us fix some k ≥ 2. The direction “1 ⇒ 2” is clear. For the direction “2 ⇒ 1”, let us assume that for all x, y ∈ P, c P (x) = c P (y) → c Q (φ(x)) = c Q (φ(y)). We define a function f : k → k as follows. If for some i ∈ k there is some x ∈ P with i = c Q (φ(x)), then we set f (i) := c P (x). Note that if there are different x, y ∈ P with i = c Q (φ(x)) and i = c Q (φ(y)) then by assumption c P (x) = c P (y). For all

Complexity Issues for Preorders on Finite Labeled Forests | 169

other i ∈ k we choose f (i) arbitrarily. The function f is well defined. It satisfies c P = f ◦ c Q ◦ φ. Definition 5. 1. A 0-morphism (resp., 1-morphism, resp., 2-morphism) φ : (P; ≤P , c P ) → (Q; ≤Q , c Q ) between k-forests is a monotone function φ : (P; ≤P ) → (Q; ≤Q ) that satisfies c P = c Q ◦ φ (resp., one and then both of the conditions in Lemma 4, resp., ∀x, y ∈ P((x ≤P y ∧ c P (x) = c P (y)) → c Q (φ(x)) = c Q (φ(y)))). 2. We write P ≤0 Q (resp. P ≤1 Q, P ≤2 Q) to denote that there exists a 0morphism (resp. 1-morphism, 2-morphism) φ : P → Q. Using Lemma 4 it is easy to see that any 0-morphism is a 1-morphism and any 1-morphism is a 2-morphism. Therefore, ≤0 implies ≤1 and ≤1 implies ≤2 . The relations ≤0 , ≤1 , ≤2 are reflexive and transitive on Fk , i.e., they are preorders. In the same way we define the preorders ≤0 , ≤1 , ≤2 on the set Fω of finite ω-labeled

forests (in which any natural number may be used as a label). Let ω∗ be the set of finite sequences (strings) of natural numbers. The empty string is denoted by ∅, the concatenation of strings σ, τ by στ, the length of σ by |σ|. By ω+ we denote the set of finite non-empty strings in ω. By σ  τ we denote that the string σ is an initial segment of the string τ. For any n, 1 < n < ω, let n∗ be the set of finite strings of elements of {0, . . . , n − 1}, n∗ ⊆ ω∗ . E.g., 2∗ is the set of finite strings of 0’s and 1’s. In computer science people often consider the sets A∗ and A+ of finite (respectively, finite non-empty) words over a finite alphabet A. Mathematically, these sets are of course the same as n∗ and n+ respectively, where n is the cardinality of A. Since we plan to deal with the complexity of the introduced preorders, we need to represent them by finite words over a finite alphabet. For k-labeled forests we use the alphabet k ∪{(, )}. A k-labeled forest consists of finitely many k-labeled trees, and we will represent it as a concatenation of these trees, in an arbitrary order. A k-labeled tree T consists of a root with the label c(T ) and a forest F consisting of the trees appended to the root. It will be represented by the string (c(T )F ). Example 6. For example, the 4-labeled forest ({1, 2, 3, 4, 5}; ≤, c), with the partial order relation ≤ given by a ≤ b : ⇐⇒ (a, b) ∈ {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (2, 1), (3, 1), (4, 1), (4, 3)}

and with the labeling c defined by c(i) := i mod 4 can be represented by the string (1(2)(3(0)))(1). We describe this forest: it consists of two trees. The node 1, labeled with 1, is the root of the first tree, the nodes 2, labeled with 2, and 3, labeled with

170 | Peter Hertling and Victor Selivanov 3, are its children, the node 4, labeled with 0, is the child of node 3, and the node 5, labeled with 1, forms the second tree, a singleton tree.

Note that any string representing a k-labeled forest with n elements has length exactly 3n, for any k < ω. In order to represent ω-labeled forests, we proceed in the same way, but represent the labels in binary form. Thus, we use the alphabet {(, ), 0, 1}. Since we wish to compare two forests, and the Turing machine expects the input on one input tape, we will give representations of the two forests to the Turing machine, separated by some special symbol, e.g., #.

3 The Complexity of ≤0 The following theorem is our first main result. It implies Theorem 1. Theorem 7. For any k < ω, there is a Turing machine which, given two k-labeled forests F and G, checks in time O(m + n + m2 n + mn2 ) (where m := |F | and n := |G|) whether F ≤0 G or not. The same is true for ω-labeled forests (where now m and n are the lengths of representations of F and G, respectively). Thus, the relation ≤0 is computable in cubic time on Fk , for any k < ω, and on Fω . Proof. First we consider the case of k-labeled forests, for some k < ω. We will show the following statement: There is an algorithm (a Turing machine) which, given two forests F and G with m := |F | and n := |G|, decides in time O(m + n + m2 n + mn2 ) whether F ≤0 G or not. The Turing machine is based on the following sequence of statements for two forests F and G, which is similar to an observation in [21]: 1. 2. 3. 4. 5. 6.

If |F | = 0 then F ≤0 G is true. If |F | ≥ 1 and |G| = 0 then F ≤0 G is false. If F = p i (F0 ) and G = p i (G0 ) then F ≤0 G iff F0 ≤0 G. If F = p i (F0 ), G = p j (G0 ), and i = j then F ≤0 G iff F ≤0 G0 . If F = p i (F0 ) and G = G0  G1 then F ≤0 G iff F ≤0 G0 ∨ F ≤0 G1 . If F = F0  F1 then F ≤0 G iff F0 ≤0 G ∧ F1 ≤0 G.

In the following we sketch the Turing machine. Furthermore, for each step we say in which time it can be executed. Let t(m, n) denote the maximum time needed by this algorithm, given a forest F with |F | = m and a forest G with |G| = n. When started with F #G on the input tape, the machine copies F and G to two work tapes.

Complexity Issues for Preorders on Finite Labeled Forests | 171

This can be done in time O(m + n). Then the recursive algorithm is started. Whenever it is called with some subforests F  and G , these subforests can be remembered on two stacks realized on some work tapes. The machine follows the case distinction above. 1. First, the machine checks whether the forest F is empty. This can be done in constant time. If this is the case then the machine says YES and stops. Otherwise it continues with Step 2. 2. The machine checks whether G is empty. This can be done in constant time. If this is the case then the machine says NO and stops. Otherwise it continues with Step 3. 3. The machine checks whether F consists of a single tree. This can be done in time O(m). If this is not the case then it continues with Step 6. If F consists of a single tree then the machine checks whether G consists of a single tree as well. This can be done in time O(n). If this is not the case then it continues with Step 5. If G consists of a single tree as well then we have F = p i (F0 ) and G = p j (G0 ) for some labels i and j and for some forests F0 and G0 . Now the machine checks whether i = j. This can be done in constant time. If this is not the case then the machine continues with Step 4. If i = j then the machine checks by a recursive call whether F0 ≤0 G. This can be done in time t(m − 1, n). If this is the case then the machine says YES and stops. Otherwise it says NO and stops. 4. In this case we have F = p i (F0 ) and G = p j (G0 ) for some labels i and j with i = j and for some forests F0 and G0 . The machine checks by a recursive call whether F ≤0 G0 . This can be done in time t(m, n − 1). If this is the case then the machine says YES and stops. Otherwise it says NO and stops. 5. In this case we have F = p i (F0 ), and G is neither empty nor a tree. The machine sets G0 := first tree in G, and G1 := the forest consisting of the remaining trees in G. All of this can be done in time O(n). Then it checks by two recursive calls whether F ≤0 G0 or F ≤0 G1 . This can be done in time O(m + n) + max{t(m, n1 ) + t(m, n2 ) | n1 ≥ 1, n2 ≥ 1, n1 + n2 = n}. If this is the case then the machine says YES and stops. Otherwise it says NO and stops.

172 | Peter Hertling and Victor Selivanov 6. In this case F is neither empty nor a tree. The machine sets F0 := first tree in F, and F1 := the forest consisting of the remaining trees in F. All of this can be done in time O(m). Then it checks by two recursive calls whether F0 ≤0 G and F1 ≤0 G. This can be done in time O(m + n) + max{t(m1 , n) + t(m2 , n) | m1 ≥ 1, m2 ≥ 1, m1 + m2 = m}. If this is the case then the machine says YES and stops. Otherwise it says NO and stops. The description and the analysis of the algorithm sketched above show that there exists a constant c > 0 such that for all m, n ≥ 0 t(0, n) ≤ c · (1 + n)

and

t(m, 0) ≤ c · (1 + m),

and for all m, n ≥ 1 t(m, n) ≤ c · (m + n)

+ max{ t(m − 1, n), t(m, n − 1), max{t(m, n1 ) + t(m, n2 ) | n1 ≥ 1, n2 ≥ 1, n1 + n2 = n}, max{t(m1 , n) + t(m2 , n) | m1 ≥ 1, m2 ≥ 1, m1 + m2 = m}}.

Using these recursive inequalities, we now show by induction that for m, n ≥ 1 t(m, n) ≤ 2c · (m2 n + mn2 ).

(1)

This proves the claim t(m, n) ∈ O(m + n + m2 n + mn2 ). In order to prove (1) it is sufficient to prove the following four statements for m, n ≥ 1: I. c · (m + n) + t(m − 1, n) ≤ 2c · (m2 n + mn2 ). II.

c · (m + n) + t(m, n − 1) ≤ 2c · (m2 n + mn2 ).

III. For any n1 , n2 with n1 ≥ 1, n2 ≥ 1 and n1 + n2 = n

c · (m + n) + t(m, n1 ) + t(m, n2 ) ≤ 2c · (m2 n + mn2 ). IV. For any m1 , m2 with m1 ≥ 1, m2 ≥ 1 and m1 + m2 = m c · (m + n) + t(m1 , n) + t(m2 , n) ≤ 2c · (m2 n + mn2 ).

Complexity Issues for Preorders on Finite Labeled Forests | 173

Now we prove these four statements. I. c · (m + n) + t(m − 1, n) ≤ 2c · (m2 n + mn2 ). In order to prove this, we distinguish two cases: Case 1: m = 1: Then

c · (m + n) + t(m − 1, n) = ≤ =

Case 2: m > 1: Then

≤

c · (m + n) + t(0, n)

c · (m + n) + c · (1 + n)

2c · (m + n)

2 c · (m2 n + mn2 ).

c · (m + n) + t(m − 1, n) ≤ = = ≤

II.

c · (m + n) + 2c · ((m − 1)2 n + (m − 1)n2 )

2c · (m2 n + mn2 ) + c · (m + n − 4mn + 2 n − 2n2 )

2c · (m2 n + mn2 ) + c · (m(1 − n) + 3n(1 − m) − 2n2 )

2c · (m2 n + mn2 ).

c · (m + n) + t(m, n − 1) ≤ 2c · (m2 n + mn2 ).

The proof is completely symmetric to the proof of the previous statement. III. For any n1 , n2 with n1 ≥ 1, n2 ≥ 1 and n1 + n2 = n c · (m + n) + t(m, n1 ) + t(m, n2 ) ≤ 2c · (m2 n + mn2 ).

Indeed, for n1 , n2 with n1 ≥ 1, n2 ≥ 1 and n1 + n2 = n one obtains c · (m + n) + t(m, n1 ) + t(m, n2 ) ≤ = = =

c · (m + n) + 2c · (m2 n1 + mn21 ) + 2c · (m2 n2 + mn22 )

2c · (m2 n + mn2 ) + c · (m + n + 2 m(n21 + n22 − n2 ))

2 c · (m2 n + mn2 ) + c · (m + n − 4mn1 n2 )

2 c · (m2 n + mn2 )

+c · (m(1 − n1 n2 ) + n1 (1 − mn2 ) + n2 (1 − mn1 ) − mn1 n2 )

≤

2c · (m2 n + mn2 ).

IV. For any m1 , m2 with m1 ≥ 1, m2 ≥ 1 and m1 + m2 = m

c · (m + n) + t(m1 , n) + t(m2 , n) ≤ 2c · (m2 n + mn2 ).

The proof is completely symmetric to the proof of the previous statement.

174 | Peter Hertling and Victor Selivanov In the case of ω-labeled forests, one sets m := the length of the forest F as a string, i.e., the length of the string representing the first forest, and, similarly, n := the length of the string representing the second forest. Note that if the forest F is empty then m = 0, and if F is not empty then m ≥ 3. Similarly with G. The algorithm works in the same way. The recursive inequalities for the time estimation have to be modified slightly. There exists a constant c > 0 such that the following holds. For 0 ≤ m ≤ 2 and n ≥ 0 one has t(m, n) ≤ c · (1 + n), for m ≥ 0 and 0 ≤ n ≤ 2 one has t(m, n) ≤ c · (1 + m), and for m, n ≥ 3 one has t(m, n) ≤ c · (m + n)

+ max{ max{t(m − i, n) | 3 ≤ i ≤ m}, max{t(m, n − i) | 3 ≤ i ≤ n}, max{t(m, n1 ) + t(m, n2 ) | n1 ≥ 3, n2 ≥ 3, n1 + n2 = n}, max{t(m1 , n) + t(m2 , n) | m1 ≥ 3, m2 ≥ 3, m1 + m2 = m}}.

Using these recursive inequalities, similarly as above one arrives at t(m, n) ∈ O(m + n + m2 n + mn2 ).

4 The Complexity of ≤0 -minimization Theorem 7 may be used to establish polynomial-time computability of some other natural relations and functions on Fk . We give some examples related to minimal k-forests. Recall that a minimal k-forest is a finite k-forest not ≤0 -equivalent to a k-forest of lesser cardinality. As observed in [21], any finite k-forest is equivalent to a unique (up to isomorphism) minimal k-forest. The next characterization of the minimal k-forests from [21] is a kind of inductive definition (by induction on the cardinality) of the minimal k-forests. Proposition 8. 1. The empty k-forest is minimal, and any singleton k-forest is minimal. 2. A non-singleton k-tree (T ; ≤, c) is minimal iff the k-forest (F ; ≤F , c|F ) obtained by deleting the root from T is minimal, and if c(root(T )) = c(y) for all immediate predecessors y of the root in T.

Complexity Issues for Preorders on Finite Labeled Forests | 175

3. A non-empty, proper (i.e., not ≤0 -equivalent to a k-tree) k-forest is minimal iff all its k-trees are minimal and pairwise incomparable under ≤0 . The first item in the next lemma was observed in [21], the other follows from the previous proposition. Lemma 9. 1. Any two ≤0 -equivalent k-trees have the same root label. 2. If F is a minimal k-forest and x ∈ F then the lower cone ↓ x (with the induced partial order and labeling) is minimal. If F = F0 · · · F n where the summands are k-trees and i < k, let q i (F ) denote the kforest G0 · · · G m where the sequence (G0 , . . . , G m ) is obtained from (F0 , . . . , F n ) by deleting all roots labeled by i (note that q i (F ) may be empty). Relate to any k-forest F = F0  · · ·  F n where the summands are k-trees the k-forest F ∗ = F i0  · · ·  F i m where i0 is the smallest i ≤ n with F i ≤0 F i+1  · · ·  F n , i1 > i0 is the smallest number i ≤ n (if any) with F i ≤0 F i+1  · · ·  F n  F i0 , i2 > i1 is the smallest number i ≤ n (if any) with F i ≤0 F i+1  · · ·  F n  F i0  F i1 , and so on. The next lemma follows from the definitions and from Theorem 7. Lemma 10. 1. For any F ∈ Fk and i < k, |q i (F )| ≤ |F | and |F ∗ | ≤ |F |. 2. For any F = F0  · · ·  F n , F i0 , . . . , F i m are pairwise ≤0 -incomparable and F ∗ ≡0 F. 3. The function F → q i (F ) is computable in linear time. 4. The function F → F ∗ is computable in cubic time. Proof. The first two statements follow directly from the definitions. In order to delete a root with label i from a tree in a forest represented as described above one has to delete first the opening bracket preceding the label of the root, then the label itself and finally the closing bracket of the tree. For the last part, one may use a stack in which one remembers the current depth of brackets when reading the forest from left to right. It is possible to do this deletion in a single run through the forest. Therefore, it can be performed in linear time. Finally, the fourth statement can be deduced from Theorem 7 as follows. Given a forest F = F0  · · ·  F m where the summands are k-trees with n i := |F i | (note that n i ≥ 1) and n := |F | = n0 + . . . + n m , the time needed for computing F ∗ is dominated by the time for ≤0 -comparing any tree F i with any other tree F j . By Theorem 7 this time is up to a multiplicative constant bounded from above by    (n2i n j + n i n2j ) < (n2i n j + n i n2j ) = 2 · n2i n j 0≤i,j≤m, i=j

0≤i,j≤m

0≤i,j≤m

176 | Peter Hertling and Victor Selivanov

=

2·

m  i=0

n2i

m  j=0

nj = 2 · n ·

m  i=0

n2i ≤ 2n3 .

That proves the assertion. Theorem 11. There is a function min : Fk → Fk computable in time O(n4 ) such that, for any F ∈ Fk , min(F ) is minimal and min(F ) ≡0 F. Proof. The function min is given by the following recursive algorithm: 1. If |F | = 0 or |F | = 1 then set min(F ) = F. 2. If F = p i (G) is not a singleton, then set min(F ) = p i (q i (min(G))∗ ). 3. If F = F0  · · ·  F m where the summands are k-trees and m > 0 then set min( F ) = (min(F0 )  · · ·  min( F m ))∗ . The correctness of the algorithm follows from the above proposition and lemmas. Let t(n) be the maximum time needed for computing min(F ) for some k-labeled forest F with n = |F |. We are going to show t(n) ∈ O(n4 ). The first step of the algorithm can be performed in constant time. For the second step, note that checking whether F is a tree but not a singleton tree can be done in linear time. If F = p i (G), then |G| = n − 1. One can compute min(G) in time t(n − 1). By Lemma 10, then one can compute q i (min(G)) in time O(n − 1). By Lemma 10, then one can compute q i (min(G))∗ in time O((n − 1)3 ). Finally, one can compute p i (q i (min(G))∗ ) in time O(n). For the third step, we assume that F = F0  · · ·  F m where the summands are k-trees with n i := |F i | and m > 0. The  3 third step can be performed in time m i=0 t ( n i ) + O ( n ). We conclude that there is a constant c > 0 such that the following holds true: t(0) ≤ c and for n ≥ 2 t(n)

≤

and

t(1) ≤ c,

 c · n + max t(n − 1) + c · (n − 1) + c · (n − 1)3 + c · n),

c · n3 + max

m  i=0

m     t(n i )  m > 0, (∀i) n i ≥ 1, ni = n . i=0

Using these recursive inequalities, by induction one shows that t(n) ≤ 2c · n4 for n ≥ 1. This proves t(n) ∈ O(n4 ).

Complexity Issues for Preorders on Finite Labeled Forests | 177

5 The Complexity of ≤1 Lemma 12. The relations ≤1 and ≤2 coincide on Cω and also, for each k < ω, on Ck . Proof. This follows from the second characterization of ≤1 in Lemma 4 and from the definition of ≤2 . Proposition 13. The relations ≤1 and ≤2 on Cω are NP-hard. Proof. By Lemma 12, it suffices to find a polynomial time reduction of 3-SAT to ≤2 on Cω (in this proof we assume familiarity of the reader with some common knowledge from complexity theory, see e.g. [1; 2]). We have to relate in polynomial time to any 3-CNF C = C(x0 , . . . , x n ) words u, v over the alphabet ω such that C is satisfiable iff u ≤2 v. In “mnemonic notation”, u will in fact be a word over the alphabet {0, 1, T, F }, and v will be a word over the alphabet {0, 1, x0 , ¯x0 . . . , x n , ¯x n }. A straightforward coding turns u, v into words over the alphabet ω. a We may assume that C = D0 ∧ · · · ∧ D m where any D j is a disjunction x p jj ∨ b

c

x q jj ∨ x r jj of exactly three literals, p j , q j , r j ≤ n, 0 ≤ a j , b j , c j ≤ 1, x1 = x and x0 = ¯x, and no disjunction contains a letter with its negation. For each j ≤ m, let ˜ j denote the word x a j x b j x c j . Define the words u and v by D pj qj rj u

:=

( TF 01)n+1 (T 01)m+1

v

:=

˜ 0 01 · · · D ˜ m 01. x0 ¯x0 x0 01 · · · x n ¯x n x n 01D

Note that |u | = 4(n + 1) + 3(m + 1) and |v | = 5(n + 1) + 5(m + 1). Finally, let u := (01)M u and v := (01)M v where M = |v | + 1. Obviously, the function C → (u, v) is computable in linear time. ˜ j 01 the j-th We call x i ¯x i x i 01 the i-th variable factor of v (0 ≤ i ≤ n), and D disjunction factor of v (0 ≤ j ≤ m). Note that the variable (resp. disjunction) factors of v are in a natural bijective correspondence with the factors TF 01 (resp. T 01) of u. For each i ≤ n, let t i (resp. l i ) be the position in u (resp. in v) of the letter T (resp. of the first entry of x i ) in the i-th factor TF 01 (resp. of the i-th variable factor). For each j ≤ m, let s j be the position in u of the letter T in the j-th factor T 01. Let C be satisfiable, i.e., true for some assignment α : {x0 , . . . , x n } → {T, F }. Define a monotone function φ : |u| → |v| as follows: φ sends the position of any entry of 0 or 1 in u to the corresponding position of the same letter in v (note that the numbers of entries of these letters in u and v coincide); for any i ≤ n, if α(x i ) = T then φ(t i ) = l i and φ(t i + 1) = l i + 1, otherwise φ(t i ) = l i + 1 and

178 | Peter Hertling and Victor Selivanov φ(t i + 1) = l i + 2; for any j ≤ m, φ sends s j to the position of the first true literal in the j-th disjunction factor. Then φ is a 2-morphism from u to v. Conversely, let φ : u → v be a 2-morphism; we have to find a satisfying assignment α for C. Since u is repetition-free, φ is injective. Since M > |v |, φ(0, . . . , 2M − 1) contains some positions for both 0 and 1, hence φ cannot send any position of T or F to a position of 0 or 1. Since any factor of v without 0, 1 has length 3, the positions of T, F in the i-th factor TF 01 can go only to positions of x i ¯x i x i in the corresponding i-variable factor (for each i ≤ n), and, for each j ≤ m, φ(s j ) is a position of some literal in the j-th disjunctive factor. For any i ≤ n, let α(x i ) = T iff φ(t i ) = l i . Then α is a desired satisfying assignment. Theorem 14. For any k < ω, the relation ≤1 on Fk is computable in cubic time. The relation ≤1 on Fω is NP-complete. Proof. Let k < ω. For any G = (Q; ≤, c) ∈ Fk and f : k → k, we define G f := (Q; ≤, f ◦ c). From the definition of ≤1 we observe that F ≤1 G iff there exists a function f : k → k such that F ≤0 G f . Let {f1 , . . . , f k k } be an enumeration without repetition of {f | f : k → k}. Since G → (G f1 , . . . , G f kk ) is computable in linear time and ≤0 is computable in cubic time, ≤1 on Fk is computable in cubic time. For Fω , ≤1 is NP-hard by the previous proposition. It remains to show that it is in NP. The corresponding nondeterministic algorithm is obvious: given ω-labeled forests F = (P; ≤P , c P ) and G = (Q; ≤Q , c Q ), guess a function f : c Q (Q) → c P (P) and a function φ : P → Q and check (in polynomial time) whether φ is a 0morphism from (P; ≤P , c P ) to (Q; ≤Q , f ◦ c Q ). Note that by Theorem 14 the relation ≤1 on Fω is fixed-parameter tractable (compare [6]) if the largest label of the input forests is taken as the parameter.

6 The Complexity of ≤2 Theorem 15. For any k < ω, the relation ≤2 on Fk is computable in time O(n k+3 ). The relation ≤2 on Fω is NP-complete. For the proof of the first assertion in Theorem 15 we need the following simple technical lemma. Lemma 16. For every k ≥ 1 and every n ≥ 2 (n − 1)k + 2−(k−1) · n k−1 ≤ n k .

Complexity Issues for Preorders on Finite Labeled Forests | 179

Proof. Fix some (natural numbers) k ≥ 1 and n ≥ 2. By the binomial formula n k − (n − 1)k = ((n − 1) + 1)k − (n − 1)k ≥ k · (n − 1)k−1 ≥ (n − 1)k−1 . Due to n ≥ 2 we have n − 1 ≥ n/2, hence n k − (n − 1)k ≥ 2−(k−1) n k−1 . By rearranging we obtain the assertion. Proof of Theorem 15. First we prove the second assertion. The relation ≤2 on Fω is NP-hard by Proposition 13. It remains to show that it is in NP. The corresponding nondeterministic algorithm is obvious: given F, G, guess a function φ : F → G and check (in polynomial time) whether φ is a 2-morphism. We come to the proof of the first assertion. Let k < ω. Let us denote by f :⊆ k → k an arbitrary function whose domain dom(f ) and whose range may be any subsets of k = {0, . . . , k −1}. For F = (P; ≤P , c P ), G = (Q; ≤Q , c Q ) ∈ Fk , and such a function f :⊆ k → k, let F ≤2f G mean that there is a 2-morphism φ : F → G such that for all x ∈ P, if c Q (φ(x)) ∈ dom(f ) then c P (x) = f (c Q (φ(x))). Note that F ≤2 G is equivalent to F ≤2f∅ G where f∅ is the partial function from k to k with dom(f ) = ∅. Therefore it suffices to show that the relation ≤2f is computable in time O(n3+k−| dom(f )| ), for any function f :⊆ k → k. Similarly to the proof of Theorem 7, we show that there exists a natural recursive algorithm based on the following sequence of statements for k-forests F and G and a function f :⊆ k → k. This sequence of statements is based on the recursive definition of ≤2 in [7; 8; 9]: If |F | = 0 then F ≤2f G is true. If |F | ≥ 1 and |G| = 0 then F ≤2f G is false. If F = p i (F0 ), G = p j (G0 ), j ∈ dom(f ), and i = f (j) then F ≤2f G iff F0 ≤2f G. If F = p i (F0 ), G = p j (G0 ), j ∈ dom(f ), and i = f (j) then F ≤2f G iff F ≤2f G0 . If F = p i (F0 ), G = p j (G0 ), and j ∈ dom(f ) then F ≤2f G iff (F ≤2f G0 ∨ F0 ≤2g G). Here g is defined by dom(g ) := dom(f )∪{j}, and by g(l) := f (l) for l ∈ dom(f ), and g (j) := i. 6. If F = p i (F0 ) and G = G0  G1 then F ≤2f G iff F ≤2f G0 ∨ F ≤2f G1 . 7. If F = F0  F1 then F ≤2f G iff F0 ≤2f G ∧ F1 ≤2f G.

1. 2. 3. 4. 5.

In the following we sketch the Turing machine. Let t l (m, n) denote the maximum time needed by this algorithm, given a forest F with |F | = m, a forest G with |G| = n and a function f :⊆ k → k with l = | dom(f )|. When started with F #G#f on the input tape (where f is given as a list of all pairs (x, f (x)) with x ∈ dom(f )), the machine copies F, G, and f to three work tapes. This can be done in time O(m + n). Then the recursive algorithm is started. Whenever it is called with some subforests

180 | Peter Hertling and Victor Selivanov F  and G , these subforests can be remembered on two stacks realized on some work tapes. The machine follows the case distinction above. 1. First, the machine checks whether the forest F is empty. This can be done in constant time. If this is the case then the machine says YES and stops. Otherwise it continues with Step 2. 2. The machine checks whether G is empty. This can be done in constant time. If this is the case then the machine says NO and stops. Otherwise it continues with Step 3. 3. The machine checks whether F consists of a single tree. This can be done in time O(m). If this is not the case then it continues with Step 7. If F consists of a single tree then the machine checks whether G consists of a single tree as well. This can be done in time O(n). If this is not the case then it continues with Step 6. If G consists of a single tree as well then we have F = p i (F0 ) and G = p j (G0 ) for some labels i and j and for some forests F0 and G0 . Now the machine checks whether j ∈ dom(f ). This can be done in constant time. If this is not the case then the machine continues with Step 5. If j ∈ dom(f ) then the machine checks whether i = f (j). If this is not the case then the machine continues with Step 4. If i = f (j) then the machine checks by a recursive call whether F0 ≤2f G. This can be done in time t l (m − 1, n). If this is the case then the machine says YES and stops. Otherwise it says NO and stops. 4. In this case we have F = p i (F0 ) and G = p j (G0 ) for some labels i and j with j ∈ dom(f ), and with i = f (j), and for some forests F0 and G0 . The machine checks by a recursive call whether F ≤2f G0 . This can be done in time t l (m, n − 1). If this is the case then the machine says YES and stops. Otherwise it says NO and stops. 5. In this case we have F = p i (F0 ) and G = p j (G0 ) for some labels i and j with j ∈ dom(f ), and for some forests F0 and G0 . The machine checks by a recursive call whether F ≤2f G0 . This can be done in time t l (m, n − 1). If this is the case then the machine says YES and stops. Otherwise it defines a function g :⊆ k → k by dom(g ) := dom(f ) ∪ {j}, and by g (l) := f (l) for l ∈ dom(f ), and g(j) := i), and checks by a recursive call whether F0 ≤2g G.

Complexity Issues for Preorders on Finite Labeled Forests | 181

This can be done in time

t l+1 (m − 1, n).

If this is the case then the machine says YES and stops. Otherwise it says NO and stops. 6. In this case we have F = p i (F0 ), and G is neither empty nor a tree. The machine sets G0 := first tree in G, and G1 := the forest consisting of the remaining trees in G. All of this can be done in time O(n). Then it checks by two recursive calls whether F ≤2f G0 and F ≤2f G1 . This can be done in time O(m + n) + max{t l (m, n1 ) + t l (m, n2 ) | n1 ≥ 1, n2 ≥ 1, n1 + n2 = n}. If this is the case then the machine says YES and stops. Otherwise it says NO and stops. 7. In this case F is neither empty nor a tree. The machine sets F0 := first tree in F, and F1 := the forest consisting of the remaining trees in F. All of this can be done in time O(m). Then it checks by two recursive calls whether F0 ≤2f G and F1 ≤2f G. This can be done in time O(m + n) + max{t l (m1 , n) + t l (m2 , n) | m1 ≥ 1, m2 ≥ 1, m1 + m2 = m}. If this is the case then the machine says YES and stops. Otherwise it says NO and stops. We wish to estimate the time needed by this Turing machine. Note that in comparison with the sequence of statements in the proof of Theorem 7, there is an additional case, namely No. 5. If f is total, i.e., if | dom(f )| = k, then this case plays no role, and one arrives at the same bound for the time as for the relation ≤0 . But if f is not total, then this case has to be taken into consideration. Similarly to the proof of Theorem 7, we will show that for any function f :⊆ k → k, the algorithm works in time O(m + n + m2 n1+k−| dom(f )| + mn2+k−| dom(f )| ), for any input forests F, G with m := |F |, n := |G|. This is shown first for all total functions f , i.e., with k −| dom(f )| = 0, then for all functions f with k −| dom(f )| = 1, then for all functions with k − | dom(f )| = 2, and so on. The additional term k − | dom(f )| in the exponent in the time bound is caused by Case no. 5 in the sequence of statements at the beginning of the proof. In the following we give a detailed proof. The description and the analysis of the algorithm sketched above show that for every l with 0 ≤ l ≤ k there exists a constant c l > 0 such that for all m, n ≥ 0 t l (0, n) ≤ c l · (1 + n)

and

t l (m, 0) ≤ c l · (1 + m),

182 | Peter Hertling and Victor Selivanov and for all m, n ≥ 1 the following is true. In case l = k, i.e., if the function f is total, the fifth step in the algorithm will never be entered. Therefore we obtain the same estimate as in the proof of Theorem 7 for ≤0 : t k (m, n) ≤ c k · (m + n)

+ max{ t k (m − 1, n), t k (m, n − 1), max{t k (m, n1 ) + t k (m, n2 ) | n1 ≥ 1, n2 ≥ 1, n1 + n2 = n}, max{t k (m1 , n) + t k (m2 , n) | m1 ≥ 1, m2 ≥ 1, m1 + m2 = m}}.

As in the proof of Theorem 15, these recursive inequalities imply t k (m, n) ≤ 2c k · (m2 n + mn2 )

(2)

for m, n ≥ 1. For l < k, the fifth step may be entered, and we obtain the following estimate for m, n ≥ 1: t l (m, n) ≤ c l · (m + n)

+ max{ t l (m − 1, n), t l (m, n − 1), t l (m, n − 1) + t l+1 (m − 1, n), max{t l (m, n1 ) + t l (m, n2 ) | n1 ≥ 1, n2 ≥ 1, n1 + n2 = n}, max{t l (m1 , n) + t l (m2 , n) | m1 ≥ 1, m2 ≥ 1, m1 + m2 = m}}.

Of course, this can be simplified to t l (m, n) ≤ c l · (m + n)

+ max{ t l (m − 1, n), t l (m, n − 1) + t l+1 (m − 1, n), max{t l (m, n1 ) + t l (m, n2 ) | n1 ≥ 1, n2 ≥ 1, n1 + n2 = n}, max{t l (m1 , n) + t l (m2 , n) | m1 ≥ 1, m2 ≥ 1, m1 + m2 = m}}

for m, n ≥ 1. We define

d k := 2 · c k

and by induction for l = k − 1, k − 2, . . . , 0

d l := max{2 · c l + c l+1 , 21+k−l · d l+1 }.

Complexity Issues for Preorders on Finite Labeled Forests | 183

We are now going to show by induction that t l (m, n) ≤ d l · (m2 n1+k−l + mn2+k−l )

(3)

for all m, n ≥ 1 and 0 ≤ l ≤ k. This implies the assertion: t l (m, n) ∈ O(m + n + m2 n1+k−l + mn2+k−l ). We come to the proof of (3). We have already seen above that (3) is true for l = k; see (2). Now we consider the case l < k. Remember that we know

and

t l (0, n) ≤ c l · (1 + n)

and

t l (m, 0) ≤ c l · (1 + m),

t l+1 (0, n) ≤ c l+1 · (1 + n)

and

t l+1 (m, 0) ≤ c l+1 · (1 + m),

for m, n ≥ 0. And by induction hypothesis, we can assume that

t l+1 (m, n) ≤ d l+1 · (m2 n1+k−(l+1) + mn2+k−(l+1) ) for all m, n ≥ 1. In order to show (3) it is sufficient to prove the following four assertions for m, n ≥ 1: I. c l · (m + n) + t l (m − 1, n) ≤ d l · (m2 n1+k−l + mn2+k−l ). II.

c l · (m + n) + t l (m, n − 1) + t l+1 (m − 1, n) ≤ d l · (m2 n1+k−l + mn2+k−l ). III. For n1 , n2 with n1 ≥ 1, n2 ≥ 1, and n1 + n2 = n: c l · (m + n) + t l (m, n1 ) + t l (m, n2 ) ≤ d l · (m2 n1+k−l + mn2+k−l ). IV. For m1 , m2 with m1 ≥ 1, m2 ≥ 1, and m1 + m2 = m: c l · (m + n) + t l (m1 , n) + t l (m2 , n) ≤ d l · (m2 n1+k−l + mn2+k−l ). Note that the left hand side in the estimates in these assertions which is really different from the left hand side in the corresponding assertion in the proof of Theorem 7 is the left hand side in II. This left hand side causes the additional term k − l in the exponent of n. In the proof of II. we will use Lemma 16. Now we prove these four statements. I. c l · (m + n) + t l (m − 1, n) ≤ d l · (m2 n1+k−l + mn2+k−l ).

184 | Peter Hertling and Victor Selivanov In order to prove this we distinguish two cases. Case 1: m = 1. Then c l · (m + n) + t l (m − 1, n) = ≤ = ≤ ≤

c l · (m + n) + t l (0, n)

c l · (m + n) + c l · (1 + n)

2c l · (m + n)

d l · (m + n)

d l · (m2 n1+k−l + mn2+k−l ).

Case 2: m > 1. Then c l · (m + n) + t l (m − 1, n) ≤ =

c l · (m + n) + d l · ((m − 1)2 n1+k−l + (m − 1)n2+k−l ) d l · (m2 n1+k−l + mn2+k−l )

+ c l · (m + n) + d l · (−2mn1+k−l + n1+k−l − n2+k−l ) ≤ =

d l · (m2 n1+k−l + mn2+k−l )

+ d l · (m + n) + d l · (−2mn1+k−l )

d l · (m2 n1+k−l + mn2+k−l )

+ d l · (m · (1 − n1+k−l ) + n · (1 − mn k−l )) ≤

d l · (m2 n1+k−l + mn2+k−l ).

II. c l · (m + n) + t l (m, n − 1) + t l+1 (m − 1, n) ≤ d l · (m2 n1+k−l + mn2+k−l ). In order to prove this, we distinguish four cases. Case 1: m = 1 and n = 1: Then c l · (m + n) + t l (m, n − 1) + t l+1 (m − 1, n) = ≤ ≤ =

c l · 2 + t l (1, 0) + t l+1 (0, 1) c l · 2 + c l · 2 + c l+1 · 2 dl · 2

d l · (m2 n1+k−l + mn2+k−l ).

Case 2: m = 1 and n > 1: Then c l · (m + n) + t l (m, n − 1) + t l+1 (m − 1, n) =

c l · (1 + n) + t l (1, n − 1) + t l+1 (0, n)

Complexity Issues for Preorders on Finite Labeled Forests | 185

c l · (1 + n) + d l · (12 · (n − 1)1+k−l + 1 · (n − 1)2+k−l )

≤

+ c l+1 · (1 + n)

d l · (1 + n) + d l · n · (n − 1)1+k−l

≤

d l · 1 + d l · n · (1 + (n − 1)1+k−l )

=

d l · 1 + d l · n · n1+k−l

≤

d l · (1 + n2+k−l )

=

d l · (m2 n1+k−l + mn2+k−l ).

≤

Case 3: m > 1 and n = 1: Then c l · (m + n) + t l (m, n − 1) + t l+1 (m − 1, n) = ≤ = ≤ ≤ ≤

c l · (m + 1) + t l (m, 0) + t l+1 (m − 1, 1) c l · (m + 1) + c l · (1 + m)

+ d l+1 · ((m − 1)2 · 11+k−l−1 + (m − 1) · 12+k−l−1 ) 2c l · (1 + m) + d l+1 · m(m − 1)

d l · (1 + m) + d l · m(m − 1) d l · (1 + m2 )

d l · (m2 n1+k−l + mn2+k−l ).

Case 4: m > 1 and n > 1: Then c l · (m + n) + t l (m, n − 1) + t l+1 (m − 1, n) ≤

c l · (m + n)

+d l · (m2 · (n − 1)1+k−l + m · (n − 1)2+k−l )

+d l+1 · ((m − 1)2 · n1+k−l−1 + (m − 1) · n2+k−l−1 )

=

c l · (m + n)

+d l · (m2 · (n − 1)1+k−l + m · (n − 1)2+k−l ) +d l+1 · ((m − 1)2 · n k−l + (m − 1) · n1+k−l )

≤

c l · (m + n)

+d l · (m2 · (n − 1)1+k−l + m · (n − 1)2+k−l ) +d l+1 · (m2 · n k−l + m · n1+k−l )

≤

c l · (m + n)

+d l · (m2 · (n − 1)1+k−l + m · (n − 1)2+k−l ) +2−(1+k−l) · d l · (m2 · n k−l + m · n1+k−l )

≤

c l · (m + n)

(here we use Lemma 16)

186 | Peter Hertling and Victor Selivanov 

=

≤ ≤



1 · m2 · (n − 1)1+k−l + n1+k−l + d l · m · n2+k−l 2 d l · (m2 n1+k−l + mn2+k−l )   1 +c l · (m + n) + d l · · m2 · (n − 1)1+k−l − n1+k−l 2 2 1+k−l d l · (m n + mn2+k−l )   1 1 + · d l · (m + n) + · d l · m2 · (n − 1)1+k−l − n1+k−l 2 2 d l · (m2 n1+k−l + mn2+k−l )

+d l ·

because due to k > l we have 1 + k − l ≥ 2, and, furthermore, due to n ≥ 2 we obtain   m2 · n1+k−l − (n − 1)1+k−l   n 1+k−l = m2 · (n − 1)1+k−l · −1 n−1   n 2 2 2 −1 ≥ m · (n − 1) · n−1   = m2 · n2 − (n − 1)2 =

= ≥

m2 · (2n − 1)

nm2 + nm2 − m2 n + 2m2 − m2

=

n + m2

≥

n + m.

III. For n1 , n2 with n1 ≥ 1, n2 ≥ 1, and n1 + n2 = n: c l · (m + n) + t l (m, n1 ) + t l (m, n2 ) ≤ d l · (m2 n1+k−l + mn2+k−l ). Indeed, for n1 , n2 with n1 ≥ 1, n2 ≥ 1 and n1 + n2 = n one obtains c l · (m + n) + t l (m, n1 ) + t l (m, n2 ) ≤

c l · (m + n)

k−l k−l +d l · (m2 n1+ + mn2+ ) 1 1 k−l k−l + mn2+ ) +d l · (m2 n1+ 2 2

≤

d l · (m2 n1+k−l + mn2+k−l )

+c l · (m + n)

k−l k−l + n1+ − n1+k−l ) +d l · m2 · (n1+ 1 2

k−l +d l · m · (n2+ + n22+k−l − n2+k−l ) 1

Complexity Issues for Preorders on Finite Labeled Forests | 187

≤

d l · (m2 n1+k−l + mn2+k−l )

1 · d l · (m + n1 + n2 ) 2 −d l · m2 · (1 + k − l) · n1 · n2 +

−d l · m · (2 + k − l) · n1 · n2

≤

d l · (m2 n1+k−l + mn2+k−l ).

where in the third estimate we used for h ≥ 2, for n1 ≥ 1, n2 ≥ 2, and for n = n1 + n2 : n h − n1h − n2h ≥ h · n1h−1 · n2 ≥ h · n1 · n2 . IV. For m1 , m2 with m1 ≥ 1, m2 ≥ 1, and m1 + m2 = m: c l · (m + n) + t l (m1 , n) + t l (m2 , n) ≤ d l · (m2 n1+k−l + mn2+k−l ). Indeed, for n1 , n2 with n1 ≥ 1, n2 ≥ 1 and n1 + n2 = n one obtains c l · (m + n) + t l (m1 , n) + t l (m2 , n) ≤

c l · (m + n)

+d l · (m21 n1+k−l + m1 n2+k−l )

+d l · (m22 n1+k−l + m2 n2+k−l ) ≤

d l · (m2 n1+k−l + mn2+k−l )

+c l · (m + n)

+d l · n1+k−l · (m21 + m22 − m2 )

≤

d l · (m2 n1+k−l + mn2+k−l )

1 · d l · (m1 + m2 + n) 2 −d l · n1+k−l · 2m1 m2

+

≤

d l · (m2 n1+k−l + mn2+k−l ).

7 Some Open Questions Some interesting questions related to the topic of this paper remain open. We have shown that on Fk the preorders ≤0 and ≤1 are computable in cubic time and ≤2 is computable in time O(n k+3 ). It would be interesting to determine the exact complexity classes of these preorders. One may also ask for the complexity of some further natural functions related to the quotient posets Fik of the preorders (Fk ; ≤i ),

188 | Peter Hertling and Victor Selivanov i ≤ 2. This applies for example to the rank and spectrum functions defined as follows. The quotient-posets Fik are known [8] to be well posets of rank ω, i.e., they have neither infinite antichains nor infinite descending chains but they have infinite ascending chains. The rank function rk i : Fk → ω measures the rank of the ≤i -equivalence class [F ]i in Fik . By the spectrum function on Fik we mean the function sp i : ω → ω where sp i (n) is the number of elements of rank n in Fik .

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J. L. Balcázar, J. Díaz, and J. Gabarró. Structural complexity. I. EATCS Monographs on Theoretical Computer Science, Vol. 11. Springer, Berlin, 1988. J. L. Balcázar, J. Díaz, and J. Gabarró. Structural complexity. II. EATCS Monographs on Theoretical Computer Science, Vol. 22. Springer, Berlin, 1990. V. Brattka and G. Gherardi. Effective choice and boundedness principles in computable analysis. Bulletin of Symbolic Logic, 17(1):73–117, 2011. V. Brattka and G. Gherardi. Weihrauch degrees, omniscience principles and weak computability. J. Symb. Log., 76(1):143–176, 2011. B. Davey and H. Priestley. Introduction to Lattices and Order. Cambridge University Press, Cambridge, second edition, 2002. R. G. Downey and M. R. Fellows. Parameterized Complexity Theory. Monographs in Computer Science. Springer, New York, 1999. P. Hertling. A topological complexity hierarchy of functions with finite range. Technical Report 223, Centre de recerca matematica, Institut d’estudis catalans, Barcelona, October 1993. Workshop on Continuous Algorithms and Complexity. P. Hertling. Topologische Komplexitätsgrade von Funktionen mit endlichem Bild. Informatik Berichte 152, FernUniversität Hagen, Hagen, December 1993. P. Hertling. Unstetigkeitsgrade von Funktionen in der effektiven Analysis. PhD thesis, Fachbereich Informatik, FernUniversität Hagen, 1996. P. Hertling and V. L. Selivanov. Complexity issues for preorders on finite labeled forests. B. Löwe (ed.) et al., Models of Computation in Context. CiE 2011, LNCS 6735, Springer, Berlin, 112-121, 2011. M. D. Hirsch. Applications of topology to lower bound estimates in computer science. Hirsch, M. W. (ed.) et al., From topology to computation: Proceedings of the Smalefest. Springer, New York, 395-418, 1993. A. S. Kechris. Classical descriptive set theory. Graduate Texts in Mathematics. 156. Springer, Berlin, 1995. S. Kosub. NP-partitions over posets with an application to reducing the set of solutions of NP problems. Theory Comput. Syst., 38(1):83–113, 2005. S. Kosub and K. W. Wagner. The Boolean hierarchy of NP-partitions (extended abstract). Reichel, Horst (ed.) et al., STACS 2000. LNCS 1770, Springer, Berlin, 157-168, 2000. O. V. Kudinov and V. L. Selivanov. Undecidability in the homomorphic quasiorder of finite labeled forests. Beckmann, Arnold (ed.) et al., Logical approaches to computational barriers. CiE 2006. LNCS 3988, Springer, Berlin, 289-296, 2006.

Complexity Issues for Preorders on Finite Labeled Forests | 189 [16] O. V. Kudinov and V. L. Selivanov. Undecidability in the homomorphic quasiorder of finite labelled forests. J. Log. Comput., 17(6):1135–1151, 2007. [17] O. V. Kudinov and V. L. Selivanov. A Gandy theorem for abstract structures and applications to first-order definability. Ambos-Spies, Klaus (ed.) et al., Mathematical theory and computational practice. CiE 2009. LNCS 5635, Springer, Berlin, 290-299, 2009. [18] O. V. Kudinov, V. L. Selivanov, and A. V. Zhukov. Undecidability in Weihrauch degrees. Ferreira, Fernando (ed.) et al., Programs, proofs, processes. CiE 2010. LNCS 6158, Springer, Berlin, 256-265, 2010. [19] L. Kwuida and E. Lehtonen. On the homomorphism order of labeled posets. Order, 2010. [20] E. Lehtonen. Labeled posets are universal. Eur. J. Comb., 29(2):493–506, 2008. [21] V. Selivanov. Boolean hierarchies of partitions over a reducible base. Algebra Logika, 43(1):77–109, 2004. Translation in Algebra and Logic 43, No. 1, 44-61. [22] V. L. Selivanov. Hierarchies of ∆02 –measurable k–partitions. Mathematical Logic Quarterly, 53(4–5):446–461, 2007. [23] T. von Stein. Vergleich nicht konstruktiv lösbarer Probleme in der Analysis. Diplomarbeit, Fakultät für Informatik, FernUniversität Hagen, March 1989. [24] K. Weihrauch. The degrees of discontinuity of some translators between representations of the real numbers. Informatik Berichte 129, FernUniversität Hagen, Hagen, July 1992. [25] K. Weihrauch. The TTE-interpretation of three hierarchies of omniscience principles. Informatik Berichte 130, FernUniversität Hagen, Hagen, September 1992. [26] K. Weihrauch. Computable Analysis. Springer, Berlin, 2000.

Anton Konovalov

Boolean Algebras of Regular Quasi-aperiodic Languages¹ Abstract: We characterize up to isomorphism the Boolean algebras of regular quasi-aperiodic languages and of regular d-quasi-aperiodic languages, and show decidability of classes of languages related to these characterizations. Keywords: Boolean algebra, Frechét ideal, regular quasi-aperiodic language, regular d-quasi-aperiodic language. Mathematics Subject Classification 2010: 03G05, 03C13, 06E05, 68Q15 || Anton Konovalov: A.P. Ershov Institute of Informatics Systems, Siberian Division of the Russian Academy of Sciences and Novosibirsk State University, Russia

1 Introduction Boolean algebras are of principal importance for several branches of mathematics. Accordingly, characterization of naturally arising Boolean algebras attracts attention of many researchers. As examples we mention characterizations of natural Boolean algebras in logic and computability theory [4; 5; 10; 11; 12]. In formal language theory, people consider many classes of languages which form Boolean algebra’s but until recently there was no attempt to characterize those Boolean algebra’s up to isomorphism. To our knowledge, the first papers in this direction are [6; 13]. Such characterizations could be of some interest because they provide a new kind information on the classes of languages. Due to Stone duality, this could contribute to understanding of the corresponding Stone spaces which, for the case of regular languages, are closely related to the important and intricate profinite topologies [8; 2]. In this note we characterize up to isomorphism the Boolean algebras of regular quasi-aperiodic languages and of regular d-quasi-aperiodic languages. These characterizations are similar to the corresponding characterizations of Boolean algebras of regular languages RΣ and of regular aperiodic languages AΣ in [13], though some proofs are different.

1 Supported by RFBR Grant 13-01-00015a

192 | Anton Konovalov If B is a Boolean algebra and α an ordinal, let F α (B) be the α-th iterated Frechét ideal of B, B(α) = B/F α (B) is the α-th Frechét derivative of B and B = B(1) . (See [3] for a detailed treatment, some definitions are recalled in the next section.) For a finite alphabet Σ, let QΣ (resp. QΣ,d ) denote the Boolean algebra of all regular quasi-aperiodic (resp. of all regular d-quasi-aperiodic) languages over Σ. The main result of this paper looks as follows: Theorem 1. 1. For any alphabet Σ, QΣ is an atomic Boolean algebra with infinitely many atoms, and QΣ is a countable atomless Boolean algebra. 2. For any d > 1 and a unary alphabet Σ, QΣ,d is an atomic Boolean algebra with infinitely many atoms, and QΣ,d is an Boolean algebra with 2d elements. 3. For any d > 1 and alphabet Σ with at least two symbols, F0 (QΣ,d ) ⊂ F1 (QΣ,d ) ⊂ (n) · · · ⊂ F ω (QΣ,d ) = F ω+1 (QΣ,d ), for each n < ω the Boolean algebra QΣ,d is (ω)

atomic with infinitely many atoms, and QΣ,d is a countable atomless Boolean algebra.

From some well-known facts on Boolean algebra’s (see Chapter 1 of [3]) it follows that items 1 – 3 characterize the corresponding Boolean algebra’s up to isomorphism. Moreover, our proofs imply decidability of the classes of regular quasiaperiodic languages related to the main theorem. The main result of this paper and the corresponding result about languages of finite words imply the following Corollary 2. For any d > 1 and alphabet Σ with at least two symbols, QΣ  RΣ and QΣ,d  AΣ where  denotes the isomorphism relation. In Sections 2 and 3 we provide the necessary Boolean algebrackground on Boolean algebra’s and on regular languages, respectively. In Sections 4 and 5 we characterize the Boolean algebra’s QΣ and QΣ,d , respectively.

2 Preliminaries on Boolean Algebras In this section we very briefly recall some notions and facts on Boolean algebra’s used in the sequel (more details are contained in [13]). We assume the reader to be familiar with Boolean algebrasic notions related to Boolean algebra’s like ideal of a Boolean algebra, quotient-algebra of a Boolean algebra modulo a given ideal, and canonical homomorphism of a Boolean algebra onto its quotient-algebra (for

Boolean Algebras of Regular Quasi-aperiodic Languages | 193

detailed treatments of Boolean algebra’s we refer to [3; 14]). Boolean algebra’s are considered in the signature {∪, ∩,¯, 0, 1}. Recall that element a of a Boolean algebra A is an atom if a = 0 and x < a implies x = 0. A Boolean algebra A is atomless if it has no atom, and it is atomic if below any non-zero element there is an atom. The ideal of a Boolean algebra A generated by atoms is called Frechét ideal of A. It consists of all finite unions (including the empty union which coincides with 0) of atoms and it is denoted by F (A). The quotient-algebra A/F (A) is called the Frechét derivative of A and is denoted by A . Define the transfinite sequence {F α (A)} of iterated Frechét ideals of a Boolean algebra A as follows: F0 (A) = {0}, F β+1 (A) = {x | x/F β (A) ∈ F (A(β) )} where  A(β) = A/F β (A), and F α (A) = β 1. A counting pattern (q, u n ) of a DFA A is called d-balanced counting pattern (where d ≥ 1) if u ≡ 0modd. Note that in [21] definitions of counting patterns of d-balanced counting patterns look slightly different but they are clearly equivalent to the definitions here for the case of minimal DFA’s.

Proposition 9. [20] For any L ∈ RΣ the following conditions are equivalent: 1. There is some d ≥ 1 such that L can be constructed from singleton languages {a}, a ∈ Σ and the language (Σ d )∗ by finite applications of operations of union, complementation, and concatenation. 2. There is some d ≥ 1 such that there is n < ω such that xy n z ∈ L iff xy n+1 z ∈ L for all x, y, z ∈ Σ ∗ with |y| ≡ 0 mod d. Languages satisfying the conditions in the last proposition are called quasiaperiodic. By quasi-aperiodicity index of a quasi-aperiodic language L we mean the least number n satisfying the condition 2 above. The class QΣ of regular quasiaperiodic languages over Σ is closed under the Boolean operations.

196 | Anton Konovalov Proposition 10. [20; 21] For any L ∈ RΣ and d ≥ 1 the following conditions are equivalent: 1. L can be constructed from singleton languages {a}, a ∈ Σ and the language (Σ d )∗ by finite applications of operations of union, complementation, and concatenation. 2. There is n < ω such that xy n z ∈ L iff xy n+1 z ∈ L for all x, y, z ∈ Σ ∗ with |y| ≡ 0 mod d. 3. The minimal DFA of L has no d-balanced counting pattern. Languages satisfying the conditions in the last proposition are called d-quasiaperiodic. Note that the class of 1-quasi-aperiodic languages coincides with the class of aperiodic laguages. By d-quasi-aperiodicity index of a d-quasi-aperiodic language L we mean the least number n satisfying the condition 2 above. The class QΣ,d of regular d-quasi-aperiodic languages over Σ is closed under the Boolean operations and under concatenation. There are several other important characterizations of the class QΣ and QΣ,d but in this paper we use only those mentioned above. For a more systematic treatment see e.g. [21; 20].

4 The Boolean Algebra QΣ First we characterize the Boolean algebra QΣ formed by the class QΣ of regular quasi-aperiodic languages over Σ. Thus, we prove item 1 of the main theorem from Introduction. Proposition 11. 1. For any alphabet Σ, QΣ is an atomic Boolean algebra with infinitely many atoms, and QΣ is a countable atomless Boolean algebra. 2. The class F (QΣ ) is decidable. Proof. 1. Obviously, the atoms of QΣ are exactly the singleton languages, hence the first assertion holds. For the second assertion, it suffices to show that for any infinite language L ∈ QΣ there is an infinite regular quasi-aperiodic language M ⊆ L such that L \ M is infinite. By Proposition 3, there are x, y, z ∈ Σ∗ such that |y| ≥ 1 and xy∗ z ⊆ L. Note that by proposition 9 the language x(y n )∗ z is quasi-aperiodic (with d = |y n |) for any n < ω. Then the language M = x(yy)∗ z has the desired properties. 2. The assertion is well-known because F (QΣ ) is the class of finite languages.

Boolean Algebras of Regular Quasi-aperiodic Languages | 197

According to Proposition 1.8.3 of [3]), the item 1 of Proposition 11 characterizes the Boolean algebra QΣ up to isomorphism. This immediately implies Corollary 12. For any alphabet Σ, QΣ  RΣ . Proof. Since Boolean algebra RΣ has the same properties as in item 1 of Proposition 11 (see [6; 13]), QΣ  RΣ .

5 The Boolean Algebra QΣ,d First we characterize the Boolean algebra QΣ,d formed by the class QΣ,d of d-quasiaperiodic regular languages over a unary alphabet Σ. Thus, we prove item 2 of the main theorem from Introduction. We recall that a non-empty word u ∈ Σ ∗ is called primitive if v n = u implies n = 1 (and hence v = u) for all v ∈ Σ + . It is well known [22] that any non-empty word is uniquely representable as a power of a primitive word. In other words if u n = v m and u, v are primitive then u = v and m = n. Lemma 13. Let L ∈ QΣ,d , A be the minimal DFA of L, (q, y) be a cycle of A, and y = v k for a primitive word v and k > 0. Then k divides dn where n is the greatest common divisor of d and |v|. d

Proof. Consider the word w = v n . Since |w| = |v| · dn and n divides |v|, both d and |v| divide |w|. Since L ∈ QΣ,d , by Proposition 10 (q, w m ) is not a counting pattern of A for each m ≥ 2, hence |y| divides |w|. Thus, |v| · dn = |w| = k · |v| · q for some integer q ≥ 1. Therefore, k divides dn . Lemma 14. Let y = v k ∈ Σ + where v is primitive and k > 0. Then y∗ ∈ QΣ,d iff k divides dn where n is the greatest common divisor of d and |v|. Proof. Clearly, the minimal DFA A of y∗ has |y| + 1 states q0 , . . . , q|y| where q0 is the initial state which is also the unique accepting state, q i · y(i) = q i+1 for each i < |y| − 1, q i · y(i) = q0 for i = |y| − 1, and q|y| is the sink state that attracts all other transitions of A. Then implication from left to right follows immediately from the previous lemma. Conversely, let k divide dn , i.e. dn = k · q for some integer q ≥ 1. By Proposition 10, it suffices to show that A has no d-balanced counting pattern, i.e., if (s, w m ) is a cycle of A and d divides |w| then already (s, w) is a cycle of A. By the structure of A, we may w.l.o.g. assume that s = q0 and q|y| is not in the cycle (q0 , w m ). Then

198 | Anton Konovalov y l = w m for some integer l ≥ 1. Let w = u j where u is primitive and j ≥ 1, then v kl = y l = u jm . Since u and v are primitive, u = v and hence w = v j . Since d and d|v| d|v| |v| divide |w| and n is the greatest common divisor of d and |v|, n divides |w|. Then for some integer p ≥ 1 we have |w| = d|nv| · p = k · q · |v| · p, hence k · |v| divides |y|. Therefore, q0 · w = q0 . Corollary 15. For any y ∈ Σ + , y∗ ∈ QΣ,|y| . Furthermore, Sω ⊆ QΣ . Proof. The first assertion follows from the previous lemma (take d = |y|). For the second assertion, it suffices to show (by Proposition 4) that L ∈ QΣ for any language L of the form xy∗1 z1 . . . y∗k z k where k ≥ 1 and y1 , . . . y k ∈ Σ + . By the first assertion, L ∈ QΣ,d ⊆ QΣ where d is the least common multiple of |y1 |, . . . , |y k |. Proposition 16. For any unary alphabet Σ = {a} and d > 1, QΣ,d is an atomic Boolean algebra with infinitely many atoms, and QΣ,d is a Boolean algebra with 2d elements. Proof. Obviously, the atoms of QΣ,d are exactly the singleton languages, hence the first assertion holds. Let L j = a j (a d )∗ , where j < ω. It is clear that L j ∈ QΣ,d , Lj = 0 for any j < ω and Li ∩ Lj =  for any distinct i, j < d, where Li = L i /F1 (QΣ,d ). For the second assertion it suffices to show that Li is atom in QΣ,d for all i < d and for any infinite L ∈ QΣ,d there is j < d such that Lj ⊆ L . We have to show that if M is an infinite d-quasi-aperiodic subset of L i then L i \ M is finite. We have a i (a d )m ∈ M for infinitely many m. By proposition 10, a i (a d )m ∈ M for all m ≥ n, where n is the d-quasi-aperiodicity index of M. Therefore L i \ M is a finite. It is clear that for any k ≥ d there is j < d such that Lk = Lj , hence Li is atom in QΣ,d for all i < d. Consider an arbitrary infinite d-quasi-aperiodic language L. By Proposition 3, there are x, y, z ∈ Σ∗ such that |y| ≥ 1 and xy∗ z ⊆ L. As Σ = {a}, then xy∗ z = a i (a k )∗ for some i, k < ω. By Lemma 14 k divides d and hence a i (a d )∗ ⊆ a i (a k )∗ ⊆ L. Thus L ⊇ Li for some i < d.

To characterize the Boolean algebra QΣ,d for non-unary alphabets Σ, we need next lemmas.

Lemma 17. Let y = v k ∈ Σ + where v is primitive and k > 0, and let y∗ ∈ QΣ,d . Then there is a word w ∈ (Σ d )+ such that y∗ is a finite union of languages of the form x · w∗ , x ∈ Σ ∗ .

Boolean Algebras of Regular Quasi-aperiodic Languages

| 199

Proof. By Lemma 14, dn = k · q for some integer q ≥ 1. By the proof of Lemma 13, d for the word w = v n we have w ∈ (Σ d )+ and |y| divides |w|. Thus, y∗ = (v k )∗ = w∗ ∪ v k · w∗ ∪ · · · ∪ v k(q−1) · w∗

is a desired representation. Lemma 18. For any k < ω and x, z i ∈ Σ ∗ , y i ∈ Σ + with y∗1 , · · · , y∗k ∈ QΣ,d , xy∗1 z1 · · · y∗k z k ∈ F k+1 (QΣ,d ). If, in addition, y i (0) = z i (0) for each i = 1, . . . , k then xy∗1 z1 · · · y∗k z k ∈ F k (QΣ,d ). Proof. Representing any y i as in the previous lemma and using the fact that concatenation is distributive with respect to ∪, we see that w.l.o.g. we may assume that y1 , . . . , y k ∈ (Σ d )+ . To prove the first assertion it suffices to check the statement by induction on k. For k = 0 this is obvious because L = {x} is an atom of (0) QΣ,d = QΣ,d . Let k = 1. The case y1 = ε is trivial, so assume y1 to be non-empty. Then L is infinite and we have to show that if A is an infinite d-quasi-aperiodic subset of L then L \ A is finite. We have xy1m z1 ∈ A for infinitely many m. By Proposition 10, xy1m z1 ∈ A for all m ≥ n where n is the d-quasi-aperiodicity index of A. Therefore, L \ A is finite. Let now k ≥ 2. For any n < ω, set L n = xy1n y∗1 z1 · · · y nk y∗k z k , then L = L0 ⊃ L1 ⊃ L2 ⊃ · · · , where L = xy∗1 z1 · · · y∗k z k . It suffices to show that for any d-quasiaperiodic language A ⊆ L at least one of languages A, L \ A is in F k (QΣ,d ). We distinguish two cases. Case 1. L n ⊆ A for some n. Then A = L \ A ⊆ L \ L n ∈ F k (QΣ,d ) by the proof for k = 1. Thus, A ∈ F k+1 (QΣ,d ). Case 2. L n ⊆ A for all n, i.e. for any n there are m1 , m2 ≥ n such that xy1m1 z1 y2m2 z2 ∈ A. By Proposition 10, xy1m1 z1 y2m2 z2 ∈ A for all m1 , m2 ≥ m where m is the d-quasi-aperiodicity index of A. Then L m ⊆ A, hence L \ A ⊆ L \ L m ∈ F k (QΣ,d ), hence L \ A ∈ F k (QΣ,d ). For the second assertion, represent any y i as y i = v ki i where v i are primitive, d n

and let, as above, w i = v i i . Then w i ∈ (Σ d )+ and xw∗1 z1 · · · w∗k z k ⊆ xy∗1 z1 · · · y∗k z k , hence it suffices to check that xw∗1 z1 · · · w∗k z k ∈ F k (QΣ,d ). The assertion xw∗1 z1 ∈ F1 (QΣ,d ) is clear because the language xw∗1 z1 is infinite while F1 (QΣ,d ) is the class of finite languages. The language xw∗1 z1 w∗2 z2 is a disjoint union of languages K n = xw1n z1 w∗2 z2 , and, for each n, K n /F1 (QΣ,d ) is an atom of QΣ,d by the first assertion. Therefore, xw∗1 z1 w∗2 z2 ∈ F2 (QΣ,d ). Continuing in this manner, we derive the desired assertions.

200 | Anton Konovalov Lemma 19. For any L ∈ F ω (QΣ,d ), the minimal DFA of L has no ω-pattern. Proof. By contraposition, assume that the minimal DFA of L has an ω-pattern with some words x, y1 , y2 , z. We assume w.l.o.g. that y1 (0) = y2 (0). We have to show that L ∈ F ω (QΣ,d ). Since x(y1 + y2 )∗ z ⊆ L, it suffices to show that xy∗1 z ∈ F ω (QΣ,d ), xy∗1 y2 y∗1 z ∈ F ω (QΣ,d ), xy∗1 y2 y∗1 y2 y∗1 z ∈ F ω (QΣ,d ) and so on. But this holds by the previous lemma. Theorem 20. For any L ∈ QΣ,d the following conditions are equivalent: 1. L ∈ F ω (QΣ,d ). 2. The minimal DFA of L has no d-balanced counting pattern and has no ωpattern. 3. L ∈ QΣ,d ∩ Sω . 4. L is a finite union of languages xy∗0 z0 · · · y∗k z k where k < ω, x, y i , z i ∈ Σ∗ and y∗0 , · · · , y∗k ∈ QΣ,d . Proof. 1→2. This follows from the previous lemma and Proposition 10. 2↔3. This follows from Propositions 10 and 5. 2→4. The desired representation of L follows from Lemmas 7 and 14 4→1. This follows from Lemma 18. Corollary 21. The class of regular d-quasi-aperiodic languages F ω (QΣ,d ) is decidable. Next we characterize F k (QΣ,d ) for any k < ω. Note that F0 (QΣ,d ) = {∅} and F1 (QΣ,d ) is the class of finite regular d-quasi-aperiodic languages over Σ. Theorem 22. For any k < ω and L ∈ RΣ the following conditions are equivalent: 1. L ∈ F k+2 (QΣ,d ). 2. The minimal DFA of L has no d-balanced counting pattern, has no ω-pattern, and there is no chain F0 < · · · < F k+1 of SCC’s of ML such that a final state of ML is reachable from F k+1 . 3. L ∈ QΣ,d ∩ Sk . 4. L is a finite union of languages xy∗0 z0 · · · y∗k z k where x, y i , z i ∈ Σ ∗ and y∗0 , · · · , y∗k ∈ QΣ,d . Proof. 1→2. This follows from Proposition 10, Lemma 19 and the proofs of Lemma 19 and Proposition 8. 2↔3. This follows from Propositions 10 and 8. 2→4. The desired representation of L follows from Lemmas 7 and 14.

Boolean Algebras of Regular Quasi-aperiodic Languages | 201

4→1. This follows from Lemma 18.

Now we are able to prove the item 3 of the main theorem in Introduction. Theorem 23. For any alphabet Σ with at least two symbols, F0 (QΣ,d ) ⊂ F1 (QΣ,d ) ⊂ · · · ⊂ F ω (QΣ,d ) = F ω+1 (QΣ,d ) , (n)

for each n < ω the Boolean algebra QΣ,d is atomic with infinitely many atoms, and (ω)

QΣ,d is a countable atomless Boolean algebra.

Proof. First we check that F k (QΣ,d ) ⊂ F k+1 (QΣ,d for each k < ω. For k = 0 the inclusion is trivial. Let a, b ∈ Σ, a = b, and let y = a d and z = b d , then y∗ , z∗ ∈ QΣ,d . By Lemma 18 y∗ ∈ F2 (QΣ,d ) \ F1 (QΣ,d ), y∗ zy∗ ∈ F3 (QΣ,d ) \ F2 (QΣ,d ),

y∗ zy∗ zy∗ ∈ F4 (QΣ,d ) \ F3 (QΣ,d ),

and so on. By Lemma 18 the elements y∗ /F1 (QΣ,d ), y∗ zy∗ /F2 (QΣ,d ), . . . are atoms re(1) (2) spectively in the Boolean algebra’s QΣ,d , QΣ,d , . . ., and the same applies to the n ∗ n ∗ ∗ elements z y /F1 (QΣ,d ), z y zy /F2 (QΣ,d ), . . . for each n < ω. Since the languages z n y∗ (as well as the languages z n y∗ zy∗ and so on) are pairwise disjoint (1) (2) for distinct n, the Boolean algebra’s QΣ,d , QΣ,d , . . . have infinitely many atoms (as

well as the Boolean algebra QΣ,d = QΣ,d ). (0)

(k)

Next we check that the Boolean algebra QΣ,d is atomic for each k < ω. For k < 2 this is again obvious, so it remains to show that for any k < ω and L ∈ QΣ,d \ F k+2 (QΣ,d ) there is a d-quasiaperiodic language A ⊆ L such that A/F k+2 (QΣ,d ) is (k+2) an atom of QΣ,d . We distinguish the cases L ∈ F ω (QΣ,d ) and L ∈ F ω (QΣ,d ). In the first case, the minimal DFA ML of L has an ω-pattern, hence A exists by Theorem 20 and the proof of Lemma 19. In the second case, by Theorem 20 and Lemma 7 there are SCC’s F0 < · · · < F k+1 and the words specified there. The words z0 , . . . , z k are non-empty and the first letters in z i , y i are distinct for each i ≤ k. Since ML has no d-balanced counting pattern, y∗0 , · · · , y∗k ∈ QΣ,d . Then by (the proof of) Lemma 18 the language A = xy∗0 z0 · · · y∗k z k has the desired property. To prove the remaining assertions for the Boolean algebra QΣ,d , it suffices to show that for any L ∈ QΣ,d \ F ω (QΣ,d ) there is a d-quasiaperiodic language M ⊆ L such that M, L \ M ∈ F ω (QΣ,d ). By Proposition 10 and Theorem 20, the minimal DFA ML of L has no d-balanced counting pattern but has an ω-pattern formed by some words x, y1 , y2 , z such that y1 (0) = y2 (0). Since ML has no d-balanced counting pattern, y∗1 , y∗2 ∈ QΣ,d . By the proof of Lemma 19, we can take M = xy1 (y1 + y2 )∗ z.

202 | Anton Konovalov Similar to Corollary 12 we obtain Corollary 24. For any d > 1 and alphabet Σ with at least two symbols, QΣ,d  AΣ . Acknowledgement: I am grateful to Victor Selivanov for stating the problem of characterization the Boolean algebra of regular languages and for providing a copy of his paper [21], and also for many useful discussions of quasi-aperiodic and d-quasi-aperiodic languages.

Bibliography [1] [2]

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14] [15] [16]

[17]

C. Choffrut and J. Karhumäki. Combinatorics of Words. A chapter of Handbook of Formal Languages, Berlin, Springer, 1997. M. Gehrke, S. Grigorieff and J.-E. Pin. Duality and equational theory of regular languages. Proc. of ICALP 2008, Lecture Notes in Computer Science, v. 5126, Springer Verlag, (2008), 246–257. S.S. Goncharov. Countable Boolean Algebras and Decidability. Plenum, New York, 1996. W. Hanf, The boolean algebra of logic. Bull. Amer. Math. Soc., 20, No 4 (1975), 456–502. S. Lempp, M. Peretyat’kin and R. Solomon. The Lindenbaum algebra of the theory of the class of all finite models. Journal of Mathematical Logic 2, No 2 (2002), 145–225. C. Marini, A. Sorbi, G. Simi, M. Sorrentino. A note on algebras of languages. Theoretical Computer Science , v. 412 (2011), 6531–6536. J.-E. Pin. Unpublished manuscript on regular languages. N. Pippenger. Regular languages and Stone duality. Theory of Computing Systems, 30, No 2 (1997), 121–134. D. Perrin and J.-E. Pin. Infinite Words. v. 141 of Pure and Applied Mathematics, Elsevier, 2004. V. L. Selivanov. Universal Boolean algebras with applications. Abstracts of Int. Conf. in Algebra, Novosibirsk, 1991, 127 (in Russian). V. L. Selivanov. Hierarchies, Numerations, Index Sets. Handwritten notes, 1992, 290 pp. V. L. Selivanov. Positive structures. In: Computability and Models, Perspectives East and West, S. Boolean algebrarry Cooper and Sergei S. Goncharov, eds., Kluwer Academic/Plenum Publishers, New York, 2003, 321–350. V. Selivanov and A. Konovalov. Boolean Algebras of Regular Languages. DLT 2011, LNCS 6795, G. Mauri and A. Leporati, eds., Springer, Heidelberg, 2011, 386–396. R. Sikorski. Boolean Algebras. Springer-Verlag, Berlin, 1964. H. Straubing. Finite automata, formal logic and circuit complexity. Birkhäuser, Boston, 1994. A. Szilard, S. Yu, K. Zhang and J. Shallit. Characterizing Regular Languages with Polynomial Densities. Proc. of MFCS, Lecture Notes in Computer Science, v. 629 (1992), 494– 503. W. Thomas. Languages, automata and logic. Handbook of Formal Language theory, v. B (1996), 133–191.

Boolean Algebras of Regular Quasi-aperiodic Languages | 203 [18] S. Yu. Regular Languages. A chapter of Handbook of Formal Languages, ed. G. Rozenberg and A. Salomaa, Springer, 1997. [19] M. Davis. Computability and Unsolvability. McGraw-Hill Book Company, 1958. [20] Z. Wu. Quasi-star-free Languages on Infinite Words. Acta Cybernetica, v. 17, No. 1, (2005), 75–93. [21] V.L. Selivanov. Hierarchies and reducibilities on regular languages related to modulo counting. RAIRO Theoretical Informatics and Applications, 41 (2009), 95–132. [22] R. C. Lyndon and M. P. Schützenberger. The Equation a M = b N c P in a Free Group. Michgan Math. J., 9(1962), 289-298.

Eryk Kopczyński, Damian Niwiński

A Simple Indeterminate Infinite Game¹ Abstract: We show a proof of the existence of an indeterminate game with perfect information, which can be taught to high school students. The winning criterion is based on an infinite XOR, i.e., a Boolean function over infinite strings of bits, such that the change of one bit in an argument changes the value. The indeterminacy follows by strategy stealing technique. Keywords: Infinite games, perfect information games, determinacy, strategy stealing Mathematics Subject Classification 2010: 03E25, 03E60 || Eryk Kopczyński and Damian Niwiński: Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland.

1 Introduction One of the simplest and yet non-trivial results in foundations of mathematics, which can be presented to high school pupils, is the Theorem of Zermelo on the determinacy of the game of chess [14]: either Black or White has a winning strategy, or both players have strategies to achieve a draw. The proof smoothly generalizes the problem to all finite games with perfect information², and reveals an algorithmic aspect of the result. The message however remains incomplete if we hide from our students that even perfect information games can be indeterminate if we allow the players to make infinite sequences of moves. Infinite games have a long history started with mathematical puzzles considered by Stefan Banach and Stanisław Mazur in the 1930s; see [10] for concise introduction and [13] for historical overview. Although such games can hardly be played in real life, they are vital for multiple areas of mathematics—in particular descriptive set theory [6; 9], and also for computer science; see [4] for a survey.

1 The authors were supported by the Polish Ministry of Science grant nr N N206 567840. 2 In a two player perfect information game both players know the current state of the game, and the future depends uniquely on their moves, see, e.g., [10].

206 | Eryk Kopczyński and Damian Niwiński Indeterminacy of perfect information two player infinite games was observed in the first published work on such games [3]. It means that, even though an actual choice of players’ strategies always determines the result of the game, a winning strategy may not exist for any of the players, which makes the result of the game indeterminate. From a teaching perspective, we can note a remote analogy with real life games. Indeed, the result of a chess match is “indeterminate” for us because the players presumably ignore the optimal strategies. But we know from the Zermelo Theorem that such optimal strategies exist. In contrast, for infinite games, the winning (or optimal) strategies may not exist at all. The existence of indeterminate games can be proved in many ways, which however usually introduce some bits of mathematics unknown to high school pupils. Indeterminate games can be constructed by a diagonal argument using transfinite induction (see, e.g., [5]); their existence can be also inferred from a classical result in general topology, namely the Bernstein partition of a Polish space into two parts, none of which contains a perfect set³ (see, e.g., [10]). Another argument, game-theoretic in flavour, can be used to show indeterminacy of a certain game based on a Fréchet ultrafilter; the authors learned this example from the unpublished lecture notes⁴ by Jacques Duparc[2]. This argument uses the strategy stealing technique, which is well known for solving finite games, in particular Hex. It is based on the following reasoning: If player B had a winning strategy, it could be used to construct a winning strategy for player A.

As finite games are determined, this implies that player A has winning strategy. In case of infinite games however, the same argument implies indeterminacy, assuming that the dependence above holds in both directions (i.e., A → B, and B → A). In the present note, we use a similar argument to show the indeterminacy of a conceptually simpler game based on an infinite XOR function, i.e., a function f : {0, 1}ω → {0, 1}, such that the change of one bit in an argument changes the value. (From {0, 1}n to {0, 1}, there are clearly only two such functions: XOR and ¬ XOR.) A bit of advanced mathematics is only needed to show that such functions exist. Here the use of the Axiom of Choice is unavoidable, but no transfinite induction is necessary. We took an idea of infinite XOR function from

3 Because the set of plays consistent with any strategy (in particular, a hypothetical winning strategy) forms a perfect set. 4 A similar proof appears in lecture notes by Alessandro Andreta [1].

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Michael Sipser [12], who showed that infinite XOR functions cannot be computed by countable Boolean circuits. This unpublished work constitutes an interesting link between set-theoretic topology and complexity theory. Recall that the celebrated Furst-Saxe-Sipser Theorem shows that (finitary) XOR function cannot be computed by polynomial-size circuits of bounded depth, and according to [12], the solution to the infinitary problem “proved to be useful to direct the search for the solution to the finitary one”.

Bibliographical note The proof presented in this paper has been invented by the first author for a course held by the second author at the University of Warsaw in 2008–2009. The second author used this proof in his talk at the congress on Square of Opposition [11]. We have been aware that the indeterminacy of the infinite XOR games is not surprising per se, as the winning criteria are non-measurable here. After having submitted the first version of this article, we have learned that infinite XOR functions are familiar in descriptive set theory under the name of flip sets (more precisely, XOR functions are characteristic functions of flip sets). More importantly, also the idea of our proof was known to the descriptive set theory community before. In particular Yurii Khomskii presented an essentially the same proof (in terms of flip sets) in his Intensive course on Infinite Games at Sofia University [7]; the origins of the idea can be traced back further, but are hard to fix at this moment [8]. The idea is so natural that could likely be found by more people independently, although we are not aware of any published source except for the Internet publications mentioned above. In any case, we do not claim priority here, but hope the example can be useful, in particular in teaching computer science students. Indeed, it uses only a tiny portion of abstract mathematics, and relies on the XOR function well familiar to programmers. And we would like to present it to the volume dedicated to Victor Selivanov, as it touches one of his favorite topics — teaching of logic.

2 Infinite XOR game Let B = {0, 1}. For two words v, w ∈ Bm , where m ≤ ω, let hd(v, w) = |{i : v i = w i }| be the Hamming distance between v and w. For v, w ∈ B ω , we let v ∼ w iff hd(v, w) < ω.

208 | Eryk Kopczyński and Damian Niwiński Definition 1. An infinite XOR function f : B ω → B is a function with the following property: if hd(w1 , w2 ) = 1 then f (w1 ) = f (w2 ). Theorem 2. There exists an infinite XOR function. Proof. We use the Axiom of Choice. Let S be a set which contains exactly one element from each equivalence class of ∼. For w ∈ B ω , let r(w) be the element of S such that w ∼ r(w). We define f (w) = hd(w, r(w)) mod 2. One easily checks that f is an infinite XOR function.

Remark In fact, there are 2c infinite XOR functions, where c is the cardinality of the set of real numbers. For, observe first that, for S as above, |S| = c, as each equivalence class of ∼ is countable. Then, for each α : S → {0, 1}, we obtain a different infinite XOR function given by f α (w) = (f (w) + α(r(w))) mod 2. Definition 3. Let f be an infinite XOR function. The infinite XOR game G f is played as follows. Player 0 picks a word w0 ∈ B+ . Then, Player 1 picks a word w1 ∈ B+ . Player 0 picks a word w2 ∈ B+ , Player 1 picks a word w3 ∈ B+ , and so on. Thus, we obtain a play which is an infinite sequence of words: w0 w1 , w2 , w3 , . . . Player i wins iff f (w0 w1 w2 w3 . . .) = i. Definition 4. A strategy for player i in G f is a function  + 2k+i (B ) → B+ . S: k∈ω

A play w0 , w1 , w2 , . . . is consistent with S iff w k+1 = S(w0 , w1 , . . . , w k ), for each suitable k (i.e., each move of player i is given by S). S is winning iff Player i wins each play consistent with S. Note that in the above we view (B+ )m as a product B+ × B+ × . . . × B+ (m times) rather than concatenation B+ B+ . . . B+ (m times). Such an identification would restrict the set of strategies, but in fact it would not affect our result. Note that, by definition, (B+ )0 = {∅}. We use the strategy stealing argument to show that no player has a winning strategy in the infinite XOR game. Intuitively, whenever our opponent answers our move v with w, we could have instead changed one bit in v to obtain another word v , and play v w instead of v. This effectively exchanges the roles of the two players, so if our opponent had a winning strategy, we can use it now for ourselves. The precise argument follows.

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Theorem 5. No player has a winning strategy in an infinite XOR game G f . Proof. Let S be a strategy for Player 1 − i. We construct two strategies for Player i, T and T  , such that one of them will win at least one play against S. Consider first i = 0, and let the first move of Player 0 (who starts the game) be T (∅) = 0. Suppose the answer of Player 1 is S(0) = w1 . We let T  (∅) = 1w1 . Now, if S(1w1 ) = w2 then we let T (0, w1 ) = w2 , and if S(0, w1 , w2 ) = w3 , we let T  (1w1 , w2 ) = w3 , and so on. In symbols, we let T  (1w1 , w2 , . . . , w2k )

=

S(0, w1 , . . . , w2k )

T (0, w1 , . . . , w2k+1 )

=

S(1w1 , w2 , . . . , w2k+1 ).

In the figure below, the dashed arrows indicate the “stealing”. Player 0

0

w2

w4

 





Strategy T

 Player 1

 

w1

w3



w5







 

 



 Player 0



 

1w1







w3







   

w5



   Player 1



w2

Strategy T 

 

w4

Note that in the two plays above Player 1 uses his strategy S, but the resulting sequences differ exactly in one bit (actually the bit number 0), hence one of the plays is lost by Player 1. The argument for i = 1 is similar. Let the starting move of Player 0 be S(∅) = w0 . We let T (w0 ) = 0. Now suppose S(w0 , 0) = w1 . We let T  (w0 ) = 1w1 . If S(w0 , 1w1 ) = w2 , we let T (w0 , 0, w1 ) = w2 , and so on, as represented on the figure below.

210 | Eryk Kopczyński and Damian Niwiński Player 0

w0

Player 1

Player 0

0

w0

Player 1

w 1              1w1

w 2    w2

w 3              w3

w 4    w4

w 5               w5

Strategy T

Strategy T 

Analogically as above, Player 0 uses her strategy S, but the resulting sequences differ exactly in one bit (namely the bit number |w0 |), hence this strategy cannot be winning. Hence the game G f is indeed indeterminate. Acknowledgement: We thank Jacques Duparc, Yurii Khomskii, and Henryk Michalewski, as well as the anonymous referees, for many insightful comments.

Bibliography [1] [2]

Andretta, A. Notes on Descriptive Set Theory. June, 2001. Duparc, J., Games and Their Application in Computer Science, Unpublished lecture notes, RWTH Aachen, 2002, http://www-mgi.informatik.rwth-aachen.de/Teaching/GamesSS02/index.html [3] Gale, D., and Stewart, F.M., Infinite games with perfect information., Annals of Mathematics 28 (1953), 245–266. [4] Grädel, E., Positional Determinacy of Infinite Games, In STACS 2004, 4–18. [5] Gurevich, Y., The Logic in Computer Science Column., Bulletin of the EATCS 38, June 1989, 93–100. [6] Kechris, A.S., Classical descriptive set theory. Springer-Verlag, New York, 1995. [7] Khomskii, Y., Intensive course on Infinite Games (lecture notes), Sofia University, 12-19 June 2010, http://www.logic.univie.ac.at/~ykhomski/infinitegames2010/index.html [8] Khomskii, Y., Private communication, August 2012. [9] Moschovakis, Y. N., Descriptive Set Theory, North Holland, 1980. [10] Mycielski, J., Games with Perfect Information, In R.J. Aumann, S. Hart, editors, Handbook of Game Theory, volume 1, North-Holland, 1992, 41–70.

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[11] Niwiński, D., Symmetry and duality in fixed-point calculus (slides)., Square of Opposition, Corté, Corsica, June 17–20, 2010. http://www.square-of-opposition.org/start2.html [12] Sipser, M., On polynomial vs. exponential growth, Unpublished manuscript, Mathematics Department MIT, 1981. [13] Telgarsky, R., Topological games: On the 50th anniversary of Banach-Mazur games, Rocky Mountain Journal of Mathematics 17, 1987, 227–276. [14] Zermelo, E., Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels, Proceedings of the Fifth Congress of Mathematicians, Cambridge 1912, 501–504.

Luca Motto Ros, Philipp Schlicht

Lipschitz and Uniformly Continuous Reducibilities on Ultrametric Polish Spaces¹ Abstract: We analyze the reducibilities induced by, respectively, uniformly continuous, Lipschitz, and nonexpansive functions on arbitrary ultrametric Polish spaces, and determine whether under suitable set-theoretical assumptions the induced degree-structures are well-behaved. Keywords: Wadge reducibility, continuous reducibility, Lipschitz reducibility, uniformly continuous reducibility, ultrametric Polish space, nonexpansive function, Lipschitz function, uniformly continuous function Mathematics Subject Classification 2010: 03E15, 03E60, 54C10, 54E40 || Luca Motto Ros: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Germany Philipp Schlicht: Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Germany

1 Introduction Throughout the paper, we work in the usual Zermelo-Frænkel set theory ZF, plus the Axiom of Dependent Choices over the reals DC(R). Let X be a Polish space, and let F be a reducibility (on X), that is a collection of functions from X to itself closed under composition and containing the identity id = idX . Given A, B ⊆ X, we say that A is reducible to B if and only if A = f −1 (B) for some f : X → X, and that A is F-reducible to B (A ≤F B in symbols) if A is reducible to B via a function in F. Notice that clearly A ≤F B ⇐⇒ ¬A ≤F ¬B (where, to simplify the notation, we set ¬A = X \ A whenever the underlying space X is clear from the context). Since F is a reducibility on X, the relation ≤F is a preorder which can be used to measure the “complexity” of subsets of X: in fact, if F consists of reasonably simple functions, the assertion “A ≤F B” may be understood as “the 1 The authors would like to congratulate Professor Victor Selivanov on the occasion of his sixtieth birthday for his wide and important contributions to mathematical logic and, in particular, to the theory of Wadge-like reducibilities and its connections with theoretical computer science.

214 | Luca Motto Ros and Philipp Schlicht set A is not more complicated than the set B” — to test whether a given x ∈ X belongs to A or not, it is enough to pick a witness f ∈ F of A ≤F B, and then check whether f (x) ∈ B or not. This suggests that the reducibility F may be used to form a hierarchy of subsets of X in the following way. Say that A, B ⊆ X are F-equivalent (A ≡F B in symbols) if A ≤F B ≤F A. Since ≡F is the equivalence relation canonically induced by ≤F , we can consider the F-degree [A]F = {B ⊆ X | A ≡F B} of a given A ⊆ X, and then order the collection Deg(F) = {[A]F | A ⊆ X } of such F-degrees using the quotient of ≤F , namely setting [A]F ≤ [B]F ⇐⇒ A ≤F B for every A, B ⊆ X. The resulting structure Deg(F) = (Deg(F), ≤) is then called F-hierarchy on X. When considering the restriction DegΓ (F) of such structure to the F-degrees of sets in a given Γ ⊆ P(X ), we speak of F-hierarchy on Γ-subsets of X. In his Ph.D. thesis [24], Wadge considered the case when X is the Baire space ω ω (i.e. the space of all ω-sequences of natural numbers endowed with the product of the discrete topology on ω) and F is either the set W = W(X ) of all con¯ ) of all functions which are nonexpansive with tinuous functions, or the set L(d ¯ respect to the usual metric d on ω ω (see Section 2 for the definition). Using gametheoretical methods, he was able to show that in both cases the F-hierarchy on Borel subsets of X = ω ω is semi-well-ordered, that is: (1) it is semi-linearly ordered, i.e. either A ≤F B or ¬B ≤F A for all Borel A, B ⊆ X; (2) it is well-founded. Notice that the Semi-Linear Ordering principle for F (briefly: SLOF ) defined in (1) implies that antichains have size at most 2, and that they are of the form {[A]F , [¬A]F } for some A ⊆ X such that A F ¬A (sets with this last property are called F-nonselfdual, while the other ones are called F-selfdual: since Fselfduality is ≡F -invariant, a similar terminology will be applied to the F-degree of A as well). This in particular means that if we further identify each F-degree [A]F with its dual [¬A]F we get a linear ordering, which is also well-founded when (2) holds. A semi-well-ordered hierarchy is practically optimal as a measure of complexity for (Borel subsets of) X: by well-foundness, we can associate to each A ⊆ X an ordinal rank (the F-rank of A), and antichains are of minimal size.² In fact,

2 Asking for no antichain at all seems unreasonable by the following considerations: let A be e.g. a proper open subset of a given Polish space X. On the one hand, checking membership in A cannot be considered strictly simpler or strictly more difficult than checking membership in its complement: this means that the degrees of A and ¬A cannot be one strictly below the other in the

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in [16; 15] it is proposed to classify arbitrary F-hierarchies on corresponding topological spaces X according to whether they provide a good measure of complexity for subsets of X. This led to the following definition. Definition 1. Let F be a reducibility on a (topological) space X, and let Γ ⊆ P(X ). The F-hierarchy DegΓ (F) on Γ-subsets of X is called: – very good if it is semi-well-ordered; – good if it is a well-quasi-order, i.e. all its antichains and descending chains are finite; – bad if it contains infinite antichains; – very bad if it contains both infinite antichains and infinite descending chains. Since the pioneering work of Wadge, many other F-hierarchies on the Baire space ω ω (or, more generally, on zero-dimensional Polish space) have been considered in the literature [23; 1; 2; 11; 13; 14], including Borel functions, ∆0α -functions,³ Lipschitz functions, uniformly continuous functions, functions of Baire class < α for a given additively closed countable ordinal α, Σ 1n -measurable functions, and so on. It turned out that all of them are very good when restricted to Borel sets, or even to larger collections of subsets of ω ω if suitable determinacy principles are assumed. In contrast, it is shown in [6; 7; 9; 18; 16] that when considering the continuous reducibility on the real line R or, more generally, on arbitrary Polish spaces with nonzero dimension, then one usually gets a (very) bad hierarchy (and the same applies to some other classical kind of reducibilities, depending on the space under consideration).⁴ Given all these results, one may be tempted to conjecture that all “natural” F-hierarchies on (Borel subsets of) a zero-dimensional Polish space X need to be very good. This conjecture is justified by the fact that every such space is homeomorphic to a closed subset (hence to a topological retract) of the Baire space, and

hierarchy. On the other hand, the fact that open sets and closed sets have in general different (often complementary) combinatorial and topological properties, strongly suggests that the degrees of A and ¬A should be kept distinct. Therefore such degrees must form an antichain of size 2. 3 Given a countable ordinal α ≥ 1 and a Polish space X, a function f : X → X is called ∆0α -function if f −1 (A) ∈ Σ 0α for every A ∈ Σ0α . 4 Of course, one can further extend the class of topological spaces under consideration, and analyze e.g. the continuous reducibility on them: for example, [19] considers the case of ω-algebraic domains (a class of spaces relevant in theoretical computer science), while [16] consider the broader class of the so-called quasi-Polish spaces. Moreover, it is possible to generalize the notion of reducibility itself by considering e.g. reducibilities between finite partitions (see e.g. [22; 6; 19; 20; 21] and the references contained therein).

216 | Luca Motto Ros and Philipp Schlicht a well-known transfer argument (see e.g. [16, Proposition 5.4]) shows that this already implies the following folklore result. Proposition 2. Let X be a zero-dimensional Polish space, and let F be an arbitrary reducibility on X which contains W(X ), i.e. all continuous functions from X to itself. Then the F-hierarchy Deg∆11 (F) on Borel subsets of X is very good. In fact, [11, Theorem 3.1] (essentially) shows that this result can be further strength¯ ened when X itself is a closed subset of ω ω: if X is equipped with the restriction d X ω ¯ of the canonical metric d on ω, then Deg∆11 (F) is very good as soon as F contains ¯ -nonexpansive functions. ¯ ) of all d the collection L(d X X Despite the above mentioned results, in [15, Theorem 5.4, Proposition 5.10, and Theorem 5.11] it is shown that there are various natural reducibilities on ω ω that actually induce (very) bad hierarchies on its Borel subsets. In particular, it is shown that ω ω can be equipped with a complete ultrametric d , still compatible with its usual product topology, such that the F-hierarchy on Borel (in fact, even just clopen) subsets of ω ω is very bad for F the collection of all the d -nonexpansive (alternatively: d -Lipschitz) functions. Motivated by these results, in the present paper we continue this investigation by considering various complete ultrametrics on ω ω (compatible with its product topology) and, more generally, the collection of all ultrametric Polish spaces X = (X, d), a very natural and interesting class which includes e.g. the space Qp of p-adic numbers (for every prime p ∈ N).⁵ On such spaces, we then consider the hierarchies of degrees induced by one of the following reducibilities⁶ on X: – the collection L(d) of all nonexpansive functions, where f : X → X is called nonexpansive if d(f (x), f (y)) ≤ d(x, y) for all x, y ∈ X; – the collection Lip(d) of all Lipschitz functions (with arbitrary constants), where f : X → X is a Lipschitz function with constant L (for a nonnegative real L) if d(f (x), f (y)) ≤ L · d(x, y) for all x, y ∈ X; – the collection UCont(d) of all uniformly continuous functions, where f : X → X is uniformly continuous if for every ε ∈ R+ there is a δ ∈ R+ such that d(x, y) < δ ⇒ d(f (x), f (y)) < ε for all x, y ∈ X (here R+ denotes the set of strictly positive reals).

5 More generally, the completion of any countable valued field K with valuation | · |K : K → R and metric d(x, y) = |x − y|K (for x, y ∈ K) is always an ultrametric Polish space. 6 Notice that since the metric topology on X is always zero-dimensional, it does not make much sense to consider reducibilities F ⊇ W(X), because by Proposition 2 they always induce a very good hierarchy on Borel subsets of X.

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The main results of the paper are the following: (A) The UCont(d)-hierarchy on Borel subsets of X is always very good (Theorem 16). Since by Proposition 10 it is possible to equip the Baire space with a ¯ is ¯ ) ⊆ UCont(d ) (where d compatible complete ultrametric d such that L(d ω ¯ the usual metric on ω), this also implies that L(d) ⊆ F is a sufficient but not necessary condition for the F-hierarchy on Borel subsets of ω ω being very good (for F a reducibility on ω ω). (B) If X is perfect, then the Lip(d)-hierarchy on the Borel subsets of X is either very good (if X has bounded diameter), or else it is very bad already when restricted to clopen subsets of X (if the diameter of X is unbounded). A technical strengthening of the property of having (un)bounded diameter (see Definition 17) works similarly for arbitrary ultrametric Polish spaces (Theorems 20 and 26, Corollary 29). (C) If the range of d contains an honest increasing sequence (see Definition 31), then the L(d)-hierarchy on clopen subsets of X is very bad (Theorem 32); in particular, this happens in the special case when X is perfect and has unbounded diameter. If instead the range of d is either finite or a decreasing ω-sequence converging to 0, then the L(d)-hierarchy on Borel subsets of X is always very good (Theorem 37). (D) It follows from the second part of (C) that if X is compact, then both⁷ the Lip(d)and the L(d)-hierarchy on Borel subsets of X are very good (Theorem 47). (E) If we assume the Axiom of Choice AC, then the F-hierarchy on (arbitrary subsets of) an uncountable X is very bad for every reducibility F such that L(d) ⊆ F ⊆ Bor(X ), where Bor(X ) is the collection of all Borel functions from X into itself (Theorem 56). If we further assume that V = L, then the F-hierarchy on X is very bad already when restricted to Π 11 , i.e. coanalytic,⁸ subsets of X (Theorem 63). In particular, the results in (A)–(D) generalize those from [15, Section 5] and answer most of the questions in [15, Section 6]. Moreover, they allow us to construct discrete ultrametric Polish spaces X = (X, d) whose Lip(d)- and L(d)-hierarchies are very bad (Corollaries 25 and 33), a fact which contradicts the conceivable conjecture that the Lip(d)- and the L(d)-hierarchy on them need to be (very) good since all subsets of such spaces are extremely simple (i.e. clopen). Notice also

7 Since on compact metric spaces continuity and uniform continuity coincide, the UCont(d)hierarchy on Borel subsets of a compact X is very good already by Proposition 2. 8 Equivalently, to Σ11 (i.e. analytic) subsets of X.

218 | Luca Motto Ros and Philipp Schlicht that the result mentioned in (E) under the assumption V = L (which is best possible for most reducibilities F by Proposition 2 and the comment following it) can be viewed as an extension of the well-know classical result that if Π 11 -determinacy fails then there are proper Π 11 sets which are not (Borel-)complete for coanalytic sets. We end this introduction with two general remarks concerning the results presented in this paper: i) to simplify the presentation, we will consider only F-hierarchies on Borel subsets of a given ultrametric Polish space X (except in Section 6): this is because in this way we can avoid to assume any axiom beyond our basic theory ZF + DC(R). However, as usual in Wadge theory, all our results can be extended to larger pointclasses Γ ⊆ P(X ) by assuming corresponding determinacy axioms (more precisely: the determinacy of subsets of ω ω which are Boolean combinations of sets in Γ). In particular, under the full Axiom of Determinacy AD (asserting that all games on ω are determined), all these results remain true when considering unrestricted F-hierarchies Deg(F) on X; ii) when showing that a given F-hierarchy on X (possibly restricted to some Γ ⊆ P(X )) is very bad, we will actually show that some very complicated partial (quasi-)order on P(ω), like the inclusion relation ⊆, or even the more complicated relation ⊆∗ of inclusion modulo finite sets, embeds into such a hierarchy. This gives much stronger results, as it implies e.g. that the F-hierarchy under consideration contains antichains of size the continuum and, in the case of ⊆∗ , that (under AC) every partial order of size ℵ1 embeds into the Fhierarchy on (Γ-subsets of) X (see [17]).

2 Basic facts about ultrametric Polish spaces Given a metric space X = (X, d), we denote by τ d the metric topology (induced by d), i.e. the topology generated by the basic open balls B d (x, ε) = {y ∈ X | d(x, y) < ε} (for some x ∈ X and ε ∈ R+ ). When considered as a topological space, the space X is tacitly endowed with such topology, and therefore we will e.g. say that the metric space X is separable if there is a countable τ d -dense subset of X, and similarly for all other topological notions. The diameter of X is bounded if there is R ∈ R+ such that sup{d(x, y) | x, y ∈ X } ≤ R, and unbounded otherwise. A metric d on a space X is called ultrametric if it satisfies the following strengthening of the triangle inequality, for all x, y, z ∈ X: d(x, z) ≤ max{d(x, y), d(y, z)}.

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Definition 3. An ultrametric Polish space is a separable metric space X = (X, d) such that d is a complete ultrametric. The collection of all ultrametric Polish spaces will be denoted by X . Every (τ d -)closed subspace C of an ultrametric Polish space X = (X, d) will be tacitly equipped with the metric d C = d  C, which is obviously a complete ultrametric compatible with the relative topology on C induced by τ d . When there is no danger of confusion, with a little abuse of notation the metric d C will be sometimes denoted by d again. Notation 4. Given an ultrametric Polish space X = (X, d), we set R(d) = {d(x, y) | x, y ∈ X, x = y}, the set of all nonzero distances realized in X. A typical example of an ultrametric Polish space is obtained by equipping the ¯ defined by Baire space with the usual metric d ¯ (x, y) = d



if x = y

0 2

−n

if n is smallest such that x(n) = y(n) :

¯ is actually an ultrametric generating the prodit is straightforward to check that d ω ¯ ) = {2−n | n ∈ ω}. We will keep denoting uct topology on ω, and obviously R(d ¯ this ultrametric by d throughout the paper. We collect here some easy but useful facts about arbitrary ultrametric (Polish) spaces X = (X, d):

(1) for every x, y, z ∈ X two of the distances d(x, y), d(x, z), d(y, z) are equal, and they are greater than or equal to the third (the “isosceles triangle” rule); (2) for every x, y, z ∈ X, if d(x, z) = d(y, z) then d(x, y) = max{d(x, z), d(y, z)}. In particular, if x, y, z, w ∈ X are such that d(x, z), d(y, w) < d(x, y) then d(z, w) = d(x, y); (3) given a (τ d -)dense set Q ⊆ X, all distances are realized by elements of Q, that is: for every x, y ∈ X there are q, p ∈ Q such that d(x, y) = d(q, p). In particular, if X is separable then R(d) is countable;⁹ (4) for every x ∈ X and r ∈ R+ the open ball B d (x, r) is actually clopen, and B d (y, r) = B d (x, r) for every y ∈ B d (x, r). In particular, the topology τ d is always zero-dimensional, and hence if X is an ultrametric Polish space, then

9 Vice versa, for every countable R ⊆ R+ there is an ultrametric Polish space X = (X, d) such that R(d) = R, for example X = R ∪ {0} with d(x, y) = max{x, y} for distinct x, y ∈ X.

220 | Luca Motto Ros and Philipp Schlicht it is homeomorphic to a closed subset of the Baire space by [10, Theorem 7.8] (see also Lemma 11); (5) given x, y ∈ X and r, s ∈ R+ , the (cl)open balls B d (x, r) and B d (y, s) are either disjoint, or else one of them contains the other. To simplify the terminology, we adapt the definition of family of reducibilities introduced in [16, Definition 5.1] to the restricted context of ultrametric Polish spaces. Definition 5. Let F be a collection of functions between any ultrametric Polish spaces. For X, Y ∈ X , denote by F(X, Y ) the collection of all functions from F with domain X and range included in Y. The collection F is called family of reducibilities (on X ) if: 1. it contains all the identity functions, i.e. idX ∈ F(X, X ) for every X ∈ X ; 2. it is closed under composition, i.e. for every X, Y , Z ∈ X , f ∈ F(X, Y ), and g ∈ F(Y , Z ), the function g ◦ f belongs to F(X, Z ); Examples of family of reducibilities are the collections of all continuous functions, of all uniformly continuous functions, of all Lipschitz functions, and of all nonexpansive functions. Notice also that if F is a family of reducibilities then F(X ) = F(X, X ) is a reducibility on the space X (for every X ∈ X ). The next simple lemma is a minor variation of [16, Proposition 5.4] and can be proved in a similar way. Lemma 6. Let F be a family of reducibilities and X, Y ∈ X . Suppose that there is a surjective f ∈ F(X, Y ) admitting a right inverse g ∈ F(Y , X ). Then there is an embedding from (P(Y ), ≤F(Y ) , ¬) into (P(X ), ≤F(X) , ¬). In particular, if F consists of Borel functions and the F(X )-hierarchy on Borel subsets of X is (very) good, then also the F(Y )-hierarchy on Borel subsets of Y is (very) good. Proof. The map P(Y ) → P(X ) : A → f −1 (A) is the desired embedding.

3 Uniformly continuous and Lipschitz reducibilities In [15, Question 6.2], it is asked whether one can equip the Baire space ω ω with ¯ ) ⊆ UCont(d ), and whether it is a compatible complete ultrametric d so that L(d

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possible to strengthen this last condition to: the UCont(d )-hierarchy on X is (very) bad. We start by answering positively the first part of this question. Notation 7. Given a function ϕ : ω → R+ , we denote by rg(ϕ) the range of ϕ, i.e. rg(ϕ) = {r ∈ R+ | ∃ n ∈ ω (ϕ(n) = r)}. Definition 8. Given a function ϕ : ω → R+ with inf rg(ϕ) > 0, define the metric d ϕ on ω ω by setting for every x, y ∈ ω ω ¯ (x, y). d ϕ (x, y) = max{ϕ(x(0)), ϕ(y(0))} · d

It is not hard to check that each d ϕ is a complete ultrametric compatible with the product topology on ω ω (and that inf rg(ϕ) > 0 is necessary for completeness). Notation 9. Given a natural number i ∈ ω and an ordinal α, we denote by i(α) the constant α-sequence with value i. ¯ ) ⊆ UCont(d ). Proposition 10. Let ϕ : ω → R+ : n → 2n . Then L(d ϕ

Proof. Consider the map f : ω ω → ω ω : n x → 3n x. We show that for every ε, δ ∈ R+ there are x, y ∈ ω ω such that d ϕ (x, y) < δ but d ϕ (f (x), f (y)) > ε. Let 0 = k ∈ ω be such that 2−k < δ. Then for every n ≥ k we get that setting x = n(2n)  0(ω) and y = n(2n)  1(ω) , d ϕ (x, y) = 2n · 2−2n = 2−n ≤ 2−k < δ. However,

d ϕ (f (x), f (y)) = 23n · 2−2n = 2n ,

hence letting n be large enough we get d ϕ (f (x), f (y)) > ε, as desired. In order to answer the second half of [15, Question 6.2], we abstractly analyze the behavior of the UCont(d)-hierarchy on an arbitrary ultrametric Polish space X = (X, d). The following lemma uses standard arguments (see e.g. the proof of [10, Theorem 7.8]), but we fully reprove it here for the reader’s convenience. Lemma 11. Let X = (X, d) be an ultrametric Polish space. Then there is a closed set ¯ ) → (X, d) such that f is uniformly continuous and C ⊆ ω ω and a bijection f : (C, d −1 f is nonexpansive. Moreover, if X has bounded diameter, then f is even Lipschitz, and if X has diameter ≤ 1 then we can alternatively require f to be nonexpansive and f −1 to be Lipschitz with constant 2. Proof. Let Q be a countable dense subset of X. Define the sets A s ⊆ X for s ∈ k such that both x and y are not ε-isolated.¹¹ Notice that if X is perfect, then the diameter of X is nontrivially unbounded if and only if it is unbounded. Example 18. Let p be a prime natural number, and let Qp be the ultrametric Polish space of p-adic numbers equipped with the usual p-adic metric d p : then Qp has unbounded diameter and is perfect (hence its diameter is nontrivially unbounded). To see the former, given k ∈ ω let n ∈ ω be such that n ≥ 2 and k < p n : setting x = p−1 and y = p−n we easily get d p (x, y) = p n > k. To see that Qp is also perfect, fix an arbitrary q ∈ Q, and given ε ∈ R+ let l ∈ ω be such that p−l < ε: then q = q − p l is distinct from q and d p (q, q ) = p−l < ε. This shows that q is not isolated, and since Q is dense in Qp we are done.

10 When working in models of AD (as it is often the case when dealing with Wadge-like hierarchies), for technical reasons it is often preferable to express “cardinality inequality” using surjections instead of injections. Therefore the stated property should be intended (in any model of ZF) as: the cardinality of UCont(d ϕ ) is not larger than that of the Baire space. Obviously, further assuming the Axiom of Choice AC this just means that UCont(d ϕ ) has cardinality ≤ 2ℵ0 . 11 Recall that a point x of a metric space is called ε-isolated (for some ε ∈ R+ ) if B d (x, ε) = {x}.

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Notation 19. We let ⊆∗ denote the relation of inclusion modulo finite sets between subsets of ω, i.e. for every a, b ⊆ ω we set a ⊆∗ b ⇐⇒ ∃k¯ ∈ ω ∀k ≥ k¯ (k ∈ a ⇒ k ∈ b). Theorem 20. Let X = (X, d) be an ultrametric Polish space, and assume that its diameter is nontrivially unbounded. Then there is a map ψ from P(ω) into the clopen subsets of X such that for all a, b ⊆ ω: 1. if a ⊆∗ b then ψ(a) ≤L(d) ψ(b); 2. if ψ(a) ≤Lip(d) ψ(b) then a ⊆∗ b. In particular, (P(ω), ⊆∗ ) embeds into both Deg∆01 (Lip(d)) and Deg∆01 (L(d)). Proof. Let (q n )n∈ω be an enumeration of a countable dense subset Q of X. We first recursively construct two sequences (r n )n∈ω , (s n )n∈ω of nonnegative reals and two sequences (x n )n∈ω , (y n )n∈ω of points of X such that for all distinct n, m ∈ ω the following properties hold: (a) d(x n , x m ) = rmax{n,m} and d(x n , y n ) = s n ; (b) r n+1 > max{n + 1, r2n } (in particular, (r n )n∈ω is strictly increasing and unbounded in R+ ); n (c) s0 < 1 and s n+1 < r ns+1 (in particular, (s n )n∈ω is a strictly decreasing sequence). Claim 21. If x ∈ X is not ε-isolated then there are at least two distinct q i , q j ∈ Q such that q i , q j ∈ B d (x, ε).

Proof of the Claim. Since x is not ε-isolated, there is y ∈ B d (x, ε) such that x = y. By density of Q, there are q i , q j ∈ Q such that q i ∈ B d (x, d(x, y)) and q j ∈ B d (y, d(x, y)). Then q i = q j since B d (x, d(x, y))∩ B d (y, d(x, y)) = ∅, while q i , q j ∈ B d (x, ε) because B d (x, d(x, y)), B d (y, d(x, y)) ⊆ B d (x, ε) by d(x, y) < ε.

Let x ∈ X be not 1-isolated (such an x exists because the diameter of X is nontrivially unbounded), and let q i , q j be as in Claim 21 for ε = 1. Then we set x0 = q i , y0 = q j , r0 = 0, and s0 = d(q i , q j ). Now assume that x n , y n , r n , and s n have been n defined. Let x, y ∈ X be such that d(x, y) > max{n + 1, r2n } and x, y are not r ns+1 isolated. Then at least one of x and y has distance greater than max{n + 1, r2n } from x n (and hence also from all the x m for m ≤ n): if not, then we would have d(x, y) ≤ max{d(x, x n ), d(y, x n )} ≤ max{n + 1, r2n }, contradicting our choice of x, y. So we may assume without loss of generality that d(x, x n ) > max{n + 1, r2n } n n and x is not r ns+1 -isolated. Let q i , q j be as in Claim 21 for ε = r ns+1 , and set x n+1 = n q i , y n+1 = q j , r n+1 = d(q i , x n ), and s n+1 = d(q i , q j ). Since d(q i , x) < r ns+1 ≤1≤

226 | Luca Motto Ros and Philipp Schlicht max{n + 1, r2n }, we have r n+1 = d(q i , x n ) = d(x, x n ) > max{n + 1, r2n }. Moreover, n n s n+1 < r ns+1 by the fact that q i , q j ∈ B d (x, r ns+1 ). Arguing by induction on n ∈ ω,

it is then easy to check that the sequences constructed in this way have all the desired properties. ˆ = {2i | i ∈ ω} ∪ {2i + 1 | i ∈ a}, so that a ˆ is always infinite Given a ⊆ ω, let a and for every a, b ⊆ ω ˆ ˆ ⊆∗ b. a ⊆∗ b ⇐⇒ a  For a ⊆ ω, set ψ(a) = i∈ˆa B d (x i , s i ). Clearly, each ψ(a) is an open subset of X. To see that it is also closed, observe that B d (x i , s i ) ⊆ B d (x i , 1) for every i ∈ ω by our choice of the s i ’s, and that for distinct i, j ∈ ω the clopen balls B d (x i , 1) and B d (x j , 1) are disjoint by our choice of the x i ’s and of the r i ’s: therefore, since the open balls in X are automatically closed we get that    X \ ψ(a) = B d (x i , 1) B d (z, 1) | z ∈/ i∈ˆ a  ˆ} ∪ {B d (x i , 1) \ B d (x i , s i ) | i ∈ a is open. ˆ ˆ ⊆∗ b, Let now a, b ⊆ ω be such that a ⊆∗ b, which in particular implies a ¯ ¯ ˆ ¯ ˆ and k ∈ a ˆ ⇒ k ∈ b for every k ≥ k. Define and let 0 = k ∈ ω be such that k ∈ a f : (X, d) → (X, d) as follows:  ¯  ˆ  x¯k if x ∈ B d (x i , s i ), i < k, i ∈ a    y if x ∈ B (x , r ) \ {B (x , s ) | i < k, ¯ i∈a ˆ} ¯ i d 0 ¯ d i k k f (x) = ¯  ˆ y i if x ∈ B d (x i , s i ), i ≥ k, i ∈/ a     x otherwise.

It is straightforward to check that f reduces ψ(a) to ψ(b), so we only need to check that f is nonexpansive, and this amounts to check that if x, y are distinct points of X which fall in different cases in the definition of f , then d(f (x), f (y)) ≤ d(x, y). A careful inspection shows that the unique nontrivial cases are the following:  ¯ i∈ ˆ }. Then case A: x ∈ B d (x0 , r¯k ), while y ∈/ B d (x0 , r¯k ) ∪ {B d (x i , s i ) | i ≥ k, / a d(x, y) ≥ r¯k (by case assumption) and d(x, f (x)) = r¯k (because either f (x) = x¯k or f (x) = y¯k , depending on whether x ∈ B d (x i , s i ) for some i ∈ ¯ or not). Since in the case under consideration f (y) = y, ˆ smaller than k a we get that either d(f (x), f (y)) ≤ r¯k , or else d(f (x), f (y)) = d(f (x), y) = d(x, y) by the isosceles triangle rule: in both cases, d(f (x), f (y)) ≤ d(x, y) as required.  ¯ i∈a ˆ }, while y ∈ B d (x i , s i ) for some case B: x ∈ B d (x0 , r¯k ) \ {B d (x i , s i ) | i < k, ¯ ˆ i ≥ k, i ∈/ a. Then since d(x, x0 ) < r¯k and d(x0 , y) = r i ≥ r¯k , we get

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d(x, y) = r i . Since by case assumption f (x) = y¯k and f (y) = y i , either ¯ or d(f (x), f (y)) = r , and hence we again get f (x) = f (y) (in case i = k) i d(f (x), f (y)) ≤ d(x, y), as required. ¯ i ∈ ˆ , while also y ∈ case C: x ∈ B d (x i , s i ) for some i ≥ k, / a / B d (x0 , r¯k )∪  ¯ ˆ }. Then d(x, y) ≥ s i , d(x, f (x)) = s i (because {B d (x i , s i ) | i ≥ k, i ∈ / a f (x) = y i ), and f (y) = y: this implies that either d(f (x), f (y)) ≤ s i or d(f (x), f (y)) = d(f (x), y) = d(x, y), so that in any case d(f (x), f (y)) ≤ d(x, y). This concludes the proof of part (1). We now prove part (2) of the theorem. Given a, b ⊆ ω, assume that f : (X, d) → (X, d) is a Lip(d)-reduction of ψ(a) to ψ(b), and let 0 = n ∈ ω be such that d(f (x), f (y)) ≤ r n · d(x, y) for every x, y ∈ X (such an n exists because (r n )n∈ω is unbounded in R+ by (b) above). Notice that, necessarily,    ˆ } = f (ψ(a)) ⊆ ψ(b) ⊆ {B d (x i , s i ) | i ∈ a B d (x j , s j ). f j∈ω

We now argue as in the proof of [15, Theorem 5.4]. ˆ . If there are x ∈ B d (x i , s i ) and j ≥ n such that Claim 22. Fix an arbitrary i ∈ a f (x) ∈ B d (x j , s j ), then f (B d (x i , s i )) ⊆ B d (x j , s j ).

Proof of the Claim. Suppose not, and let y ∈ B d (x i , s i ) and j = j be such that f (y) ∈ B d (x j , s j ). Then d(f (x), f (y)) = max{r j , r j } ≥ r j ≥ r n · 1 > r n · s i > r n · d(x, y),

contradicting the choice of n. ˆ such that i > n, f (B d (x i , s i )) ⊆ B d (x j , s j ) for some j ≥ i. Claim 23. For every i ∈ a

Proof. Suppose towards a contradiction that there are x ∈ B d (x i , s i ) and j < i such ˆ because x ∈ ψ(a) and f reduces that f (x) ∈ B d (x j , s j ), so that, in particular, j ∈ b ψ(a) to ψ(b). Then since d(x, y i ) = s i , by our choice of the s i ’s we get d(f (x), f (y i )) ≤ r n · s i ≤ r i−1 · s i < s i−1 ≤ s j , and hence f (y i ) ∈ B d (f (x), s j ) = B d (x j , s j ) ⊆ ψ(b): but this contradicts the fact that f is a reduction of ψ(a) to ψ(b), because y i ∈/ ψ(a) while B d (x j , s j ) ⊆ ψ(b) ˆ Thus, given an arbitrary x ∈ B (x , s ) there is j ≥ i > n such that since j ∈ b. i d i f (x) ∈ B d (x j , s j ): by Claim 22, we then get f (B d (x i , s i )) ⊆ B d (x j , s j ), as required.

228 | Luca Motto Ros and Philipp Schlicht ˆ . By Claim 22, either f (B d (x¯ı , s¯ı )) ⊆ Let now ¯ı be the smallest element of a  B ( x , s ) , or f ( B ( x s )) ⊆ B ( x , s ¯ ¯ ı ı j j ) for some j ≥ n. Therefore, in both d d j j max{n, ¯ı} such that f (B d (x¯ı , s¯ı )) ⊆ j≤¯k B d (x j , s j ): we claim ˆ for every k ≥ k, ¯ which also implies a ⊆∗ b. ˆ⇒k∈b that k ∈ a ¯ > n, there is j ≥ k such that ˆ . By Claim 23 and k Fix k ≥ k¯ such that k ∈ a f (B d (x k , s k )) ⊆ B d (x j , s j ). Assume towards a contradiction that j > k: then

d(f (x¯ı ), f (x k )) = r j > r k · r k > r n · r k = r n · d(x¯ı , x k ), contradicting the choice of n. Therefore f (B d (x k , s k )) ⊆ B d (x k , s k ), which in particular implies that ψ(b) ∩ B d (x k , s k ) = ∅ (since x k ∈ ψ(a) and f reduces ψ(a) to ˆ and hence we are done. ψ(b)): but this means that k ∈ b,

Applying Theorem 20 to the space Qp of p-adic numbers (which is possible by Example 18) we get the following corollary. Corollary 24. Let p be a prime natural number, and let d p be the p-adic metric on the space Qp . Then both the Lip(d p )- and the L(d p )-hierarchies are very bad already when restricted to clopen subsets of Qp .

The condition on the diameter of X = (X, d) used to prove Theorem 20 is very weak: this allows us to construct extremely simple (in fact: discrete) ultrametric Polish spaces X = (X, d) with the property that their Lip(d)- and L(d)-hierarchies are both very bad, despite the fact that all their subsets are topologically simple (i.e. clopen). Corollary 25. There exists a discrete (hence countable) ultrametric Polish space X0 = (X0 , d0 ) such that (P(ω), ⊆∗ ) embeds into both the Lip(d0 )- and the L(d0 )-hierarchy on (the clopen subsets of) X0 . In particular, Deg(Lip(d0 )) = Deg∆01 (Lip(d0 )) and Deg(L(d0 )) = Deg∆01 (L(d0 )) are both very bad. Proof. Let X0 = {x in | n ∈ ω, i = 0, 1} and set   if n = m and i = j  0 j i d 0 ( x n , x m ) = 2− n if n = m and i = j   max{n, m} if n =  m.

It is easy to check that X0 = (X0 , d0 ) is a discrete ultrametric Polish space. Now observe that the diameter of X0 is nontrivially unbounded. In fact, given n ∈ ω and ε ∈ R+ , let k be minimal such that 2−k < ε and l = max{n, k}: then d0 (x0l , x0l+1 ) = l + 1 > n, and the points x1l and x1l+1 witness that x0l and x0l+1 are not ε-isolated. Therefore X0 is as desired by Theorem 20.

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The next proposition extends Theorem 16 and shows that the condition on X in Theorem 20 is optimal. Theorem 26. Let X = (X, d) be an ultrametric Polish space whose diameter is not nontrivially unbounded. Then the Lip(d)-hierarchy Deg∆11 (Lip(d)) on Borel subsets of X is very good. Proof. Let n ∈ ω and ε ∈ R+ be such that for every x, y, if d(x, y) > n then at least one of x and y is ε-isolated. Let us first consider the degenerate case in which all points of X are ε-isolated. Since constant functions are always (trivially) Lipschitz, we get that the sets X and ∅ are Lip(d)-incomparable, and that they are both (strictly) ≤Lip(d) -below any other set ∅, X = A ⊆ X. Assume now that B ⊆ X is another set which is different from both ∅ and X: we claim that then A ≡Lip(d) B. To see this, fix ¯x ∈ B and ¯ y ∈ ¬B, and for every x ∈ X set f (x) = ¯x if x ∈ A and f (x) = ¯y if x ∈ ¬A. Then f : (X, d) → (X, d) reduces A to B. Moreover, since for all distinct x, y ∈ X we have d(x, y) ≥ ε (because both x and y are ε-isolated), we get d(f (x), f (y)) ≤ d(¯x , ¯y) =

d(¯x , ¯y) d(¯x , ¯y) ·ε≤ · d(x, y), ε ε

so that f is Lipschitz with constant d(¯xε, y) . This shows that A ≤Lip(d) B. Switching the role of A and B, we get that also B ≤Lip(d) A, and hence we are done. Therefore we have shown that the Lip(d)-hierarchy on X is constituted by the two Lip(d)incomparable degrees [∅]Lip(d) = {∅} and [X ]Lip(d) = {X }, plus a unique Lip(d)degree above them containing all other subsets of X, and is thus (trivially) very good. Assume now that there is a non-ε-isolated point x0 ∈ X, and set X  = B d (x0 , n + 1). By our choice of n and ε, we get that d(x, y) ≥ n + 1 for every x ∈ X  and y ∈ X \ X  , and that each y ∈ X \ X  is ε-isolated (because d(x0 , y) > n and x0 is not ε-isolated). We first prove the following useful claim. ¯

Claim 27. Let A, B ⊆ X be such that B = ∅, X. If there is a Lipschitz reduction f : (X  , d X ) → (X  , d X ) of A = A ∩ X  to B = B ∩ X  , then A ≤Lip(d) B.

Proof. Let f be as in the hypothesis of the claim, and let 1 ≤ k ∈ ω be such that d(f (x), f (y)) ≤ k · d(x, y) for every x, y ∈ X  . Fix ¯x ∈ B and ¯y ∈ ¬B, and extend f to the map ˆf : (X, d) → (X, d) by letting ˆf (x) = ¯x if x ∈ A \ X  and ˆf (x) = ¯y if x ∈ X \ (X  ∪ A). Clearly, ˆf reduces A to B, and we claim that ˆf is Lipschitz with constant c, where c is   d(¯x , ¯y) d(x0 , ¯x) d(x0 , ¯y) , c = max k, , . ε n+1 n+1

230 | Luca Motto Ros and Philipp Schlicht Fix arbitrary x, y ∈ X. If x, y ∈ X  , then d(ˆf (x), ˆf (y)) = d(f (x), f (y)) ≤ k · d(x, y) ≤ c · d(x, y) by our choice of k ∈ ω. If x, y ∈ X \ X  , then d(x, y) ≥ ε because both x and y are ε-isolated, and either ˆf (x) = ˆf (y) or d(ˆf (x), ˆf (y)) = d(¯x , ¯y). Therefore in both cases d(¯x , ¯y) d(ˆf (x), ˆf (y)) ≤ · ε ≤ c · d(x, y). ε Let now x ∈ X  and y ∈ X \ X  , and assume without loss of generality that ˆf (y) = ¯x (the case ˆf (y) = ¯y is analogous, just systematically replace ¯x with ¯y in the argument below). Then either ¯x ∈ X  , in which case d(ˆf (x), ˆf (y)) < n + 1 ≤ d(x, y) ≤ c · d(x, y) (since c ≥ k ≥ 1), or else d(ˆf (x), ˆf (y)) = d(x0 , ¯x) =

d(x0 , ¯x) · n + 1 ≤ c · d(x, y). n+1

The case x ∈ X \ X  and y ∈ X  can be treated similarly, so in all cases we obtained d(ˆf (x), ˆf (y)) ≤ c · d(x, y), as required.

We now want to show that the SLOLip(d) principle holds for Borel subsets of X, so let us fix arbitrary Borel A, B ⊆ X. Assume first that B = X. Then either A = X, in which case the identity map on X witnesses A ≤Lip(d) B, or else ¬A = ∅, in which case any constant map with value ¯x ∈ ¬A witnesses B ≤Lip(d) ¬A. The symmetric case B = ∅ can be dealt with in a similar way, so in what follows we can assume without loss of generality that B = ∅, X. Moreover, switching the role of A and B in the argument above we may further assume that A = ∅, X. Set A = A ∩ X  and B = B ∩ X  . Since X  has bounded diameter, by Theorem 16 there is a Lipschitz function f : (X  , d) → (X  , d) such that either f −1 (B ) = A or f −1 (X  \ A ) = B . Since ¬A ∩ X  = X  \ A , applying Claim 27 we get that either A ≤Lip(d) B or B ≤Lip(d) ¬A, as desired. Finally, let us show that the Lip(d)-hierarchy on Borel subsets of X is also wellfounded. Suppose not, and let (A n )n∈ω be a sequence of Borel subsets of X such that A n+1 0. Then (P(ω), ⊆∗ ) embeds into both the Lip(d ϕ )- and L(d ϕ )-hierarchy on clopen subsets of ω ω, and therefore both Deg∆01 (Lip(d ϕ )) and Deg∆01 (L(d ϕ )) are very bad. Conversely, if ϕ has bounded range, then the Lip(d ϕ )-hierarchy Deg∆11 (Lip(d ϕ )) on Borel subsets of ω ω is very good. Proof. Observe that (ω ω, d ϕ ) is a perfect ultrametric Polish space, and that it has unbounded diameter if and only if the rg(ϕ) is unbounded in R+ ; then apply Theorems 20 and 16.

4 Nonexpansive reducibilities Definition 31. Let X = (X, d) be an ultrametric Polish space. We say that R(d) contains an honest increasing sequence if it contains a strictly increasing sequence (r n )n∈ω such that for some sequences (x n )n∈ω , (y n )n∈ω of points in X the following conditions holds: (i) d(x n , x m ) = rmax{n,m} for all distinct n, m ∈ ω; (ii) d(x0 , y0 ) < r0 and d(x n+1 , y n+1 ) < d(x n , y n ) for all n ∈ ω.

232 | Luca Motto Ros and Philipp Schlicht The above condition is somewhat technical, but in case X = (X, d) is a perfect ultrametric Polish space it is immediate to check that R(d) contains an honest increasing sequence if and only if one of the following equivalent¹² conditions are satisfied: 1. there is X  ⊆ X such that R(d X ) has order type ω (with respect to the usual ordering on R); 2. there is a sequence (x n )n∈ω of points in X and a strictly increasing sequence (r n )n∈ω of distances in R(d) such that d(x n , x m ) = rmax{n,m} for all distinct n, m ∈ ω. Notice also that if the diameter of an ultrametric Polish space X = (X, d) is nontrivially unbounded, then R(d) contains an honest increasing sequence by the first part of the proof of Theorem 20. Theorem 32. Let X = (X, d) be a ultrametric Polish space such that R(d) contains an honest increasing sequence. Then there is a map ψ from P(ω) into the clopen subsets of X such that for all a, b ⊆ ω a ⊆∗ b ⇐⇒ ψ(a) ≤L(d) ψ(b). Proof. Argue similarly to Theorem 20, with the following variations: (a) let the sequences (x n )n∈ω , (y n )n∈ω , and (r n )n∈ω constructed at the beginning of the proof of Theorem 20 be witnesses of the fact that R(d) contains an honest increasing sequence (forgetting about the extra properties required in Theorem 20), and set s n = d(x n , y n );¹³  (b) given a ⊆ ω, define ψ(a) as before, i.e. set ψ(a) = i∈ˆa B d (x i , s i ), where ˆ = {2i | i ∈ ω} ∪ {2i + 1 | i ∈ a}; a (c) to prove the backward direction, use an argument similar to that of Theorem 20, but dropping any reference to the integer n (this simplification can be adopted here because we have to deal only with nonexpansive functions). More precisely: let f be a nonexpansive reduction of ψ(a) to ψ(b). Then for evˆ there is a unique j ∈ ω such that f (B d (x i , s i )) ⊆ B d (x j , s j ) (because ery i ∈ a of the choice of the x i , y i ’s and the fact that f is nonexpansive). Arguing as in Claim 23, one immediately sees that we cannot have j < i because in such case

12 To see that these two conditions are indeed equivalent, argue as in the first part of the proof of Theorem 20. 13 Clearly, the points x n and y n can again be chosen in any given countable dense set Q ⊆ X.

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s i ≤ s j . Conclude as in the final part of the proof of Theorem 20, using the fact that r k < r j for every j > k. Corollary 33. There is an ultrametric Polish space X1 = (X1 , d1 ) whose set of nonzero distances R(d1 ) is bounded away from 0 (hence it is countable and discrete) such that (P(ω), ⊆∗ ) embeds into the L(d1 )-hierarchy on (clopen subsets of) X1 . Therefore Deg(L(d1 )) = Deg∆01 (L(d1 )) is very bad. Proof. Let X1 = {x in | n ∈ ω, i = 0, 1} and set    0 j i d1 (x n , x m ) = 12 + 2−(n+1)   2 − 2− max{n,m}

if n = m and i = j if n = m and i = j if n = m.

It is easy to check that X1 = (X1 , d1 ) is an ultrametric Polish space. Moreover r ≥ 12 for every r ∈ R(d1 ), hence R(d1 ) is bounded away from 0. Moreover, the sequences obtained by setting r n = 2 − 2−n , x n = x0n , and y n = x1n witness that R(d1 ) contains an honest increasing sequence. Hence the result follows from Theorem 32. Remark 34. Notice that if an ultrametric Polish space X = (X, d) satisfies the hypothesis of Corollary 33 (i.e. it is such that R(d) is bounded away from 0), then its Lip(d)-hierarchy is always (trivially) very good by Theorem 26 and the fact that all its points are ε-isolated for ε = inf R(d) > 0. Corollary 35. Given ϕ : ω → R+ such that inf rg(ϕ) > 0, if rg(ϕ) contains an increasing ω-sequence then (P(ω), ⊆∗ ) embeds into the L(d ϕ )-hierarchy on clopen subsets of ω ω, and therefore Deg∆01 (L(d ϕ )) is very bad. Proof. Notice that (ω ω, d ϕ ) is always a perfect Polish space, and that R(d ϕ ) has an honest increasing sequence if and only if rg(ϕ) contains an increasing ωsequence. Then apply Theorem 32. Proposition 36. Suppose that X = (X, d) is an ultrametric Polish space such that R(d) is either finite or a descending (ω-)sequence converging to 0, let I ≤ ω be the cardinality of R(d), and let ρ be the unique order-preserving map from {2−i | i < I } and R(d). Then there is a closed set C ⊆ ω ω and a bijection f : C → X such that for all x, y ∈ X ¯ (f −1 (x), f −1 (y))). d(x, y) = ρ(d (∗ )

In particular, the structures (P(X ), ≤L(d) , ¬) and (P(C), ≤L(¯d) , ¬) are isomorphic.

234 | Luca Motto Ros and Philipp Schlicht Proof. Let us first assume that I = ω, i.e. that R(d) is a descending (ω-)sequence converging to 0. Inductively define the family (A s )s∈