Symbolic and Quantitative Approaches to Reasoning and Uncertainty: European Conference, ECSQARU'99, London, UK, July 5-9, 1999, Proceedings (Lecture Notes in Computer Science, 1638) 354066131X, 9783540661313

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Lecture Notes in Artificial Intelligence Subseries of Lecture Notes in Computer Science Edited by J. G. Carbonell and J. Siekmann

Lecture Notes in Computer Science Edited by G. Goos, J. Hartmanis and J. van Leeuwen

1638

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Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo

Anthony Hunter

Simon Parsons (Eds.)

Symbolic and Quantitative Approaches to Reasoning and Uncertainty European Conference, ECSQARU’99 London, UK, July 5-9, 1999 Proceedings

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Series Editors Jaime G. Carbonell, Carnegie Mellon University, Pittsburgh, PA, USA J¨org Siekmann, University of Saarland, Saarbr¨ucken, Germany

Volume Editors Anthony Hunter University College London, Department of Computer Science Gower Street, London WC1E 6BT, UK E-mail: [email protected] Simon Parsons University of London, Queen Mary and Westfield College Department of Electronic Engineering London E1 4NS, UK E-mail: [email protected]

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Symbolic and quantitative approaches to reasoning and uncertainty : European conference ; proceedings / ECSQARU’99, London, UK, July 5 - 9, 1999. Anthony Hunter ; Simon Parsons (ed.). - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 1999 (Lecture notes in computer science ; Vol. 1638 : Lecture notes in artificial intelligence) ISBN 3-540-66131-X

CR Subject Classification (1998): I.2.3, F.4.1 ISBN 3-540-66131-X Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 Printed in Germany Typesetting: Camera-ready by author SPIN: 10703448 06/3142 - 5 4 3 2 1 0 – Printed on acid-free paper

Preface Uncertainty is an increasingly important research topic in many areas of computer science. Many formalisms are being developed, with much interest at the theory level directed at developing a better understanding of the formalisms and identifying relationships between formalisms, and at the technology level directed at developing software tools for formalisms and applications of formalisms. The main European forum for the subject is the European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty (ECSQARU). Following the success of the previous ECSQARU conferences, held in Marseilles (1991), Granada (1993), Fribourg (1995), and Bonn (1997), the fifth conference in the series was held at University College London in July 1999. This volume contains papers accepted for presentation at ECSQARU’99. In addition to the main conference, two workshops were held. The first was on Decision Theoretic and Game Theoretic Agents, chaired by Simon Parsons and Mike Wooldridge, and the second was on Logical and Uncertainty Models for Information Systems, chaired by Fabio Crestani and Mounia Lalmas. Selected papers from the workshops are also included in these proceedings. We are indebited to the programmme committee for their effort in organising the programme, to the invited speakers, and to the presenters of the tutorials. Furthermore, we gratefully acknowledge the contribution of the many referees who were involved in the reviewing process. Finally we would like to thank the Department of Computer Science at University College London for administrative support. Programme Committee The programme committee was chaired by Anthony Hunter (University College London), and comprised Dov Gabbay (King’s College London), Finn Jensen (Aalborg University), Rudolf Kruse (University of Magdeburg), Simon Parsons (Queen Mary, University of London) Henri Prade (IRIT, Toulouse), Torsten Schaub (University of Potsdam), and Philippe Smets (ULB, Bruxelles). Reviewers The programme committee is very grateful for all the hard work contributed by the reviewers. Hopefully, we have not missed anyone from the following list: Bruce D’Ambrosio, Florence Bannay, Salem Benferhat, Philippe Besnard, Hugues Bersini, Christian Borgelt, Rachel Bourne, Stefan Brass, Laurence Cholvy, Roger Cooke, Adnan Darwiche, Yannis Dimopoulos, Jurgen Dix, Didier Dubois, Uwe Egly, Linda van der Gaag, Joerg Gebhardt, Siegfried Gottwald, Rolf Haenni, Jean-Yves Jaffray, Radim Jirousek, Ruth Kempson, Uffe Kjaerulf, Frank Klawonn, Aljoscha Klose, Juerg Kohlas, Paul Krause, Gerhard Lakemeyer, Mounia Lalmas, Jerome Lang, Kim G. Larsen, Norbert Lehmann, T. Y. Lin, Thomas Linke, Khalid Mellouli, Jerome Mengin, J.-J. Ch. Meyer, Sanjay Modgil, Yves Moinard, Serafin Moral, Detlef Nauck, Ann Nicholson, Pascal Nicolas, Dennis

VI

Preface

Nilsson, Kristian G. Olesen, Rainer Palm, Zdzislaw Pawlak, Vincent Risch, Regis Sabbadin, Camilla Schwind, Prakash P. Shenoy, Milan Studeny, Heiko Timm, Hans Tompits, Marco Valtorta, and Cees Witteven.

April 1999

Anthony Hunter and Simon Parsons

Table of Contents

On the dynamics of default reasoning Grigoris Antoniou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Non-monotonic and paraconsistent reasoning: From basic entailments to plausible relations Ofer Arieli and Arnon Avron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 A comparison of systematic and local search algorithms for regular CNF formulas Ram´ on B´ejar and Felip Many` a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Query-answering in prioritized default logic Farid Benhammadi, Pascal Nicolas and Torsten Schaub . . . . . . . . . . . . . . . . . . . . 32 Updating directed belief networks Boutheina Ben Yaghlane and Khaled Mellouli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Inferring causal explanations Philippe Besnard and Marie-Odile Cordier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 A critique of inductive causation Christian Borgelt and Rudolf Kruse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Connecting lexicographic with maximum entropy entailment Rachel A. Bourne and Simon Parsons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Avoiding non-ground variables Stefan Br¨ uning and Torsten Schaub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Anchoring symbols to vision data by fuzzy logic Silvia Coradeschi and Alessandro Saffiotti. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Filtering vs revision and update: Let us debate! Corine Cossart and Catherine Tessier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Irrelevance and independence axioms in quasi-Bayesian theory Fabio G. Cozman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Assessing the value of a candidate: A qualitative possibilistic approach Didier Dubois, Michel Grabisch and Henri Prade . . . . . . . . . . . . . . . . . . . . . . . . . 137

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Table of Contents

Learning default theories B´eatrice Duval and Pascal Nicolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Knowledge representation for inductive learning Peter A. Flach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Handling inconsistency efficiently in the incremental construction of stratified belief bases Eric Gr´egoire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Rough knowledge discovery and applications J. W. Guan and D. A. Bell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Gradient descent training of Bayesian networks Finn V. Jensen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Open default theories over closed domains: An extended abstract Michael Kaminski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Shopbot economics Jeffrey O. Kephart and Amy R. Greenwald . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Optimized algorithm for learning Bayesian network from data F´edia Khalfallah and Khaled Mellouli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Merging with integrity constraints S´ebastien Konieczny and Ram´ on Pino P´erez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Boolean-like interpretation of Sugeno integral Ivan Kramosil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 An alternative to outward propagation for Dempster-Shafer belief functions Norbert Lehmann and Rolf Haenni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 On bottom-up pre-processing techniques for automated default reasoning Thomas Linke and Torsten Schaub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Probabilisitc logic programming under maximum entropy Thomas Lukasiewicz and Gabriele Kern-Isberner . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Lazy propagation and independence of causal influence Anders L. Madsen and Bruce D’Ambrosio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Table of Contents

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A Monte Carlo algorithm for combining Dempster-Shafer belief based on approximate pre-computation Seraf´ın Moral and Antonio Salmer´ on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 An extension of a linguistic negation model allowing us to deny nuanced property combinations Daniel Pacholczyk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Argumentation and qualitative decision making Simon Parsons and Shaw Green . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Handling different forms of uncertainty in regression analysis: A fuzzy belief structure approach Simon Petit-Renaud and Thierry Denœux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 State recognition in discrete dynamical systems using Petri nets and evidence theory Mich`ele Rombaut, Iman Jarkass and Thierry Denœux . . . . . . . . . . . . . . . . . . . . . 352 Robot navigation and map building with the event calculus Murray Shanahan and Mark Witkowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 Information fusion in the context of stock index prediction Stefan Siekmann, J¨ org Gebhardt and Rudolf Kruse . . . . . . . . . . . . . . . . . . . . . . . . 363 Defeasible goals Leendert van der Torre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Logical deduction using the local computation framework Nic Wilson and J´erˆ ome Mengin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

On the Dynamics of Default Reasoning Grigoris Antoniou Griffith University, QLD 4111, Australia University of Macedonia, Thessaloniki, Greece [email protected]

Abstract. Default logic is a prominent rigorous method for reasoning with incomplete information based on assumptions. It is a static reasoning approach, in the sense that it doesn’t reason about changes and their consequences. On the other hand, its nonmonotonic behaviour appears when changes to a default theory are made. This paper studies the dynamic behaviour of default logic in the face of changes. We consider the operations of contraction and revision, present several solutions to these problems, and study their properties.

1

Introduction

Nonmonotonic reasoning [12,2] comprises qualitative techniques for reasoning with incomplete information. Default logic [14] is a specific nonmonotonic reasoning method which is based on the use of default rules representing plausible assumptions. The use of default rules in a knowledge base is beneficial for practical purposes. It is well-recognized that humans maintain multiple sets of beliefs which are often mutually contradictory. The choice of a belief set is then driven by the context. Default logic supports this phenomenon since its semantics is based on alternative extensions of a given default theory. The use of default rules has been proposed for the maintenance of software [4,11] in general, and requirements engineering [3,17] in particular. When the requirements of a software product are collected it is usual that potential stakeholders have conflicting requirements for the system to be developed. As has been shown, default logic representations may adequately model various facets of this problem; in particular they allow for the representation of conflicting requirements within the same model, and they highlight potential conflicts and possible ways of resolving them. These are central problems in the area of requirements engineering [6,9]. One central issue of requirements engineering that cannot be addressed by pure default logic is that of evolving requirements. For example, requirements may evolve because requirements engineers and users cannot possibly foresee all the ways is which a system may be used, or because the software context changes over time. While default logic supports reasoning with incomplete (and contradictory) knowledge it does not provide means for reasoning with change. The purpose of A. Hunter and S. Parsons (Eds.): ECSQARU’99, LNAI 1638, pp. 1–10, 1999. c Springer-Verlag Berlin Heidelberg 1999


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Grigoris Antoniou

this paper is to study ways in which default logic can be enhanced by dynamic operators. In particular we shall investigate two operations: revision, which is concerned with the addition of new information in a way that does not lead to conflicting knowledge (in the sense described in later sections); and contraction, which achieves the deletion of some information, without necessarily adding its negation to the current knowledge. While belief revision [1,7] has been studying change in classical logic knowledge bases since the mid 80s, the problem of revising default theories has attracted attention only recently, and not much has been achieved in this direction. A notable exception is [16] which reports on the implementation of a specific revision and contraction strategy as part of the CIN project. In our exposition we will present two different families of approaches. It is easy to see that if we use default logic and its standard variants, there is an asymmetry between revision and contraction: while revision can be achieved by manipulating the set of facts only (if we are interested in adding a formula to all extensions), this cannot be done in the case of contraction, because there exist no “negative facts”. Thus the contraction has to be implemented by manipulating the defaults. If this manipulation of defaults is undesirable (for example because defaults are meant to express long term rules) then we propose the use of preconstrained default theories. The idea is similar to the use of constraints in the Theorist framework [13] and involves the formulation of formulae which should not be included in any extension. When constraints are used, revision and contraction can be achieved by manipulating only the facts and constraints, while the defaults remain unaltered. Revision in the context of Theorist was recently studied in [8]. Finally we wish to point out that this paper follows the basic idea of avoiding extensive computations with defaults, relying instead on a static analysis of default theories. This is done at the expense of somewhat ad hoc solutions, as well as the neglect of minimal change ideas.

2

Basics of Default Logic

n A default δ has the form ϕ:ψ1 ,...,ψ with closed formulae ϕ, ψ1 , . . .,ψn , χ. ϕ is the χ prerequisite pre(δ), ψ1 , . . . , ψn the justifications just(δ), and χ the consequent cons(δ) of δ. A default theory T is a pair (W, D) consisting of a set of formulae W (the set of facts) and a countable set D of defaults. In case D is finite and W is finitely axiomatizable, we call the default theory T = (W, D) finite. In this paper we consider finite default theories, since they are the most relevant for practical purposes. A preconstrained default theory is a triple T = (W, D, C) such that W and C are sets of formulae, and D a countable set of defaults. We assume that T h(W ) ∩ C = ∅. n Let δ = ϕ:ψ1 ,...,ψ be a default, and E a deductively closed set of formulae. χ We say that δ is applicable to E iff ϕ ∈ E, and ¬ψ1 , . . . , ¬ψn ∈ E.

On the Dynamics of Default Reasoning

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Let Π = (δ0 , δ1 , δ2 , . . .) a finite or infinite sequence of defaults from D without multiple occurrences (modelling an application order of defaults from D). We denote by Π[k] the initial segment of Π of length k, provided the length of Π is at least k. – In(Π) = T h(W ∪{cons(δ) | δ occurs in Π}), where T h denotes the deductive closure. – Out(Π) = {¬ψ | ψ ∈ just(δ), δ occurs in Π} (∪C in case T is preconstrained). Π is called a process of T iff δk is applicable to In(Π[k]), for every k such that δk occurs in Π. Π is successful iff In(Π) ∩ Out(Π) = ∅, otherwise it is failed. Π is closed iff every default that is applicable to In(Π) already occurs in Π. [2] shows that Reiter’s original definition of extensions is equivalent to the following one: A set of formulae E is an extension of a default theory T iff there is a closed and successful process Π of T such that E = In(Π). Finally we define: – T sc ϕ iff ϕ is included in all extensions of T . – T cr ϕ iff there is an extension E of T such that ϕ ∈ E.

3

Dynamic Operators For Standard Default Theories

Suppose we have a default theory T = (W, D) and wish to add new information in the form of a formula ϕ. First we have to lay down which interpretation of default logic we will be using: extensions, credulous reasoning, or sceptical reasoning. In this paper we view a default theory as a top-level specification which supports several alternative views (the extensions). Therefore we wish to make changes at the level of a default theory, rather than revising single extensions. One advantage of this design decision is our approach incorporates lazy evaluation: computation is deferred until it becomes necessary. In our case we make changes to the default theory without the necessity to compute extensions at every step. Default logic supports this approach since a default theory represents a set of alternative extensions. 3.1

Revision - Adding a Formula to All Extensions

This task can be achieved by adding ϕ to the set of facts W . Of course if we do it in the naive way of just adding it as a new fact, the set of facts may become inconsistent. In that case the new default theory has only one extension, the set of all formulae. This is undesirable, and we resolve the problem by revising the set of facts W with ϕ in the sense of belief revision. This ensures that ϕ is added to W in such a way that consistency is maintained and the changes made are minimal. Definition 1. Let T = (W, D) be a default theory, and ϕ a formula. Define Tϕ+ = (Wϕ∗ , D), where ∗ is a theory base revision operator.

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Note that in the definition above we are using a theory base revision operator instead of an AGM revision operator. The reason is that AGM revision operators [1] apply to deductively closed (classical) theories. Theoretically we can assume that W is deductively closed, indeed according to the definition of default logic in section 2, there is no difference between the use of W and T h(W ). But in practice W will be a finite axiomatization of the facts, thus we need to use operators which work on finite bases rather than deductively closed theories. More information on these issues, including the relationship between theory base revision and AGM belief revision are found in [15]. Theorem 2. (a) If ϕ is not a contradiction, then Tϕ+ sc ϕ. (b) If E is an extension of T such that ϕ ∈ E, then E is an extension of Tϕ+ . 3.2

Revision - Adding a Formula to At Least One Extension

In this case we don’t necessarily want to add ϕ to every extension. On the other hand, it is difficult to identify statically, that means without actually computing the extensions, a default δ contributing to an extension, and add ϕ to its consequent. The solution we propose is to add a new default which leads to a new extension containing ϕ. The other extensions should remain the same. The naive is not sufficient. For example, the approach of just adding the default true:ϕ ϕ ϕ:true existence of another default f alse would destroy any extension containing ϕ. Thus we must take care to separate the default to be added from all previous defaults. In addition we need to remove ¬ϕ from the set of facts W . In case ¬ϕ ∈ T h(W ) we turn the formulae removed from W into a default1 . Let us first outline this construction before giving the formal definition: 1. Remove ¬ϕ from T h(W ) using theory base contraction. This is necessary, otherwise ϕ will be excluded from all extensions. 2. Add the default δ = true:ϕ∧p ϕ∧p , where p is a new atom. Note that δ is applicable to the new set of facts, due to step 1 and the condition on p. According to the steps 3 and 4, all other defaults will be inapplicable after the application of δ. Thus δ is guaranteed to lead to an extension which contains ϕ. 3. Add ¬p to the justification and consequent of every (old) default in D. This ensures that δ does not interfere in any way with the other defaults. 4. Turn all formulae deleted from W in step 1 into defaults, and take care that they work in conjunction with the old defaults only (using ¬p as in step 3). Definition 3. Let T = (W, D) be a default theory and ϕ a formula. Define an pre(δ):just(δ)∪{¬p} − , { true:ϕ∧p |δ∈ operator cre Tϕ+1 as follows: cre Tϕ+1 = (W¬ϕ ϕ∧p }∪{ cons(δ)∧¬p true:¬p − − D} ∪ { ¬p∧ψ | ψ ∈ W − W¬ϕ }), where p is a new atom and a theory base contraction operator. 1

In general, contraction of a formula α from a theory T may result in the contraction of other formulae, too. For example, if β and β → α are included in T , then at least one of them has to be removed.

On the Dynamics of Default Reasoning

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Theorem 4. Let T be a default theory over the signature (logical language) Σ, and ϕ a formula over Σ which is not a contradiction. Then: (a) cre Tϕ+1 cr ϕ. (b) E is an extension of T iff there is an extension E of cre Tϕ+1 such that p ∈ E and E /Σ = E, where /Σ denotes restriction to the signature Σ. (b) states that if we forget about the new symbol p, all extensions of T are preserved. Together with (a) it says that we have kept the old extensions and just added a new one which contains ϕ. This construction looks rather complicated. We note that there are cases in which it can be simplified significantly. For example in case we restrict attention to normal default theories, or if we consider a semi-monotonic variant of default logic, such as Constrained Default Logic [5] and Justified Default Logic [10]. In those cases addition of the default true:ϕ (almost) does the work, since, ϕ due to semi-monotonicity, once the default is applied it will ultimately lead to an extension which contains ϕ [2]. The only thing needed is to treat the facts adequately, as done before. In the rest of this subsection we present the definitions and results for normal default theories in standard default logic, but note that similar results can be derived for semi-monotonic variants of default logic without the restriction to normal theories. Definition 5. Let T = (W, D) be a normal default theory and ϕ a formula. Define an operator cre Tϕ+2 as follows: – If ¬ϕ ∈ T h(W ) then cre Tϕ+2 = (W, { true:ϕ ϕ } ∪ D). true:¬p +2 − – If ¬ϕ ∈ T h(W ) then cre Tϕ = (W¬ϕ , { true:ϕ∧p ϕ∧p } ∪ D ∪ { ¬p∧ψ | ψ ∈ W − − W¬ϕ }), where p is a new atom and − a theory base contraction operator. Theorem 6. Let T be a default theory over the signature (logical language) Σ, and ϕ a formula over Σ which is not a contradiction. Then: (a) cre Tϕ+2 cr ϕ. (b) If E is an extension of T then there is an extension E of E /Σ ⊇ E; if ¬ϕ ∈ T h(W ) then even E /Σ = E;

+2 cre Tϕ

such that

The opposite direction of (b) is not necessarily true. Take ϕ to be q, Σ = {q, r}, and T to consist of the fact ¬q and the default q:r r . T has the single extension E = T h({¬q}), while cre Tq+2 has two extensions, E1 = T h({¬p, ¬q}) and E2 = T h({p, q, r}). E1 /Σ = E, as predicted by theorem 3, but E2 /Σ = T h({q, r}) is not an extension of the original theory T . For normal default theories we can propose an alternative approach. In the beginning of this subsection we said that identifying a default which is guaranteed to lead to an extension is nontrivial. In case semi-monotonicity is guaranteed, any default which is initially applicable will do.

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Definition 7. Let T = (W, D) be a normal default theory, and ϕ a noncontradictory formula. Suppose that there is a default δ ∈ D such that (i) δ is applicable to T h(W ); and (ii) W ∪ just(δ) ∪ {ϕ} is consistent. Then we define pre(δ):cons(δ)∧ϕ +3 }). cre Tϕ = (W, (D − {δ}) ∪ { cons(δ)∧ϕ Theorem 8. Suppose that the conditions of the previous definition hold. Then: (a) cre Tϕ+3 cr ϕ. (b) Every extension E of T , such that δ is not applicable to E, is also an extension of cre Tϕ+3 . (c) Every extension E of T such that δ is applicable to E, T h(E∪{ϕ}) is included in an extension of cre Tϕ+3 . 3.3

Contraction - Removing a Formula From All Extensions

Of course we could remove ϕ by adding ¬ϕ to the set of facts, but that it not what we wish to achieve: Our aim is to delete ϕ without adding ¬ϕ. Here is the approach we propose. First we need to remove ϕ from the facts, using a classical theory base contraction strategy. Then we need to make sure that ϕ is not reintroduced by the application of some defaults. One way of ensuring this is to add the default ϕ:true f alse which destroys (that is, turns to failed) any process which includes ϕ in its In-set. This approach is not unappealing from the viewpoint of maintainability and naturalness: All defaults are kept as they are, and a new is added which has a clear meaning (it resembles the use of the “cut/fail” combination in logic programming). Definition 9. Let T = (W, D) be a default theory, and ϕ a formula. We define: ϕ:true − − cre Tϕ = (Wϕ , D ∪ { f alse }). Theorem 10. Let T be a default theory, and ϕ a non-tautological formula. Then: (a) cre Tϕ−  cr ϕ. (b) If E is an extension of T with ϕ ∈ E, then E is also an extension of

− cre Tϕ .

Alternatively, one may add ¬ϕ to the justifications of all defaults in D. 3.4

Contraction - Removing a Formula From At Least One Extension

This task causes similar problems to the revision case. We propose instead to do something different, which has formally the right effect: we add a new extension which does not include ϕ. This way ϕ is not contained in at least one extension of the new default theory. Thus we achieve, at least, that ϕ is not included in the sceptical interpretation of the default theory at hand.

On the Dynamics of Default Reasoning

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Definition 11. Let T = (W, D) be a default theory and ϕ a formula. Define true:p − Tϕ− = (Wϕ− , { pre(δ):just(δ)∪{¬p} | δ ∈ D} ∪ { true:¬p cons(δ)∧¬p ψ∧¬p | ψ ∈ W − Wϕ } ∪ { p }), where p is a new atom. Theorem 12. Let Σ be a signature, T a default theory over Σ, and ϕ a formula over Σ. Then: (a) Tϕ−  sc ϕ. (b) E is an extension of T iff there is an extension E of Tϕ− such that p ∈ E and E /Σ = E. Thus if we forget about the new symbol p, all extensions of T are preserved, but there is a new one which does not contain ϕ.

4

Dynamic Operators for Preconstrained Theories

We mentioned already the asymmetry inherent to approaches such as those described above; it is caused by the asymmetry of the default theories themselves: it is possible to express which formulae should be included in all extensions, but not which formulae to exclude. This asymmetry is removed when we study preconstrained default theories. New possibilities arise for the contraction and revision of theories, and we will describe some of them below. But first we make the following remark: if we manipulate facts and constraints only and leave the defaults unaltered, then revision and contraction operators will achieve results for all extensions. Thus in the following we will restrict our attention to the addition of a formula to all extensions, and the removal of a formula from all extensions. 4.1

Revision

We define a revision operator as follows: 1 Definition 13. (W, D, C)+ ϕ is a set (W , D, C ) such that:

(a) W ⊆ W ∪{ϕ} and for all W ⊆ W ∪{ϕ}, if W ⊃ W then T h(W )∩C = ∅. (b) C ⊆ C, and for all C ⊆ C, if C ⊃ C then T h(W ) ∩ C = ∅. (c) ϕ ∈ W . In other words, W must include ϕ, and W and C are minimal such that the property T h(W ) ∩ C = ∅ is preserved. Theorem 14. Let ϕ be satisfiable. 1 (a) ϕ is included in all extensions of (W, D, C)+ ϕ . (b) If E is an extension of (W, D, C) and ϕ ∈ E, then E is also an extension of 1 (W, D, C)+ ϕ .

8

Grigoris Antoniou

We note that this approach is different from building the revision on facts based on classical belief revision. For example, consider the theory ({p → q}, D, {q}), and suppose that we wish to add p. The approach we propose admits a solution, ({p}, D, {q}), which is not admitted by the pure belief revision approach on the set of facts. Negative information has the same weight as positive information! In the approach described above any facts retracted are lost forever. But it is possible to store them as defaults, which cannot be applied at present, but could become applicable later after further revisions of the default theory (which, for example, may cause ϕ to be retracted). Thus the following approach may have advantages for iterated revision. But note that this way we reintroduce an asymmetry, albeit of a different kind: whereas retracted facts can be kept as defaults, retracted constraints are lost. 2 Definition 15. (W, D, C)+ ϕ is a set (W , D, C ) such that properties (a), (b) and (c) from the previous definition and the following condition hold:

: ψ ∈ (W ∪ {ϕ})-W }. (d) D = D ∪ { true:ψ ψ +2 1 Theorem 16. (W, D, C)+ ϕ and (W, D, C)ϕ have the same extensions.

4.2

Contraction

Here we reap the benefits which derive from the symmetry between positive and negative information: contraction of ϕ is achieved simply by adding ϕ to the constraints (followed, as before, by the reduction of the sets of facts and constraints in such a way that the consistency condition T h(W ) ∩ C = ∅ is satisfied). Definition 17. (W, D, C)− ϕ = (W , D, C ) such that:

(a) W ⊆ W , and for all W ⊆ W , if W ⊃ W then T h(W ) ∩ C = ∅. (b) C ⊆ C ∪ {ϕ}, and for all C ⊆ C ∪ {ϕ}, if C ⊃ C then T h(W ) ∩ C = ∅. (c) ϕ ∈ C . Theorem 18. Let ϕ be non-tautological. (a) ϕ is not included in any extension of (W, D, C)− ϕ. (b) If E is an extension of (W, D, C) and ϕ ∈ E, then E is an extension of (W, D, C)− ϕ. A variant for iterated change is possible as in the case of revision.

On the Dynamics of Default Reasoning

5

9

Conclusion

In this paper we defined a problem that has attracted little attention so far: the revision of default theories. We believe that belief revision techniques should move beyond classical representations and consider more rich knowledge structures. We considered the two main operators of revising knowledge bases, revision (addition of information) and contraction (deletion of information), and did this in the two main approaches to default reasoning: the sceptical case and the credulous case. In most cases we gave more than one possible solutions, and discussed their respective advantages and disadvantages. In a companion paper we provide a deeper study of the revision of preconstrained default theories. Two specific techniques have already been implemented in the system (see [16]). The focus of that work was how to make the change without having to recalculate the extensions of the revised default theory from scratch. Our work has shown that considerable computational effort can be avoided by carrying over part of the computation from the old to the new theory. This paper is orthogonal to [16] in the sense that the techniques developed there will be used to save computational effort when the revision methods of this paper are implemented. We intend to study other approaches of revising default theories, beginning from desirable abstract postulates and moving towards concrete realizations that satisfy them. Such an approach may require more computational effort, but is more in line with classical belief revision.

References 1. C.E. Alchourron, P. Gardenfors and D. Makinson. On the Logic of Theory Change: Partial Meet Contraction and Revision Functions. Journal of Symbolic Logic 50 (1985), 510-530. 2. G. Antoniou. Nonmonotonic Reasoning. MIT Press 1997. 3. G. Antoniou. The role of nonmonotonic representations in requirements engineering. International Journal of Software Engineering and Knowledge Engineering, 1998 (in press). 4. D.E. Cooke and Luqi. Logic Programming and Software Maintenance. Annals of Mathematics and Artificial Intelligence 21 (1997): 221-229. 5. J.P. Delgrande, T. Schaub and W.K. Jackson. Alternative approaches to default logic. Artificial Intelligence 70(1994): 167-237. 6. S. Easterbrook and B. Nuseibeh. Inconsistency Management in an Evolving Specification. In Proc. 2nd International Symposium on Requirements Engineering, IEEE Press 1995, 48-55. 7. P. Gardenfors. Knowledge in Flux: Modeling the Dynamics of Epistemic States. The MIT Press 1988. 8. A. Ghose and R. Goebel. Belief states as default theories: Studies in non-prioritized belief change. In Proc. ECAI’98. 9. A. Hunter and B. Nuseibeh. Analysing Inconsistent Specifications. In Proc. Third International Symposium on Requirements Engineering, IEEE Press 1997.

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10. W. Lukaszewicz. Considerations on Default Logic. Computational Intelligence 4(1988): 1-16. 11. Luqi and D.E. Cooke. ”How to Combine Nonmonotonic Logic and Rapid Prototyping to Help Maintain Software. International Journal on Software Engineering and Knowledge Engineering 5,1 (1995): 89-118. 12. V. Marek and M. Truszczynski. Nonmonotonic Logic, Springer 1993. 13. D. Poole. A Logical Framework for Default Reasoning. Artificial Intelligence 36 (1988): 27-47. 14. R. Reiter. A logic for default reasoning. Artificial Intelligence 13(1980): 81-132. 15. M.A. Williams. Iterated Theory Base Change: A Computational Model. In Proc. 14th International Joint Conference on Artificial Intelligence, 1541-1550, Morgan Kaufmann 1995 . 16. M.A. Williams and G. Antoniou. A Strategy for Revising Default Theory Extensions. in Proc. 6th International Conference on Principles of Knowledge Representation and Reasoning, Morgan Kaufmann 1998 (in press). 17. D. Zowghi and R. Offen. A Logical Framework for Modeling and Reasoning about the Evolution of Requirements. In Proc. Third International Symposium on Requirements Engineering, IEEE Press 1997.

Nonmonotonic and Paraconsistent Reasoning: From Basic Entailments to Plausible Relations Ofer Arieli and Arnon Avron Department of Computer Science, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel

fofera,[email protected]

Abstract. In this paper we develop frameworks for logical systems

which are able to re ect not only nonmonotonic patterns of reasoning, but also paraconsistent reasoning. For this we consider a sequence of generalizations of the pioneering works of Gabbay, Kraus, Lehmann, Magidor and Makinson. Our sequence of frameworks culminates in what we call plausible, nonmonotonic, multiple-conclusion consequence relations (which are based on a given monotonic one). Our study yields intuitive justi cations for conditions that have been proposed in previous frameworks, and also clari es the connections among some of these systems. In addition, we present a general method for constructing plausible nonmonotonic relations. This method is based on a multiple-valued semantics, and on Shoham's idea of preferential models.

1 Introduction Our main goal in this paper is to get a better understanding of the conditions that a useful relation for nonmonotonic and paraconsistent [5] reasoning should satisfy. For this we consider a sequence of generalizations the pioneering works of Gabbay [7], Kraus, Lehmann, Magidor [8] and Makinson [12]. These generalizations are based on the following ideas:

{ Each nonmonotonic logical system is based on some underlying monotonic one. { The underlying monotonic logic should not necessarily be classical logic,

but should be chosen according to the intended application. If, for example, inconsistent data is not to be totally rejected, then an underlying paraconsistent logic might be a better choice than classical logic. { The more signi cant logical properties of the main connectives of the underlying monotonic logic, especially conjunction and disjunction (which have crucial roles in monotonic consequence relations), should be preserved as far as possible. { On the other hand, the conditions that de ne a certain class of nonmonotonic systems should not assume anything concerning the language of the system (in particular, the existence of appropriate conjunction or disjunction should not be assumed). A. Hunter and S. Parsons (Eds.): ECSQARU’99, LNAI 1638, pp. 11-21, 1999.  Springer-Verlag Berlin Heidelberg 1999

12

Ofer Arieli and Arnon Avron

Our sequence of generalizations culminates in what we call (following Lehmann [9]) cautious plausible consequence relations (which are based on a given monotonic one.1) Our study yields intuitive justi cations for conditions that have been proposed in previous frameworks and also clari es the connections among some of these systems. Moreover, while the logic behind most of the systems which were proposed so far is supraclassical (i.e., every rst-degree inference rule that is classically sound remains valid in the resulting logics), the consequence relations considered here are also capable of drawing conclusions from incomplete and inconsistent theories in a nontrivial way. In the last part of this paper we present a general method for constructing such plausible nonmonotonic and paraconsistent relations. This method is based on a multiple-valued semantics, and on Shoham's idea of preferential models [21].2

2 General Background We rst brie y review the original treatments of [8] and [12]. The language they use is based on the standard propositional one. Here, ; denotes the material implication and  denotes the corresponding equivalence operator. The classical propositional language, with the connectives :, _, ^, ;, , and with a propositional constant t, is denoted here by cl.

De nition 1. [8] Let `cl be the classical consequence relation. A binary relation3 j0 between formulae in cl is called cumulative if it is closed under the following inference rules: re exivity: j0 . cautious monotonicity: if j0  and j0  , then ^  j0  . cautious cut: if j0  and ^  j0  , then j0  . left logical equivalence: if `cl   and j0  , then  j0  . right weakening: if `cl ;  and  j0 , then  j0 .

De nition 2. [8] A cumulative relation j0 is called preferential if it is closed under the following rule: left _-introduction (Or) if

j0 

and  j0  , then

_  j0  .

The conditions above might look a little-bit ad-hoc. For example, one might ask why ; is used on the right, while the stronger  is used on the left. A See [4] for another formalism with non-classical monotonic consequence relations as the basis for nonmonotonic consequence relations. 2 Due to a lack of space, proofs of propositions are omitted; They will be given in the full version of the paper. 3 A \conditional assertion" in terms of [8]. 1

Nonmonotonic and Paraconsistent Reasoning

13

discussion and some justi cation appears in [8, 11].4 A stronger intuitive justi cation will be given below, using more general frameworks. In what follows we consider several generalizations of the basic relations presented above: 1. Allowing the use of nonclassical logics (for example: paraconsistent logics) as the basis, instead of classical logic. 2. Allowing the use of a set of premises rather than a single one. 3. Allowing the use of multiple conclusions relations rather than single conclusion ones. The key logical concepts which stand behind these generalizations are the following:

De nition 3. a) An ordinary Tarskian consequence relation (tcr , for short) [22] is a binary

relation ` between sets of formulae and formulae that satis es the following conditions:5 s-TR strong T-re exivity: , ` for every 2 , . TM T-monotonicity: if , ` and ,  , 0 then , 0 ` . TC T-cut: if ,1 ` and ,2 ; `  then ,1 ; ,2 ` . b) An ordinary Scott consequence relation (scr , for short) [19, 20] is a binary relation ` between sets of formulae that satis es the conditions below: s-R strong re exivity: if , \
6= ; then , `
. M monotonicity: if , `
and ,  , 0 ,

0 , then , 0 `
0 . C cut: if ,1 ` ;
1 and ,2 ; `
2 then ,1 ; ,2 `
1 ;
2 .

De nition 4. a) Let j be a relation between sets of formulae. 

A connective ^ is called internal conjunction (w.r.t. j) if: [^ j]I



,; ;  j
,; ^  j


[^ j]E

,; ^  j
,; ;  j


A connective ^ is called combining conjunction (w.r.t. j) if:

[j ^]E , ,jj ^;;

 A connective _ is called internal disjunction (w.r.t. j) if: [j ^]I ,

j

;
, j ;
, j ^ ;


[j _]I

, j ; ;
, j _ ;


[j _]E

, j ^ ;
, j ;


, j _ ;
, j ; ;


Systems that satisfy the conditions of De nitions 1, 2, as well as other related systems, are also considered in [6, 13, 18, 10]. 5 The pre x \T" re ects the fact that these are Tarskian rules. 4

14



Ofer Arieli and Arnon Avron

A connective _ is called combining disjunction (w.r.t. j) if: j
,;  j
[_ j]I ,; ,; _  j


[_ j]E ,;,; _ jj



,; _  j
,;  j


b) Let j be a relation between sets of formulae and formulae. The notions of combining conjunction, internal conjunction, and combining disjunction are de ned for j exactly like in case (a).

Note: If ` is an scr (tcr) then ^ is an internal conjunction for ` i it is a combining conjunction for `. The same is true for _ in case ` is an scr. This, however, is not true in general.

3 Tarskian Cautious Consequence Relations De nition 5. A Tarskian cautious consequence relation (tccr , for short) is a binary relation j between sets of formulae and formulae in a language  that satis es the following conditions:6 s-TR strong T-re exivity: , j for every 2 , . TCM T-cautious monotonicity: if , j and , j , then ,; j . TCC T-cautious cut: if , j and ,; j , then , j .

Proposition 6. Any tccr j is closed under the following rules for every n: TCM[ ] if , j (i =1;: : :; n) then ,; 1 ; : : : ; ,1 j . TCC[ ] if , j (i =1;: : :; n) and ,; 1 ; : : : j , then , j . n

i

n

n

i

n

n

We now generalize the notion of a cumulative entailment relation. We rst do it for Tarskian consequence relations ` that have an internal conjunction ^.

De nition 7. A tccr j is called f^; `g-cumulative if it satis es the following

conditions:  if `  and  ` and j  , then  j  . (weak left logical equivalence )  if `  and  j , then  j . (weak right weakening )  ^ is also an internal conjunction w.r.t. j. If, in addition, ` has a combining disjunction _, then j is called f_; ^; `gpreferential if it also satis es the single-conclusion version of [_ j]I . j0  i 0 j . Then w.r.t. cl , j is cumulative [preferential] in the sense of [8]. Conversely: if j0 is cumulative [preferential] in the sense of [8] and we de ne 1 ;: : :; j  i 1 ^ : : : ^ j0 , then j is `cl -cumulative [`cl -preferential].

Proposition 8. Suppose j is `cl-cumulative [`cl-preferential]. Let n

6

n

This set of conditions was rst proposed in [7].

Nonmonotonic and Paraconsistent Reasoning

15

We next generalize the de nition of a cumulative tccr to make it independent of the existence of an internal conjunction.

Proposition 9. Let ` be a tcr, and let j be a tccr in the same language. The following connections between ` and j are equivalent: TCum T-cumulativity for every , 6= ;, if , ` then , j . TLLE T-left logical equiv. if ,; `  and ,;  ` and ,; j  , then ,;  j  . TRW T-right weakening if ,; `  and , j , then , j . TMiC T-mixed cut: for every , 6= ;, if , ` and ,; j , then , j .

De nition 10. Let ` be a tcr. A tccr j in the same language is called `-

cumulative if it satis es any of the conditions of Proposition 9. If ` has a combining disjunction _, and j satis es [_ j]I , then j is called f_; `g-preferential .

Note: Since , ` for every 2 , , TCum implies s-TR, and so a binary relation that satis es TCum, TCM, and TCC is a `-cumulative tccr. Proposition 11. Suppose that ` is a tcr with an internal conjunction ^. A

tccr j is a f^; `g-cumulative i it is `-cumulative. If ` has also a combining disjunction _, then j is f_; ^; `g-preferential i it is f_; `g-preferential.

Proposition 12. Let j be a `-cumulative relation, and let ^ be an internal

conjunction w.r.t. `. Then ^ is both an internal conjunction and a combining conjunction w.r.t. j.

4 Scott Cautious Consequence Relations De nition 13. A Scott cautious consequence relation (sccr , for short) is a binary relation j between nonempty7 sets of formulae that satis es the following conditions: s-R strong re exivity: if , \
6= ; then , j
. CM cautious monotonicity: if , j and , j
then ,; j
. CC[1] cautious 1-cut: if , j and ,; j
then , j
. A natural requirement from a Scott cumulative consequence relation is that its single-conclusion counterpart will be a Tarskian cumulative consequence relation. Such a relation should also use disjunction on the r.h.s. like it uses conjunction on the l.h.s. The following de nition formalizes these requirements.

De nition 14. Let ` be an scr with an internal disjunction _. A relation j

between nonempty nite sets of formulae is called f_; `g-cumulative sccr if it 7

The condition of non-emptiness is just technically convenient here. It is possible to remove it with the expense of complicating somewhat the de nitions and propositions. It is preferable instead to employ (whenever necessary) the propositional constants t and f to represent the empty l.h.s. and the empty r.h.s., respectively.

16

Ofer Arieli and Arnon Avron

is an sccr that satis es the following two conditions: a) Let `T and jT be, respectively, the single-conclusion counterparts of ` and j. Then jT is a `T -cumulative tccr. b) _ is an internal disjunction w.r.t. jT as well. Following the line of what we have done in the previous section, we next specify conditions that are equivalent to those of De nition 14, but are independent of the existence of any speci c connective in the language. De nition 15. Let ` be an scr. An sccr j in the same language is called weakly `-cumulative if it satis es the following conditions: Cum cumulativity: if ,;
6= ; and , `
, then , j
. RW[1] right weakening: if ,; `  and , j ;
, then , j ;
. RM right monotonicity: if , j
then , j ;
. Proposition 16. Let ` and _ be as in De nition 14. A relation j is a f_; `gcumulative sccr i it is a weakly `-cumulative sccr. Proposition 17. If ` has an internal disjunction, then j is a weakly `-cumulative sccr if it satis es Cum, CM, CC[1], and RW[1] . We turn now to examine the role of conjunction in the present context. Proposition 18. Let ` be an scr with an internal conjunction ^, and let j be a weakly `-cumulative sccr. Then: a) ^ is an internal conjunction w.r.t. j. b) ^ is a \half" combining conjunction w.r.t. j. I.e, it satis es [j ^]E. De nition 19. Suppose that an scr ` has an internal conjunction ^. A weakly `-cumulative sccr j is called f^; `g-cumulative if ^ is also a combining conjunction w.r.t. j. As usual, we provide an equivalent notion in which one does not have to assume that an internal conjunction is available: De nition 20. A weakly `-cumulative sccr j is called `-cumulative if for every nite n the following condition is satis ed: RW[ ] if , j ;
(i =1; : : : ; n) and ,; 1 ; : : : ; `  then , j ;
. Proposition 21. Let ^ be an internal conjunction for `. An sccr j is f^; `gcumulative i it is `-cumulative. Corollary 22. If ` is an scr with an internal conjunction ^ and j is a `cumulative sccr, then ^ is a combining conjunction and an internal conjunction w.r.t. j. Let us return now to disjunction, examining it this time from its combining aspect. Our rst observation is that unlike conjunction, one direction of the combining disjunction property for j of _ yields monotonicity of j: n

i

n

Nonmonotonic and Paraconsistent Reasoning

17

Lemma 23. Suppose that _ is an internal disjunction for ` and j is a weakly

`-cumulative

sccr in which [_ j]E is satis ed. Then j is (left) monotonic.

It follows that requiring [_ j]E from a weakly `-cumulative sccr is too strong. It is reasonable, however, to require its converse.

De nition 24. A weakly `-cumulative sccr j is called weakly f_; `g-preferential if it satis es [_ j]I .

Unlike the Tarskian case, this time we are able to provide an equivalent condition in which one does not have to assume that a disjunction is available:

De nition 25. Let ` be an sccr. A weakly `-cumulative sccr is called weakly if it satis es the following rule: CC cautious cut : if , j ;
and ,; j
then , j
.

`-preferential

Proposition 26. Let ` be an scr with an internal disjunction _. An sccr j is weakly f_; `g-preferential i it is weakly `-preferential.

Some characterizations of weak `-preferentiality are given in the following proposition:

Proposition 27. Let ` be an scr. a) j is a weakly `-preferential sccr i it satis es Cum, CM, CC, and RM. b) j is a weakly `-preferential sccr i it is a weakly `-cumulative sccr and for

every nite n it satis es cautious n-cut : CC[ ] if ,; j
(i =1; : : : ; n) and , j 1; : : : ; , then , j
. n

i

n

Note: By Proposition 6, the single conclusion counterpart of CC[ ] is valid for n

any sccr (not only the cumulative or preferential ones).

We are now ready to introduce our strongest notions of nonmonotonic Scott consequence relation:

De nition 28. Let ` be an scr. A relation j is called `-preferential i it is both `-cumulative and weakly `-preferential. Proposition 29. Let ` be an scr. j is `-preferential i it satis es Cum, CM, CC, RM, and RW[ ] for every n. n

Proposition 30. Let ` be an scr and let j be a `-preferential sccr. a) An internal conjunction ^ w.r.t. ` is also an internal conjunction and a combining conjunction w.r.t. j. b) An internal disjunction _ w.r.t. ` is also an internal disjunction and \half" combining disjunction w.r.t. j.8 8 I.e., j satis es [_j]I (but not necessarily [_j]E .

18

Ofer Arieli and Arnon Avron

CC[ ] (n  1), which is valid for `-preferential sccrs, is a natural generalization of cautious cut. A dual generalization, which seems equally natural, is given in the following rule from [9]: n

LCC[

n

]

, j 1 ;
: : : , j ;
; ,; 1 ; : : : ; , j
n

n

j


De nition 31. [9] A binary relation j is a plausibility logic if it satis es Inclusion (,;

j

), CM, RM, and LCC[ ] (n  1). n

De nition 32. Let ` be an scr. A relation j is called `-plausible if it is a `-preferential

sccr and a plausibility logic.

A more concise characterization of a `-plausible relation is given in the following proposition:

Proposition 33. Let ` be an scr. A relation j is `-plausible i it satis es Cum, CM, RM, and LCC[ ] for every n. n

Proposition 34. Let ` be an scr with an internal conjunction ^. A relation j is `-preferential i it is `-plausible.

Table 1 and Figure 1 summarize the various types of Scott relations considered in this section and their relative strengths. ` is assumed there to be an scr, and _, ^ are internal disjunction and conjunction (respectively) w.r.t. `, whenever they are mentioned.

Table 1. Scott relations consequence relation

sccr

general conditions valid conditions with ^ and _

s-R, CM, CC[1]

weakly `-cumulative Cum, CM, CC[1] , RW[1] , RM sccr [^j]I , [^j]E , [j ^]E , [j _]I , [j _]E `-cumulative sccr Cum, CM, CC[1] , RW[n] , RM [^j]I , [^j]E , [j ^]I , [j ^]E , [j _]I , [j _]E weakly `-preferential Cum, CM, CC, RM sccr [^j]I , [^j]E , [j ^]E , [_j]I , [j _]I , [j _]E , `-preferential sccr Cum, CM, CC, RW[n] , RM [^j]I , [^j]E , [j ^]I , [j ^]E , [_j]I , [j _]I , [j _]E `-plausible sccr Cum, CM, LCC[n] , RM [^j]I , [^j]E , [j ^]I , [j ^]E , [_j]I , [j _]I , [j _]E scr extending ` Cum, M, C [^j]I , [^j]E , [j ^]I , [j ^]E , [_j]I , [_j]E , [j _]I , [j _]E

Nonmonotonic and Paraconsistent Reasoning

19

weakly `-cumulative sccr + CC

 



u

 HH

HH

HH+ RW n HH HH [ ]

  H Hj Hweakly `-preferential u `-cumulativeH HH sccr sccr  HH  HH  + CC  + RW n HHH HH ju 

u

[ ]

`-preferential sccr

(if a combining conjunction is available)

6

+ LCC[n]

?

u

`-plausible sccr

?u

+M

An scr that extends ` Fig. 1. Relative strength of the Scott relations

5 A Semantical Point of View In this section we present a general method of constructing nonmonotonic consequence relations of the strongest type considered in the previous section, i.e., preferential and plausible sccrs. Our approach is based on a multiple-valued semantics. This will allow us to de ne in a natural way consequence relations that are not only nonmonotonic, but also paraconsistent (see examples below). A basic idea behind our method is that of using a set of preferential models for making inferences. Preferential models were introduced by McCarthy [14] and later by Shoham [21] as a generalization of the notion of circumscription. The essential idea is that only a subset of its models should be relevant for making inferences from a given theory. These models are the most preferred ones according to some conditions or preference criteria.

De nition 35. Let  be an arbitrary propositional language. A preferential multiple-valued structure for  (pms, for short) is a quadruple (L; F ; S ; ), where L is set of elements (\truth values"), F is a nonempty proper subset of L, S is

20

Ofer Arieli and Arnon Avron

a set of operations on L that correspond to the connectives in  , and  is a well-founded partial order on L. The set F consists of the designated values of L, i.e., those that represent true assertions. In what follows we shall assume that L contains at least the classical values t; f , and that t 2 F , f 62 F . De nition 36. Let (L; F ; S ; ) be a pms. a) A (multiple-valued) valuation  is a function that assigns an element of L to each atomic formula. Extensions to complex formulae are done as usual. b) A valuation  satis es a formula if  ( ) 2 F . c) A valuation  is a model of a set , of formulae, if  satis es every formula in , . The set of the models of , is denoted by mod(, ). De nition 37. Let (L; F ; S ; ) be a pms. Denote , `L F
if every model of , satis es some formula in
. Proposition 38. `L F is an scr. De nition 39. Let (L; F ; S ; ) be a pms for a language  . a) An operator ^ 9 is conjunctive if 8x; y 2 L, x ^ y 2 F i x 2 F and y 2 F . b) An operator _ is disjunctive if 8x; y 2 L, x _ y 2 F i x 2 F or y 2 F . Proposition 40. Let (L; F ; S ; ) be a pms for  , and let ^ (_) be in  . If the operation which corresponds to ^ (_) is conjunctive (disjunctive), then ^ (_) is both an internal and a combining conjunction (disjunction) w.r.t. `L F . De nition 41. Let P be a pms and , a set of formulae in  . A valuation M 2 mod(, ) is a P -preferential model of , if there is no other valuation M 0 2 mod(, ) s.t. for every atom p, M 0 (p)  M (p). The set of all the P -preferential models of , is denoted by !(,; P ). De nition 42. Let P be a pms.L AF set of formulae , P -preferentially entails a set of formulae
(notation: , `
) if every M 2 !(,; P ) satis es some  2
. ;

;

;

;

Proposition 43. Let (L; F ; S ; ) be a pms. Then `L F is `L F -plausible. Corollary 44. Let P =(L; F ; S ; ) be a pms for a language  . a) If ^ is a conjunctive connective of  (relative to P ), then it is a combining ;

;

conjunction and an internal conjunction w.r.t. `L F . b) If _ is a disjunctive connective of  (relative to P ), then it is an internal disjunction w.r.t. `L F , which also satis es [_ j]I . Examples. Many known formalisms can be viewed as based on preferential multiple-valued structures. Among which are classical logic, Reiter's closed-world assumption [17], the paraconsistent logic LPm of Priest [15, 16], and the paraconsistent bilattice-based logics `L F and `LI F [1, 2]. 9 We use here the same symbol for a connective and its corresponding operation in S . ;

;

;

k

;

Nonmonotonic and Paraconsistent Reasoning

21

Acknowledgment This research was supported by The Israel Science Foundation founded by The Israel Academy of Sciences and Humanities.

References 1. O.Arieli, A.Avron. Reasoning with logical bilattices. Journal of Logic, Language, and Information 5(1), pp.25{63, 1996. 2. O.Arieli, A.Avron. The value of the four values. Arti cial Intelligence 102(1), pp.97-141, 1998. 3. A.Avron. Simple consequence relations. Journal of Information and Computation 92, pp.105{139, 1991. 4. P.Besnard. Axiomatizations in the metatheory of non-monotonic inference systems. Proc. of the 7th Canadian AI Conference, (R. Goebel { Ed.), Edmonton, Canada, pp.117{124, 1988. 5. N.C.A.da-Costa. On the theory of inconsistent formal systems. Notre Damm Journal of Formal Logic 15, pp.497{510, 1974. 6. M.Freund, D.Lehmann, P.Morris. Rationality, transitivity and Contraposition . Arti cial Intelligence 52, pp.191{203, 1991. 7. D.M.Gabbay. Theoretical foundation for non-monotonic reasoning in expert systems. Proc. of the NATO Advanced Study Inst. on Logic and Models of Concurent Systems (K.P.Apt - Ed.) Springer, pp.439{457, 1985. 8. S.Kraus, D.Lehmann, M.Magidor. Nonmonotonic reasoning, preferential models and cumulative logics. Arti cial Intelligence 44, pp.167{207, 1990. 9. D.Lehmann. Plausibility logic. Proc. CSL'91, pp.227{241, Springer, 1992. 10. D.Lehmann. Nonmonotonic logics and semantics. Technical report TR-98-6, Institute of Computer Science, The Hebrew University, 1998. 11. D.Lehmann, M.Magidor. What does a conditional knowledge base entail? Arti cial Intelligence 55, p.1{60, 1992. 12. D.Makinson. General theory of cumulative inference . Non-Monotonic Reasoning (M.Reinfrank { Ed.), LNAI 346, pp.1{18, Springer, 1989. 13. D.Makinson. General patterns in nonmonotonic reasoning. Handbook of Logic in Arti cial Intelligence and Logic Programming, Vol.3 (D.Gabbay, C.Hogger, J.Robinson { Eds.), Oxford Science Pub., pp.35{110, 1994. 14. J.M.McCarthy. Circumscription { A form of non monotonic reqsoning. Arti cial Intelligence 13(1{2), pp.27{39, 1980. 15. G.Priest. Reasoning about truth. Arti cial Intelligence 39, pp.231{244, 1989. 16. G.Priest. Minimally inconsistent LP. Studia Logica 50, pp.321{331, 1991. 17. R.Reiter. On closed world databases. Logic and Databases (H.Gallaire, J.Minker { Eds.), Plenum Press, pp.55-76, 1978. 18. K.Schlechta. Nonmonotonic logics { Basic concepts, results, and techniques. LNAI 1187, Springer, 1996. 19. D.Scott. Rules and derived rules. Logical Theory and Semantical Analysis (S.Stenlund { Ed.), Reidel, Dordrecht, pp.147{161, 1974. 20. D.Scott. Completeness and axiomatization in many-valued logic. Proc. of the Tarski Symposium, Proc. of Symposia in Pure Mathematics, Vol.XXV, American Mathematical Society, Rhode Island, pp.411{435, 1974. 21. Y.Shoham. Reasoning about change. MIT Press, 1988. 22. A.Tarski. Introduction to logic. Oxford University Press, N.Y., 1941.

A Comparison of Systematic and Local Search Algorithms for Regular CNF Formulas? Ramon Bejar and Felip Manya Departament d'Informatica Universitat de Lleida Jaume II, 69, E-25001 Lleida, Spain framon,[email protected]

Abstract This paper reports on a series of experiments performed with

the aim of comparing the performance of Regular-DP and Regular-GSAT. Regular-DP is a Davis-Putnam-style procedure for solving the propositional satis ability problem in regular CNF formulas (regular SAT). Regular-GSAT is a GSAT-style procedure for nding satisfying interpretations in regular CNF formulas. Our experimental results provide experimental evidence that Regular-GSAT outperforms Regular-DP on computationally dicult regular random 3-SAT instances, and suggest that local search methods can extend the range and size of satis ability problems that can be eciently solved in many-valued logics.

1 Introduction In this paper we investigate the use of both systematic and local search algorithms for solving the propositional satis ability problem in regular CNF formulas (regular SAT). Concerning systematic search algorithms we focus on Regular-DP. It is a Davis-Putnam-style procedure for regular CNF formulas de ned by Hahnle [4], which we have implemented in C++ and equipped with suitable data structures for representing formulas [7]. Here, we describe Regular-DP, de ne a new branching heuristic and report on experimental results that indicate that Regular-DP with our heuristic outperforms Regular-DP with Hahnle's heuristic [4]. Concerning local search algorithms we describe a new procedure, Regular-GSAT. It is an extension of GSAT [10] to the framework of regular CNF formulas that we have designed and implemented in C++. As far as we know, Regular-GSAT is the rst local search algorithm developed so far for many-valued logics. In order to compare the performance of Regular-DP and Regular-GSAT we performed a series of experiments in a class of computationally dicult instances we identi ed in a previous work [7]. They are satis able instances of the hard region of the phase transition that we observed in the regular random 3-SAT ?

Research partially supported by the project CICYT TIC96-1038-C04-03 and \La Paeria". The rst author is supported by a doctoral fellowship of the Comissionat per a Universitat i Recerca (1998FI00326).

A. Hunter and S. Parsons (Eds.): ECSQARU’99, LNAI 1638, pp. 22-31, 1999.  Springer-Verlag Berlin Heidelberg 1999

A Comparison of Systematic and Local Search Algorithms for Regular CNF Formulas

23

problem (cf. Section 5). Our experimental results provide experimental evidence that Regular-GSAT outperforms Regular-DP on that kind of instances, and suggest that local search methods can extend the range and size of satis ability problems that can be eciently solved in many-valued logics. Regular CNF formulas are a relevant subclass of signed CNF formulas (cf. Section 2). Both kinds of formulas are a suitable formalism for representing and solving problems in many-valued logics. In fact, they can be seen as the clause forms of nitely-valued logics. Given any nitely-valued formula, one can derive a satis ability equivalent signed CNF formula in polynomial time [3]. Interestingly, Hahnle [2] identi ed a broad class of nitely-valued logics, so-called regular logics, and showed in [3] that any formula of one of such logics can be transformed into a satis ability equivalent regular CNF formula whose length is linear in both the length of the transformed formula and the cardinality of the truth value set. Recently, he has found a method for translating a signed CNF formula into a satis ability equivalent regular CNF formula where the length of the latter is polynomial in the length of the former [5]. Thus, an instance of the satis ability problem in any nitely-valued logic is polynomially reducible to an instance of regular SAT. In a sense, regular CNF formulas play in nitely-valued logics the same role as (classical) CNF formulas play in classical logic. This paper is organized as follows. In Section 2 we de ne the logic of signed and regular CNF formulas. In Section 3 we describe Regular-DP and a new branching heuristic. In Section 4 we describe Regular-GSAT. In Section 5 we explain the generation of regular random 3-SAT instances and the phase transition phenomenon. In Section 6 we report on our experimental investigation. We nish the paper with some concluding remarks.

2 Signed CNF formulas In this section we de ne the logic of signed CNF formulas. Regular CNF formulas are presented as a subclass of signed CNF formulas. De nition 1. A truth value set N is a nite set fi1; i2; : : : ; ing, where n 2 N . The cardinality of N is denoted by jN j. De nition 2. Let S be a subset of N (S  N ) and let p be a propositional variable. An expression of the form S : p is a signed literal and S is its sign. The complement of the signed literal L = S : p, denoted by L = S : p, is (N n S ) : p. A signed literal S : p subsumes a signed literal S 0 : p0 , denoted by S : p  S 0 : p0 , i p = p0 and S  S 0 . A signed clause is a nite set of signed literals. A signed clause containing exactly one literal is a signed unit clause. A signed CNF formula is a nite set of signed clauses. De nition 3. The length of a sign S , denoted by jS j, is the number of truth values that occur in S . The length of a signed clause C , denoted by jC j, is the total number of occurrences of signed literals in C . The length of a signed CNF formula , , denoted by j, j, is the sum of the lengths of its signed clauses.

24

Ramon Bejar and Felip Manya

De nition 4. An interpretation is a mapping that assigns to every propositional variable an element of the truth value set. An interpretation I satis es a signed literal S : p i I (p) 2 S . An interpretation satis es a signed clause i it

satis es at least one of its signed literals. A signed CNF formula , is satis able i there exists at least one interpretation that satis es all the signed clauses in , . A signed CNF formula that is not satis able is unsatis able. The signed empty clause is always unsatis able and the signed empty CNF formula is always satis able. De nition 5. Let " i denote the set fj 2 N j j  ig and let # i denote the set fj 2 N j j  ig, where  is a total order on the truth value set N and i 2 N . If a sign S is equal to either " i or # i, for some i, then it is a regular sign. De nition 6. Let S be a regular sign and let p be a propositional variable. An expression of the form S : p is a regular literal. If S is of the form " i ( # i ), then we say that S : p has positive (negative) polarity. A regular clause ( CNF formula) is a signed clause (CNF formula) whose literals are regular.

3 The Regular-DP procedure In this section we rst describe Regular-DP [4] and then a new branching heuristic. Regular-DP is based on the following rules: Regular one-literal rule: Given a regular CNF formula , containing a regular unit clause fS : pg, 1. remove all clauses containing a literal S 0 : p such that S  S 0 ; 2. delete all occurrences of literals S 00 : p such that S \ S 00 = ;. Regular branching rule: Reduce the problem of determining whether a regular CNF formula , is satis able to the problem of determining whether , [ fS : pg is satis able or , [ fS : pg is satis able, where S : p is a regular literal occurring in , . The pseudo-code of Regular-DP is shown in Figure 1. It returns true (false) if the input regular CNF formula , is satis able (unsatis able). First, it applies repeatedly the regular one-literal rule and derives a simpli ed formula , 0 . Once the formula cannot be further simpli ed, it selects a regular literal S : p of , 0 , applies the branching rule and solves recursively the problem of deciding whether , 0 [ fS : pg is satis able or , 0 [ fS : pg is satis able. As such subproblems contain a regular unit clause, the regular one-literal rule can be applied again. Regular-DP terminates when some subproblem is shown to be satis able by deriving the regular empty CNF formula or all the subproblems are shown to be unsatis able by deriving the regular empty clause in all of them. In the pseudo-code, ,S:p denotes the formula obtained after applying the regular oneliteral rule to a regular CNF formula , using the regular unit clause fS : pg. Regular-DP can be viewed as the construction of a proof tree using a depth rst strategy. The root node contains the input formula and the remaining nodes

A Comparison of Systematic and Local Search Algorithms for Regular CNF Formulas

25

procedure Regular-DP Input: a regular CNF formula , and a truth value set N Output: true if , is satis able and false if , is unsatis able begin if , ; then return true if 2 , then return false

= ; 2 ; /* regular one-literal rule*/ if , contains a unit clause fS : pg then Regular-DP(,S :p ); let S : p be a regular literal occurring in , ; /* regular branching rule */ if Regular-DP(,S :p ) then return true; else return Regular-DP(,S :p ); 0

0

end

Figure1. The Regular-DP procedure represent applications of the regular one-literal rule. Otherwise stated, there is one node for each recursive call of Regular-DP. When all the leaves contain the regular empty clause, the input formula is unsatis able. When some leaf contains the regular empty formula, the input formula is satis able. Observe that the (classical) Davis-Putnam procedure is a particular case of Regular-DP. In order to obtain the Davis-Putnam procedure, we should take N = f0; 1g and represent a classical positive (negative) literal p (:p) by the regular literal " 1 : p (# 0 : p). Example 1. Let N = f0; 1; 2g and let , be the following regular CNF formula:

ff# 0 : p1 ; # 1 : p2 g; f" 1 : p1 ; # 0 : p2 g; f# 0 : p1 ; " 2 : p3 g; f" 2 : p2 ; " 1 : p3 g; f" 2 : p2 ; # 0 : p3 gg Figure 2 shows the proof tree created by Regular-DP when the input is , . Edges labelled with a regular CNF formula indicate the application of the regular branching rule and the regular one-literal rule using the regular unit clause in the label. Edges labelled with a regular unit clause indicate the application of the regular one-literal rule using that clause. The performance of Regular-DP depends dramatically on the heuristic that selects the next literal to which the branching rule is applied. Hahnle de ned a branching heuristic which is an extension of the two-sided Jeroslow-Wang rule [4]: given a regular CNF formula , , such a heuristic selects a regular literal L occurring in , that maximizes J (L) + J (L), where

26

Ramon Bejar and Felip Manya

, , [ f# 0 : p1 g

, [ f" 1 : p1 g

ff# 0 : p2 g; f" 2 : p2 ; " 1 : p3 g; f" 2 : p2 ; # 0 : p3 gg

ff# 1 : p2 g; f" 2 : p3 g; f" 2 : p2 ; " 1 : p3 g; f" 2 : p2 ; # 0 : p3 gg;

f# 0 : p2 g

f# 1 : p2 g

ff" 1 : p3 g; f# 0 : p3 gg

ff" 2 : p3 g; f" 1 : p3 g; f# 0 : p3 gg

f" 1 : p3 g

f" 2 : p3 g

2

2

Figure2. A proof tree created by Regular-DP

J (L) =

X 9L

0

L0

L 2C2, :L

2,jC j :

(1)

0

Taking into account the work of [6] on the Jeroslow-Wang rule in the classical setting, we propose the following de nition of J (L): J (L) =

X 9L L0

0

L 2C2, : L0

1 0 Y j N j , j S j A: @ S :p2C

2(jN j , 1)

(2)

Hahnle's de nition of J (L) assigns a bigger value to those regular literals L subsumed by regular literals L0 that appear in many small clauses. This way, when Regular-DP branchs on L, the probability of deriving new regular unit clauses is bigger. Our de nition of J (L) takes into account the length of regular signs as well. This fact is important because regular literals with small signs have a bigger probability of being eliminated during the application of the regular one-literal rule. Observe that in the case that jN j = 2 we get the same equation. In the following we will refer to the branching heuristic that uses (1) to calculate J (L) + J (L) as RH-1, and the branching heuristic that uses (2) as RH-2. In Section 6 we provide experimental evidence that RH-2 outperforms RH-1 on a class of randomly generated instances.

A Comparison of Systematic and Local Search Algorithms for Regular CNF Formulas

27

4 The Regular-GSAT procedure Regular-GSAT is an extension of GSAT [10] to the framework of regular CNF formulas; its pseudo-code is shown in Figure 3. Regular-GSAT tries to nd a satisfying interpretation for a regular CNF formula , performing a greedy local search through the space of possible interpretations. It starts with a randomly generated interpretation I . If I does not satisfy , , it creates a set, say S , formed by the variable-value pairs (p; k) that, when the truth value that assigns I to p is changed to k, give the largest decrease (it may be zero or negative) in the total number of unsatis ed clauses of , . Then, it randomly chooses a propositional variable p0 that appears in S . Once p0 is selected, it randomly chooses a truth value k0 from those that appear in variable-value pairs of S that contain p0 . Next, it changes the assignment of the propositional variable p0 to the truth value k0 . Such changes are repeated until either a satisfying interpretation is found or a pre-set maximum number of changes (MaxChanges) is reached. This process is repeated as needed, up to a maximum of MaxTries times.

procedure Regular-GSAT Input: a regular CNF formula , , MaxChanges and MaxTries Output: a satisfying interpretation of , , if found begin for i

:= 1 to MaxTries

I := a randomly generated interpretation for , ; for j := 1 to MaxChanges if I satis es , then return I ; Let S be the set of variable-value pairs of the form (p; k) that, when the truth value that assigns I to p is changed to k, give the largest decrease in the total number of clauses of , that are unsatis ed; Pick, at random, one variable p from the set fp j (p; k) 2 S g; Pick, at random, one value k from the set fk j (p ; k) 2 S g; I := I with the truth assignment of p changed to k ; 0

0

0

0

0

end for end for return end

\no satisfying interpretation found";

Figure3. The Regular-GSAT procedure One diculty with local search algorithms for satis ability testing, and in particular with Regular-GSAT, is that they are incomplete and cannot prove unsatis ability. Another diculty is that they can get stuck in local minima. To cope with this problem, the most basic strategy is to restart the algorithm after a xed number of changes like in Regular-GSAT. More strategies to escape

28

Ramon Bejar and Felip Manya

from local minima are described in [8, 9]. We plan to incorporate such strategies into Regular-GSAT in the near future. As far as we know, Regular-GSAT is the rst local search algorithm that has been de ned so far for nding models in many-valued logics. In the classical setting, GSAT has successfully solved problems that cannot be solved with the fastest systematic algorithms. In Section 6 we provide experimental evidence that Regular-GSAT outperforms Regular-DP on a class of computationally dicult instances.

5 Regular random 3-SAT problem In order to compare the performance of the algorithms and heuristics described above, we used instances of the regular random 3-SAT problem as benchmarks. This election was motivated by the fact that it is known that solving some of those instances is computationally dicult [1, 7]. Moreover, a large number of instances can be generated easily. In this section, we rst describe how to generate regular random 3-SAT instances and then how to identify computationally dicult instances. Given a number of clauses (C ), a number of propositional variables (V ) and a truth value set (N ), a regular random 3-SAT instance is produced by generating C non-tautological regular clauses. Each regular clause is produced by uniformly choosing three literals with dierent propositional variable from the set of regular literals of the form # i : p or " j : p, where p is a propositional variable, i 2 N n f>g, j 2 N n f?g, and > and ? denote the top and bottom elements of N . Observe that regular literals of the form # > : p or " ? : p are tautological. In [7] we reported experimental results on testing the satis ability of regular random 3-SAT instances with Regular-DP using heuristic RH1. We observed that (i) there is a sharp phase transition from satis able to unsatis able instances for a value of the ratio of the number of clauses to the number of variables ( VC ). At lower ratios, most of the instances are under-constrained and are thus satis able. At higher ratios, most of the instances are over-constrained and are thus unsatis able. The value of VC where 50% of the instances are satis able is referred to as the crossover point; (ii) there is an easy-hard-easy pattern in the computational diculty of solving problem instances as VC is varied; the hard instances tend to be found near the crossover point; and (iii) the location of the crossover point increases as the cardinality of the truth value set increases. Recently, we have shown experimentally that such an increase is logarithmic [1]. Given the great diculty of solving hard regular random 3-SAT instances of moderate size with systematic search algorithms, we believe that they are a suitable testbed for evaluating and comparing regular satis ability testing algorithms. Figure 4 shows graphically the phase transition phenomenon for the regular random 3-SAT problem when jN j = 7 and V = 60. Along the vertical axis is the average number of nodes in the proof tree created by Regular-DP. Along the horizontal axis is the ratio VC in the instances tested. One can clearly observe the

A Comparison of Systematic and Local Search Algorithms for Regular CNF Formulas

29

easy-hard-easy pattern as VC is varied. The dashed line indicates the percentage of instances that were found to be satis able scaled in such a way that 100% corresponds to the maximum average number of nodes. 1200 Average number of nodes % of SAT instances

1000 800 600 400 200 0 5

6

7

8

9

10 11 C/V

12

13

14

15

16

Figure4. Phase transition in the regular random 3-SAT problem

6 Experimental Results In this section we report on a series of experiments performed in order to compare (i) Regular-DP with heuristic RH-1 against Regular-DP with heuristic RH-2, and (ii) Regular-DP with heuristic RH-2 against Regular-GSAT. Such experiments were performed on a PC with a 400 Mhz ALPHA Processor under Linux Operating System. As already mentioned, the algorithms were implemented in C++. All the instances we refer to below are regular random 3-SAT instances from the hard region of the phase transition. To be more precise, the ratio of the number of clauses to the number of variables in the instances tested corresponds to the crossover point.

Experiment 1

Table 1 shows the results of an experiment performed in order to compare the performance of Regular-DP with heuristic RH-1 and Regular-DP with heuristic RH-2. We considered regular random 3-SAT instances with jN j = 3. Each procedure was run on 100 satis able instances for each one of the values of C and V shown in the table. The average time needed to solve an instance, as well as the average number of applications of the branching rule, are shown in the table. The results obtained provide experimental evidence that, at least for the kind of instances tested, Regular-DP with heuristic RH-2 outperforms Regular-DP with heuristic RH-1.

30

Ramon Bejar and Felip Manya

Regular-DP with RH-1 Regular-DP with RH-2 V C branching rule time (secs.) branching rule time (secs.) 80 487 424 1 281 0.3 120 720 4872 20 2655 4 160 972 55144 290 23566 45 200 1230 482028 3152 210560 523

Table1. Comparison between Regular-DP with heuristic RH-1 and Regular-DP with heuristic RH-2.

Experiment 2

Table 2 shows the results of an experiment performed in order to compare the performance of Regular-DP with heuristic RH-2 and Regular-GSAT on satis able, regular random 3-SAT instances with jN j = 3 and jN j = 4. Both procedures were applied to blocks of 100 satis able instances with 80, 120, 160 and 200 propositional variables. In order to obtain more accurate results, each instance was run 50 times with Regular-GSAT. The rst column contains the cardinality of the truth value set (jN j), the second contains the number of propositional variables (V ), and the third the number of clauses (C ) in the instances tested. The remaining columns display the settings of MaxTries (MT) and MaxChanges (MC) employed by Regular-GSAT, the average number of tries needed to nd a solution, the average number of changes performed in the last try, the average time needed to solve an instance with Regular-GSAT and the average time needed to solve an instance instance with Regular-DP. The running time of each instance solved with Regular-DP corresponds to the time needed to solve that instance, whereas the running time of each instance solved with Regular-GSAT is the average running time of 50 runs on that instance. Regular-GSAT Regular-DP V C MT MC tries changes l.t. time (secs.) time (secs.) 3 80 487 100 1000 13 526 0.3 0.3 3 120 720 200 2800 66 1742 6 4 3 160 972 260 6200 97 4015 26 45 3 200 1230 400 12000 136 7781 100 523 4 80 566 150 2000 31 1024 1 0.5 4 120 848 280 2800 95 2529 12 8 4 160 1133 400 8000 168 5596 52 126 4 200 1432 600 28000 270 17969 311 1013

jN j

Table2. Comparison between Regular-GSAT and Regular-DP. Looking at the average times, it is clear that Regular-GSAT scales better than Regular-DP when the number of variables in the problem instances increases.

A Comparison of Systematic and Local Search Algorithms for Regular CNF Formulas

31

Such results suggest that local search methods can extend the range and size of satis ability problems that can be eciently solved in many-valued logics.

7 Concluding remarks In this paper we have presented several original results: (i) a new branching heuristic for Regular-DP, (ii) an extension of GSAT [10] to the framework of regular CNF formulas, and (iii) an experimental comparison of the algorithms and heuristics described in the preceding sections. For regular random 3-SAT instances of the hard region of the phase transition, our results provide experimental evidence that (i) Regular-DP with our heuristic outperforms Regular-DP with Hahnle's heuristic [4], and (ii) Regular-GSAT outperforms Regular-DP. Such results suggest that local search methods can extend the range and size of satis ability problems that can be eciently solved in many-valued logics. In the near future, we plan to compare the performance of Regular-DP and Regular-GSAT for solving problems other than regular random 3-SAT and equip Regular-GSAT with more powerful strategies to escape from local minima.

References 1. R. Bejar and F. Manya. Phase transitions in the regular random 3-SAT problem. In Proceedings of the International Symposium on Methodologies for Intelligent Systems, ISMIS'99, pages 292{300. Springer LNAI 1609, 1999. 2. R. Hahnle. Uniform notation of tableaux rules for multiple-valued logics. In Proceedings International Symposium on Multiple-Valued Logic, Victoria, pages 238{ 245. IEEE Press, 1991. 3. R. Hahnle. Short conjunctive normal forms in nitely-valued logics. Journal of Logic and Computation, 4(6):905{927, 1994. 4. R. Hahnle. Exploiting data dependencies in many-valued logics. Journal of Applied Non-Classical Logics, 6:49{69, 1996. 5. R. Hahnle. Personal communication, 1998. 6. J. N. Hooker and V. Vinay. Branching rules for satis ability. Journal of Automated Reasoning, 15:359{383, 1995. 7. F. Manya, R. Bejar, and G. Escalada-Imaz. The satis ability problem in regular CNF-formulas. Soft Computing: A Fusion of Foundations, Methodologies and Applications, 2(3):116{123, 1998. 8. D. McAllester, B. Selman, and H. Kautz. Evidence for invariants in local search. In Proceedings of the 14th National Conference on Arti cial Intelligence, AAAI'97, Providence/RI, USA, pages 321{326, 1997. 9. B. Selman, H. A. Kautz, and B. Cohen. Noise strategies for local search. In Proceedings of the 12th National Conference on Arti cial Intelligence, AAAI'94, Seattle/WA, USA, pages 337{343, 1994. 10. B. Selman, H. Levesque, and D. Mitchell. A new method for solving hard satis ability problems. In Proceedings of the 10th National Conference on Arti cial Intelligence, AAAI'92, San Jose/CA, USA, pages 440{446, 1992.

Query-answering in Prioritized Default Logic

F. Benhammadi1, P. Nicolas1 , and T. Schaub2 1 LERIA, Université d'Angers 2, bd Lavoisier, F-49045 Angers cedex 01 {farid.benhammadijpascal.nicolas}@univ-angers.fr 2

Institut für Informatik, Universität Potsdam Postfach 60 15 53, D-14415 Potsdam [email protected]

Abstract. The purpose of this paper is to investigate a new methodology for default reasoning in presence of priorities which are represented by an arbitrary partial order on default rules. In a previous work, we have dened an alternative approach for characterizing extensions in prioritized default theories initially proposed by Brewka. We present here another approach for computing default proofs for query answering in prioritized normal default logic. 1

Introduction

Representing and reasoning with incomplete knowledge has gained growing importance in the recent decades. The literature contains many dierent approaches. Most of them are directly based on non-monotonic reasoning and Reiter's default logic [7] seems to be one of the best known and most popular one. In default logic, knowledge is represented by means of a set W of closed formulas and a set D of default rules of the form  : ; where ; ; are formulas respectively called prerequisite, justication and consequent. The defaults in D induce one or multiple extensions of the facts in W . Any such extension, E say, is a deductively closed set of formulas containing W such that for any  : 2 D; if  2 E and : 62 E then 2 E: Despite default logic's popularity, we encounter deciencies in the most general setting. For instance, default logic does not guarantee the existence of extensions. Moreover, it does not address the notion of specicity, which is a fundamental principle in common sense reasoning according to which more specic defaults should be preferred over less specic ones. For avoiding the rst problem, it is usually sucient to restrict oneself to a subclass of default logic given by so-called normal default theories. Such theories consist of defaults of the form  : : This class enjoys several desirable properties not present in the general case. Apart from guaranteeing the existence of extensions, normal default theories are semi-monotonic (ie. monotonic wrt the addition of defaults) and the resulting extensions are always mutually contradictory to each other. In particular the former property is very desirable from a computational point of view, since it A. Hunter and S. Parsons (Eds.): ECSQARU’99, LNAI 1638, pp. 32-42, 1999.  Springer-Verlag Berlin Heidelberg 1999

Query-Answering in Prioritized Default Logic

33

allows us to restrict our attention to default rules relevant for proving a query (cf. [7]). Unfortunately, the restriction to normal default theories (or any other subclass) does not work for addressing the failure of specicity. As an example, consider the statements typically penguins do not y,  penguins are typically birds and  typically birds y along with the corresponding default theory 1 :

  p : :f p : b b : f (1) (D; W ) = :f ; b ; f ; fpg Intuitively, starting from W , we should like to derive b and :f (a penguin is a bird that does not y). We do not want to derive f , since this violates the 

principle of specicity. In fact, being a penguin is more specic than being a bird and so we want to give the rst default priority over the third one. However, this theory gives two extensions, Th(fp; b; :f g) and Th(fp; b; f g), showing that also normal default theories cannot deal with specicity. A rst attempt to deal with specicity has been proposed in [8] by means of semi-normal defaults 2 but the problem of non-existence of extension arises again. Recently, some approaches were developed for isolating specicity information (cf. [1, 3, 4]) out of a given set of defaults. In the above example these methods allow us to detect automatically that the default p :: :f f is more specic than the default b f: f : This corresponds to the intuition sketched above. In [3], this specicity information is expressed by means of a strict partial order Pold then Pold + Pnew Jci n,u {z) else NotDone +- False End {While)

I

(2))

-

In order to find the parent set of each node, it first assumes that all the nodes have no parents. Then for each node x, it selects a node z among the predecessors of x in the ordering and adds incrementally that node to the parent set of x which increases the probability of the resultant structure by the largest amount. It stops adding parents to the node x if the number of parents goes up the limit u or when no additional single parent can increase the probability of the structure. 2.2

B Algorithm Wuntine, 911

Like K2, algorithm B is a greedy search heuristic that exploits the same function g(x,, n,) to evaluate the relationship between a node xi and its parent set ni. The main difference with K2 is that B does not require a previous ordering on the variables. For the pseudo-code of B, we used the same variables and structures as used in K2. Let B, be the network structure at a moment during the execution of the algorithm. The objective is to find which link to add in order to increase the quality of the structure, without introducing a cycle. Algorithm I3 calculates a function A which represents the difference between structure B, and structure B,' obtained by adding a link vj -. vi to B,. In other words, function A allows us to compare two relationships and to give the most probable one. The relations are the node xi with its parent set n and on the other hand, the node xi with its parent set IT, enforced by the node xj (ni ni U {xi)) : A[i, jl g(x,, ni U {x,)) - g(x, 70 (2)

-

+

If A[i, j] > 0 then xj is considered as a parent of x, and it is added to the parent set a,. This link added needs to avoid cycles, otherwise, function A is saturated (that means A[i, j] + -m). If an arc vm+ v, is added, only values A[k, m], m = l , ...,n, need to be recalculated. In the pseudo-code, A, is the set of indices of the ascendants of a node xi and D, is the set of indices of descendants of node xi including i.

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225

Al~orithmB For each i E {l..n) do xi = 0 For i = 1 to n, j = 1 to n do if i * j then A[i, j] g(xl, {x,)) - g(x,, 0) else A[i, j] + - m {elimination x, -. x,) Repeat Choose i, j that maximize A[i, j] If A[i, j] > 0 then xi + q U {xj) for a E A,, b E Di do A[a, b] - oo {elimination of loops) fork = 3, t o n do if A[i, k] > - oo then A[i, k] g(x,, 5 U {xk)) - g(x,, ni) A[i, j] s 0 or A[i, j] = - m for all i and j Yntil +

+

+

As K2,in order to find the parent set of each node, it first assumes that all the parent

sets are empty. Then for each node x, instead of selecting a node z from the predecessors, it chooses the node among all the other nodes (z E V - {x)). It adds incrementally to the parent set of x the node which increases the probability of the resulting structure and does not produce loops.

3

New proposed solution

The starting point of this new algorithm are the two previous algorithms K2 and B presented in section 2. It is an algorithm that does not require a total ordering on the nodes. Nevertheless, we ask the expert to give his point of view concerning subsets of nodes which we call clusters. Therefore, we have to apply algorithm K2 inside the clusters to obtain several structures (one structure per cluster). Next step consists in applying algorithm B between variables in different clusters detected previously. So algorithm B is not applied on all the variables but just on the variables between clusters in order to link them. In order to improve our algorithm, we use a heuristic based on limiting the number of parents that a node can possess when applying algorithm B. We have also opted not to eliminate completely the expert in order to remain more efficient. Hence, we ask the expert about his point of view on links he is sure of and also about links which he is sure they do not exist. The experts affirmations help us to avoid many iterations. So in the initialization phase, the parent sets are not empty, but we have an initial structure. The next source of information is to ask the expert about an eventual ordering of the clusters. It means that when a cluster C, is parent of cluster C,, then we eliminate all links going from cluster C, to cluster C,. This ordering can be given by the expert whenever a link is detected between two clusters. We present successively the description of the variables and structures used, the algorithm with heuristics and the execution followed by an example.

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Datadescription

For our algorithm, we consider the same variables and structures used in K2 and B. In addition we use the following ones: C : Set of nc clusters (C,,..., C,J such that C, r l Cj = 0 , V i # j and Ui-,,no Ci = V, C,: The cluster number t, 1 s t r nc, and Pred,: Predecessors set of x, in the fhcluster. The algorithm considers the same function g(xi, xi) as K2 and B for evaluating a relationship between a node and its parent set xi,and the same function A[i, j] to compare two relationships and to give the most probable one. We use a matrix M of n + 1 rows and n columns for representing all existing binary links among n variables. The last row is allowed to compute the number of parents (U) that a node x, possesses at a moment. This matrix is initialized to 0 which corresponds to an initial structure where no variables are linked yet. We still don't have any information about the existence or the non existence of a link (only for this type of relations xi + xi which are cycles that we can eliminate from the beginning). M[i, j] corresponds to the relationship between node x, and node x,. We denote by zero (0)if the arc is still not evaluated, by one (1) if the arc is detected and means that x, is a parent of x, (q + q U {xi)) and by mines one (-1) if the arc is eliminated definitely and is excluded from the structure. Excluded arcs enable us to avoid useless calculations since we know that these links should not exist. Later on, as we go along the procedure of the algorithm, we present the process of updating matrix M. Sometimes, some links are added and other times they are excluded. So the tests are exclusively done on the arcs which have not been treated. The matrix M is the state of the structure at any time during the algorithm process, until we have the last iteration and consequently the best structure.

3.2

Initialization

In this section, we present the initialization phase (Fig.1.) that consists first on asking the expert to form the different clusters. He selects variables that he thinks have relationships between each other. Once these clusters formed, he can give his opinion about an ordering of these variables. The next step is to ask the expert about the links that he is sure of the existence and about those he is sure of the non existence. We have from the start an initial structure that allows us to avoid useless computations. The algorithm completes the structure by the links that are not known and not evident. For the matrix M, we add some links (M[i, j] = 1) and we exclude others (M[i, j] = -1). So before starting the algorithm, some parent sets are not empty. A third step consists of asking the expert about an eventual ordering between the several clusters previously formed. For example, if the expert says that cluster 1 is parent of cluster 2, it means that we will eliminate all links going from cluster 2 to cluster 1. So we eliminate such links and avoid to calculate them. Of course this ordering between clusters is given only when the expert is sure about his statements and, when it is not the case, this step is skipped in the initialization phase. But, after

Optimized Algorithm for Learning Bayesian Network from Data

227

applying the algorithm, when the first arc linking any two clusters appears, we ask the question again to the expert using an interactive process and apply all what should be done in that case. We also ask about the maximum number of parents which a node can have during the algorithm process.

I I

Berrin Initialization

v

1 1

Initialization of Matrix M 1

I

Composition of Clusters

v

Ordering variables in the clusters 1

Detection of sure links 1

I

Ordering Clusters w

I

Maximum number of parents

[

End Initialization

I

1

Fig. 1. Initialization phase of K28

3.3

Description of the algorithm K2B

Once the initialization phase achieved, we describe now the algorithm process (Fig.2.). The following step consists in applying the algorithm K2 on all the clusters (nc times). We obtain several sub-structures that must be Linked between each other. In the case where we have only one unit (nc = 1) and consequently we have an ordering to all the variables, the algorithm ends since we have obtained the desired structure and links between all the variables, thanks to K2 algorithm. In the opposite case, when there is more than one cluster (nc > I), we continue the algorithm process with the initialization phase of algorithm B. We compute function A @reviously determined in section 2.2) for all nodes between clusters (example: node x, E C, with all nodes x, E C,,and so on). Meanwhile, we exclude the links not detected in the same cluster and consider only links not treated (M[i, j] = 0). The last step consists in selecting nodes xi and xiwhich maximize function A. If A[i, j] is positive, we must update the structure. Then, we must exclude links which can introduce cycles in the structure and recalculate new values of function A, considering all changes. This task is repeated until all values of function A are saturated or negative. It means that no more updating can improve the structure. Once this state is achieved, it means that we have the most probable structure and then we can list for each node, its parent set n,. For the construction of the network we can read the final matrix M. Each time we have the value M[i, j] = 1, we create an arc going from node xito node xi.

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Fedia Khalfallah and Khaled Mellouli

I

j

Begin - K2B w

I

I

Initialization

-

Step 1 : Initialization phase

I

Step 2

Procedure K2

no

, no

I

I

I

J'"

,"

I

Print ni for all 1 < i < n

[

Step 3 : Apply B

* End

I

StepI 4 : Developing structure

]

Fig. 2. Algorithm process of K2B

4

Example

In this section, we present an example illusbating the execution of our algorithm. Let V be composed of 9 variables {x,, ..., x,), D be the database of cases and M the matrix of n + l rows and n columns. The different evaluations provided by the database are, in the following example fictive. The maximal number of parents that a node can possess is u = 3. All new modifications are written in bold.

Optimized Algorithm for Learning Bayesian Network from Data

4.1

229

Initialization phase

Initialization of matrix M k t M be the matrix previously initialized at 0 (since no arcs have been treated yet) except 1-cycles (links of type: xi xi) which can be excluded from the start (M[i, j]= -1 for all i = j). Composing the clusters and ordering the variables The expert has proposed three clusters. He was able to give his opinion about the ordering of the variables in these three sets. But he does not have any information about the relationship of the variables between clusters. For example, he is sure that X, is not a parent of x,, and, x, and x, can be parents of x,. But he can not say anything about x, and x, or x,. C I = [XI,X2,XP x ~ ] ! 2' = '91' 3' = IxS, '63 XI] Since we have these orderings, we can exclude the following links in the matrix M : M[1, 21 = -1 ; M[1,3] = -1 ;M[1,4] = -1 ; M[2,3] = -1 ; MI2, 41 = -1 ; M[3,4] - -1. fxI19

Detection of sure links According to the expert, we add to the parent sets (n,, ..., n,) the nodes that he is sure of. If it is not the case, the parent sets stay empty. For this example, the expert is sure of links going from node x, to x, and from x, to x,. The list of parent sets is as follows : n, =0,q = {x,), x, =0,x, =0,n, =0,n, = (x,), n, =0,JC,= 0,JC, = 0. On the another hand, he is sure of the non existence of the link between variable x, to x,. The matrix is then updated as follows : M[2, 1]=1 ; M[6,5]=1 ; M[9, I]=-1. Ordering on the clusters The expert says that cluster C, is parent of cluster C,. This information permits to exclude all links going from cluster C, to cluster C,. We have no information about their relationship with cluster C,. M[5, 81 = -1 ; M[5, 91 = -1; M[6, 81 = -1 ; M[6, 91 - -1 ; M[7,8] = -1 ; M[7,9] = -1.

Table 1. Matrix after the initialization phase

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Fedia Khalfallah and Khaled Mellouli

Step2: Apply K2

For each cluster, we apply algorithm K2 to determine the structures according the orderings of the variables given by the expert. The following results are obtained: JC1 =a, n; ={XI), 5 ={xz)* JC, "{X~},X*=a,JCb ={x5), q ={x,), Jill -0, q ={x*).

Table 2. Matrix after step 2

4.3

Step 3: Apply B

The next step consists in computing function A for all nodes between clusters. The number of links calculated is 52 and is obtained by this formula : ~ x ( X ~ = ~ . ~ ; ~ = ~ + ~=~2 IxC( I~cI iXI xI CI C~~I +I) I C ~ I X I C+~ II C > I x [ C ~ O =2x(4x2+4x3+2x3)=52 If we had to apply algorithm B for all the nodes, we would have been obligcd to compute 72 links. For this example, we have only 45 computations to do thanks to the heuristics applied before. We have to evaluate all links such that M[i, j] = 0 and then apply algorithm B. The values of A[i, j j are computed from the database using formula (2). The maximum value is selected in each iteration, and gives us the link to be added to the structure. Once selected, we must exclude all links able to introduce cycles in the structure. Then we must recalculate function A taking into account the modifications. Iteration 1: A[6, 41 is the maximum value. It means that x, is a parent of x, (n, + n, U{x,)). Tn avoid introducing cycles, we must exclude all the links A[a, b] such that a belongs to the set of ascendants of x, (4= (5, 4, 2, 1)) and b belongs to the set of descendants of x,, with node x,, (D, = (6, 7)). These are links A[5,6], A[5,7], A[4,6], A[4,7], A[2,6], A[2,7], A[1,6], A[1,7]. The two first links are already excluded since the nodes in question belong to the same cluster (C3. After these modifications, we recalculate the following links A[6,1], A[6,8], A[6,2], A[6,9], A[6,3], A[6,4]. Matrix M is updated by: M[1,6]=-1 ; M[1,7] = -1 ; M[2,6] = -1 ; M[2,7] = -1 ; M[4,6] =-1; M[4,7] =-1; M[6,4] = 1. Iteration 2: The next maximum value is A[9, 71. It means that x, is parent of x, (n, tn, U {x,)). It is the same process as before. Matrix M is updated by these values : M[1, 91 = -1 ; M[2,9] = -1 ; M[4,9] = -1 ; M[9,7] = 1.

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Iteration 3: Optimal value A[7, 31. It means that x, is a parent of x,. Matrix M is updated by : M[3,7]=-1 ; M[3,9]=-1 ; M[7, 3]=1. After detecting this link, the expert confirmed that cluster C, is parent of cluster C,. Consequently, we can exclude all links going from cluster C, to cluster C,. Matrix M is updated by : M[1,5]=-1 ; M[2,5]=-1 ; M[3,5]=-1 ; M[3,6]=-1 ; M[4,5] = -1. Iteration 4: Optimal value A[9, 61. It means that x, is a parent of x,. Matrix M is updated by : M[9, 61 = 1. After this modification, we notice that the number of parents that node x, has is 1 $1 = U = 3. It is the maximum value, so we can eliminate all links having as child node x9. In matrix M, we just put (-I) instead of all the (0) in column 9 (M[9,2] = -1 ; Mi9,3] = -1 ; M[9,4] = -1 ; M[9,5] = -1).

Table 3. Matrix after step 3

If all links not treated have a negative or saturated function A which is the case for the example, then the algorithm ends. We say that we have obtained the optimal structure and that no other changes can improve the structure. 4.4

Step 4: Developing structure

From the last matrix, we obtain the following final structure:

Fig. 3. The final structure

5

Conclusion

The problem of identification a structure of a Bayesian network for a given problem is a difficult and time-consuming task. Especially when it is a complex domain or when experts do not exist. We can use databases to overcome the difficulty raised by the absence of experts and by complexity of the domains. Several methods are developed to find the best structure that represents a problem [Herskovits, 911,

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[Cooper, 911, [Heckeman, 941, [Bouckaert, 941. But this task is still complex because w e must determine the more expressive, coherent and probable structure in less time as possible. It will be an efficient method for assisting (in some specid cases replacing) the manual expert method. In this paper, we proposed a method to detect the most probable structure. Some heuristics are used to improve the implementation of our algorithm.

Acknowledgments The authors are indebted to Professor Jean-Yves Jaffray and anonymous referees for their careful reading and valuable comments on this paper.

References [ABRAMSON, 911 Abramson, B. and Finim, A. Using belief networks to forecast oil prices. International Journal of Forecasting, 7, pp 299-315, 1991. [BERZUINI, 911 Berzuini, C., Bellmi, R. and Spiegelhalter, D. Bayesian Networks Applied to Therapy Moniforing. Proceedings of the seventh conference, July 13-15,1991, pp35-43. [BEINLICH, 891 Beinlich, I., A, Suermondt, H., J., Chavez, R., M., and Cooper, G., F. The ALARM monitoring system: A case study with two probabilistic inference techniques for belief networks. Proc. of the 2" Europ. Conf. on Ai in Medecine, London, pp 247-256, 1989. [BOUCKAERT, 941 Bouckaert, R. Probabilistic Network Construcfion Using Ihe Minimum Description Length Principle. Technical Report RW-CS-94-27,1994. [BUNTINE, 911 Buntine. W., L. Theory refinement on Bayesian network. In Proc. UAl, Vol. 7, pp 52-60, San Mateo (CA), 1991. Morgan Kaufrnann. [BUNTINE, 941 Buntine, W . L. Operations for learning with graphicrtl modeIs. Journal of Artificial Intelligence Research 2, 1994, pp 159-225. [COOPER, 911 Cooper, G. F., and Herskovits, E. A Bayesian Method for Constructing Bayesian Belief N e ~ u r k from s Databases. Proc. of the 7" conference, July 13-15,1991, p86. [HECKERMAN, 951 Heckeman, D. A Bayesian Approach to Learning Causal Networks. Technical Report MSR-TR-95-04, March 1995 (Revised May 1995). [HERSKOVITS, 911 Herskovits, E. H. Computer-Based Probabilistic-Network Construction. Ph.D. thesis, Section of Medical Informatiw, University of Pittsburgh, 1991. [UURITZEN, 881 Lauritzen, S., L., and Spiegelhalter, D., J. Local Computation with probabilities in Graphical structures and their ~pplicaiionto Expert Systems. Journal of the Royal Statistical Society B, Vol. 50, No 2, 1988. [MICHALSKI, 881 Michalski, R., S., and Chilausky, R., L. Learning by being told and learning from eramples: An experimental comparison of the two methods of knowledge acquisition in the c o n f a t of developing an experf .yvstem for soybean disease diagnosis. Internat. J. Policy Anal. and Inform. Systems 4(2), 1980. [PEARL, 881 Pearl, J. Probabilistic Reasoning in intelligent Systems. Morgan Kaufmann, San Mateo, California, 1988. [SINGH, 951 Singh, M,, and Valtorta, M. Construction of Bayesian Nefwork Struchrres From Data; A Brief Survey and an Eficient Algorirhrn. International Journal of Approximate Reasoning 12, 1995, ppl11-131.

Merging with Integrity Constraints ? Sebastien Konieczny and Ramon Pino Perez Laboratoire d'Informatique Fondamentale de Lille Universite de Lille 1, 59655 Villeneuve d'Ascq, France fkonieczn,[email protected]

We consider, in this paper, the problem of knowledge base merging with integrity constraints. We propose a logical characterization of those operators and give a representation theorem in terms of preorders on interpretations. We show the close connection between belief revision and merging operators and we show that our proposal extends the pure merging case (i.e. without integrity constraints) we study in a previous work. Finally we show that Liberatore and Schaerf commutative revision operators can be seen as a special case of merging. Abstract.

1

Introduction

An important issue of distributed knowledge systems is to be able to determine a global consistent state (knowledge) of the system. Consider, for example, the problem of the combination of several expert systems. Suppose that each expert system codes the knowledge of an human expert. To build an expert system it is reasonable to try to combine all these knowledge bases in a single knowledge base that expresses the knowledge of the expert group. This process allows to discover new pieces of knowledge distributing among the sources. For example if an expert knows that a is true and another knows that a b holds, then the \synthesized" knowledge knows that b is true whereas none of the expert knows it. This was called implicit knowledge in [8]. However, simply put these knowledge bases together is a wrong way since there could be contradictions between some experts. Some logical characterizations of merging have been proposed [18, 19, 13, 14, 16, 15, 11]. In this paper we extend these works by proposing a logical characterization when the result of the merging has to obey to a set of integrity constraints. We de ne two subclasses of merging operators, namely majority merging and arbitration operators. The former striving to satisfy a maximum of protagonists, the latter trying to satisfy each protagonist to the best possible degree. In other words majority operators try to minimize global dissatisfaction whereas arbitration operators try to minimize individual dissatisfaction.

!

? The proofs have been omitted for space requirements but can be found in the ex-

tended version of this work [12].

A. Hunter and S. Parsons (Eds.): ECSQARU’99, LNAI 1638, pp. 233-244, 1999.  Springer-Verlag Berlin Heidelberg 1999

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We also provide a representation theorem a la Katsuno Mendelzon [9] and we show the close connections between belief revision and merging operators. In section 2 we state some notations. In section 3 we propose a logical definition of merging operators with integrity constraints, we de ne majority and arbitration operators and give a model-theoretic representation of those operators. In section 4 we de ne two families of merging operators illustrating the logical de nition. In section 5 we show the connections with other related works, rst we show the close connection between belief revision and merging operators, then we show that this work extends the one of [11]. Finally we show that Liberatore and Schaerf commutative revision operators can be seen as a special case of merging operators. In section 6 we give some conclusions and discuss future work.

2

Preliminaries

We consider a propositional language L over a nite alphabet P of propositional atoms. An interpretation is a function from P to f0; 1g. The set of all the interpretations is denoted W . An interpretation I is a model of a formula if and only if it makes it true in the usual classical truth functional way. Let ' be a formula, mod(') denotes the set of models of ', i.e. mod(') = fI 2 W I j= 'g. A knowledge base K is a nite set of propositional formulae which can be seen as the formula ' which is the conjunction of the formulae of K . Let '1 ; : : : ; 'n be n knowledge bases (not necessarily dierent). We call knowledge set the multi-set consisting of those n knowledge bases: = V the conjunction f'1 ; : : : ; 'n g. We note bases of , i.e. V = ' ^


^ ' . The union of multi-sets ofwillthebe knowledge noted t . Knowledge bases 1 n will be denoted by lower case Greek letters and knowledge sets by upper case Greek letters. Since an inconsistent knowledge base gives no information for the merging process, we will suppose in the rest of the paper that the knowledge bases are consistent.

V

De nition 1. A knowledge set isVconsistent if and only if is consistent. We will use mod( ) to denote mod( ) and write I j= for I 2 mod( ). De nition 2. Let 1 ; 2 be two knowledge sets. 1 and 2 are equivalent, noted 1 $ 2 , i there exists a bijection f from 1 = f'11 ; : : : ; '1n g to 2 = f'21 ; : : : ; '2n g such that ` f (') $ '. A pre-order  over W is a re exive and transitive relation on W . A pre-order is total if 8I; J 2 W I  J or J  I . Let  be a pre-order over W , we de ne < as follows: I < J i I  J and J 6 I , and ' as I ' J i I  J and J  I . We wrote I 2 min(mod('); ) i I j= ' and 8J 2 mod(') I  J . By abuse if ' is a knowledge base, ' will also denote the knowledge set = f'g. For a positive integer n we will denote n the multi-set f ; : : : ; g.

| {z } n

Merging with Integrity Constraints

3

235

Merging with Integrity Constraints

We rst state a logical de nition for merging with integrity constraints operators (IC merging operators for now on), that is we give a set of properties an operator has to satisfy in order to have a rational behaviour concerning the merging. In this work, the result of the merging has to obey a set of integrity constraints where each integrity constraint is a formula. We suppose that these constraints do not contradict each others and we bring them together in a knowledge base . This knowledge base represents the constraints for the result of merging the knowledge set , not for the knowledge bases in . Thus a knowledge base ' in does not obey necessarily to the constraints. We consider that integrity constraints has to be true in the knowledge base resulting of merging , that is the result is not only consistent with the constraints (as it is often the case in databases), but it has to imply the constraints. We will consider operators mapping a knowledge set and a knowledge base  to a knowledge base  ( ) that represents the merging of the knowledge set according to the integrity constraints .

4

De nition 3. ing postulates:

(IC0) (IC1) (IC2) (IC3) (IC4) (IC5) (IC6) (IC7) (IC8)

4

4 is an IC merging operator if and only if it satis es the follow-

4( ) ` 

4

If  V is consistent, then ( ) is consistentV If is consistent with , then  ( ) =  If 1 2 and 1 2 , then 1 ( 1 ) 2 ( 2 ) 0 0 If '  and '0 , then  (' '0 ) ' 0  (' ' ) ' 0  ( 1 )  ( 2 )  ( 1 2 ) If  ( 1 )  ( 2 ) is consistent, then  ( 1 2 )  ( 1 )  ( 2 ) 1 ( ) 2 1 ^2 ( ) If 1 ( ) 2 is consistent, then 1 ^2 ( ) 1 ( )

4 ^ $ $ 4 $4 ` ` 4 t ^ ?)4 t ^ ? 4 ^4 `4 t 4 ^4 4 t `4 ^4 4 ^ `4 4 ^ 4 `4

The meaning of the postulates is the following: (IC0) assures that the result of the merging satis es the integrity constraints. (IC1) states that if the integrity constraints are consistent, then the result of the merging will be consistent. (IC2) states that if possible, the result of the merging is simply the conjunction of the knowledge bases with the integrity constraints. (IC3) is the principle of irrelevance of syntax, i.e. if two knowledge sets are equivalent and two integrity constraints bases are logically equivalent then the knowledge bases result of the two merging will be logically equivalent. (IC4) is the fairness postulate, the point is that when we merge two knowledge bases, merging operators must not give preference to one of them. (IC5) expresses the following idea: if a group 1 compromises on a set of alternatives which I belongs to, and another group 2 compromises on another set of alternatives which contains I , so I has to be in the chosen alternatives if we join the two groups. (IC5) and (IC6) together state that if you could nd two subgroups which agree on at least one alternative, then the result of the global arbitration will be exactly those alternatives the two groups agree on. (IC5) and (IC6) have been proposed by Revesz [19] for

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weighted model- tting operators. (IC7) and (IC8) are a direct generalization of the (R5-R6) postulates for revision. They states that the notion of closeness is well-behaved (see [9] for a full justi cation). Now we de ne two merging operators subclasses, namely majority merging operators and arbitration operators. A majority merging operator is an IC merging operator that satis es the following majority postulate: (Maj) 9n 4 ( 1 t 2 n ) ` 4( 2 ) This postulate expresses the fact that if an opinion has a large audience, it will be the opinion of the group. An arbitration operator is an IC merging operator that satis es the following postulate:

9 >= 2) >; ) 41_2 ('1 t '2) $ 41 ('1 )

4 ' $ 4 ' 4 $: ' t ' $  $ : 1 ( 1)

(Arb)  01  2 ( 1 2

2 ( 2)

1

2)

( 1

2 0 1

This postulate ensures that this is the median possible choices that are preferred. Now that we have a logical de nition of IC merging operators, we will de ne a representation theorem that give a more intuitive way to de ne IC merging operators. More precisely we show that to each IC merging operator corresponds a family of pre-orders on possible worlds. Let's rst de ne the following:

De nition 4. A syncretic assignment is a function mapping each knowledge set to a total pre-order  over interpretations such that for any knowledge sets ; 1 ; 2 and for any knowledge bases '1 ; '2 : 1. 2. 3. 4. 5. 6.

If I j= and J j= , then I ' J If I j= and J 6j= , then I < J If 1  2 , then  1 = 2 8I j= '1 9J j= '2 J '1 t'2 I If I  1 J and I  2 J , then I  1 t 2 J If I < 1 J and I  2 J , then I < 1 t 2 J

A majority syncretic assignment is a syncretic assignment which satis es the following: 7. If I < 2 J , then 9n I < 1 t 2 n J A fair syncretic assignment is a syncretic assignment which satis es the following:

I . So there is no direct relationship between arbitration in the sense of [11] and IC arbitration. Notice that (A7) expresses only a kind of non-majority rule and thus is not a direct characterization of arbitration, whereas (Arb) de nes in a more positive manner the arbitration behaviour.

4 is an IC merging operator, then 4> is a pure merging operator (i.e. it satis es (A1-A6)). Furthermore if 4 is an IC majority merging operator, then 4> is a pure majority merging operator. Theorem 8. If

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Sebastien Konieczny and Ramon Pino Perez

5.3 Liberatore and Schaerf Commutative Revision This section addresses the links between merging operators and those de ned by Liberatore and Schaerf. The postulates given by Liberatore and Schaerf [13, 14] for commutative revision are the following:

(LS1) '   $   ' (LS2) ' ^  implies '   (LS3) If ' ^  is satis able then '   implies ' ^  (LS4) '   is unsatis able i both ' and  are unsatis able (LS5) If '1 $ '2 and 8 1 $ 2 then '1  1 $ '2  2 (LS6)

'  ( _ )

=

or y. As this inequality holds for each y 2 I; y < x, we obtain that W !2 [f (! ) ^ (f! g)]  x holds. Consequently, W

!2 [f (! ) ^ (f! g)] 

W



I



I

x 2 I : x < f d = f d:

(3) H

To arrive at a contradiction, suppose that W!2 [f (!) ^ (f!g)] > f d holds. Then there exists !0 2 such that I

f (!0) ^ (f!0 g) > f d = W2I [ ^ (f! 2 : f (!)  g)]

(4)

holds. Setting  = f (!0) we obtain that (f! 2 : f (!)  g)  (f!0 g) holds. Consequently, I

f d = W2I [ ^ (f! 2 : f (!)  g)]   W=f (!0 ) [ ^ (f! 2 : f (!)  g)] = = f (!0 ) ^ (f! 2 : f (!)  f (!0)g)  f (!0 ) ^ (f!0g);

(5)

but this result contradicts the inequality (4). So, the equality in (3) holds and the assertion is proved. 2

3 Boolean Algebras | Preliminaries and Some Cases of Particular Interest There are many equivalent de nitions of the notion of Boolean algebra outgoing from various lists of primitive notions and various sets of axioms. For the sake of de niteness let us introduce the de nition from [4].

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Ivan Kramosil

De nition 3.1. Boolean algebra B is a quadruple hB; _; ^; :i, where (i) B is a nonempty set called the support (set) of B, (ii) _ (^, resp.) is a total binary operation called supremum (in mum, resp.) which takes B  B into B , and (iii) : is a total unary operation called complement, which takes B into B, such that the following axioms hold true for all x; y; z 2 B : (A1) x _ y = y _ x; x ^ y = y ^ x, (A2) x _ (y _ z ) = (x _ y) _ z; x ^ (y ^ z ) = (x ^ y) ^ z , (A3) x _ (y ^ z ) = (x _ y) ^ (x _ z ); x ^ (y _ z ) = (x ^ y) _ (x ^ z ), (A4) x ^ (x _ y) = x; x _ (x ^ y) = x, (A5) (x ^ (:x)) _ y = y; (x _ (:x)) ^ y = y. 2 It follows easily from axioms (A1) { (A5) that setting 0B = x ^ (:x) and 1B = x _ (:x), for an x 2 B, both 0B (the zero element of the Boolean algebra B) and 1B (the unit element of B) are de ned uniquely no matter which x 2 B may be taken. In what follow, we shall always suppose that cardB  2 holds. The binary relation B on B de ned by x B y i x ^ y = x (or, what turns to be the same, i x _ y = y), x; y 2 B , can be easily proved to possess all the properties of partial ordering on B such that _ (^, resp.) is just the supremum (in mum, resp.) operation generated by B in B . The indices B in symbols like 0B ; 1B ; B , etc. will be omitted, if no misunderstanding menaces, x B y means that x B y and x 6= y holds simulatensously. Remark. We take into consideration the menacing ambiguity resulting from the fact that we use for supremum and in mum operations over Boolean algebras the same symbols as for the standard supremum and in mum operations on the unit interval of real numbers. Let us hope that this danger will be avoided or at least minimized due to the fact that, since now on till the end of this paper, the above introduced standard operations on h0; 1i will not be used.

Let us denote by N + = f1; 2; : : :g the set of all positive integers. Given n 2 N + , let [n] denote the subset f1; 2; : : :; ng  N + and let Bn = f0; 1gn denote the space of all binary n-tuples. For each n 2 N + we de ne three Boolean algebras Bn0 ; Bn1 and Bn2 as follows. (i) Bn0 = hP ([n]); [; \; [n] ,
i, hence, Bn0 is the Boolean algebra of all subsets of the set [n] of positive integers with respect to the usual set-theoretic operations of union, intersection and complement. (ii) Bn1 = hBn ; _1 ; ^1 ; 1n ,
i, where hx1 ; : : : ; xn i _1 hy1 ; : : : ; yn i =df hx1 _ y1 ; x2 _y2 ; : : : ; xn _yn i and hx1 ; : : : ; xn i^1 hy1 ; : : : ; yn i =df hx1 ^y1 ; x2 ^y2 ; : : : ; xn ^ yn i for each x = hx1 ; : : : ; xn i; y = hy1 ; : : : ; yn i 2 f0; 1gn, So, the i-th component of x _1 y is 1 i at least one of the i-th components of x and y is 1, and the i-th component of x ^ y is 1 i both the i-th components

Boolean-Like Interpretation of Sugeno Integral

249

of x and y are 1. Finally, 1n , x = h1 , x1 ; 1 , x2 ; : : : ; 1 , xn i, so that 1n = h1; 1; : : : ; 1i is the unit element and 0n = h0; 0; : : : ; 0i is the zero element of Bn1 . Evidently, both Bn0 and Bn1 are isomorphic, their isomorphism being de ned by the mapping  : P (A) ! f0; 1gn ascribing to each A  [n] the n-tuple h(A)1 ; (A)2 ; : : : ; (A)n i such that (A)i = 1, if i 2 A; (A)i = 0 otherwise. Using this mapping, the operations _1 ; ^1 and 1n ,
can be de ned by ,




x _1 y =  ,1 (x) [ ,1 (y) ; ,
x ^1 y =  ,1 (x) \ ,1 (y) ; ,
1n , x =  [n] , ,1 (x) ;

(6)

for every x; y 2 f0; 1gn. (iii) Given n 2 N + and x = hx1 ; : : : ; xn i 2 f0; 1gn, set ,
Xn '(x) = 1 , 2,n ,1 i=1 xi 2,i :

(7)

Evidently, '(x) 2 I = h0; 1i for all x 2 f0; 1gn; '(0n ) = 0 and '(1n ) = 1. Denoting by Dn = f 2 I :  = '(x) for some x 2 f0; 1gng the set of possible values taken by the mapping ', we can easily see that ' de nes a 1 , 1 mapping between f0; 1gn and Dn . So, given ; 2 Dn , we may de ne operations _2 ; ^2 and 1
in this way: ,




 _2 = ' ',1 () _1 ',1 ( ) ; ,
 ^2 = ' ',1 () ^2 ',1 ( ) ; ,
1  = ' 1n , ',1 () ;

(8)

as ',1 (1) = 1n obviously holds. Evidently, the operations _1 ; ^1 ; _2 ; ^2 ; 1n ,
; 1
, as well as the mappings  and ' depend also on the parameter n, but in order to simplify our notations we shall not introduce this dependence explicitly unless a misunderstanding menaces. The quadruple Bn2 = hDn ; _2 ; ^2 ; 1
i is obviously a Boolean algebra isomorphic to Bn0 and Bn1 . Two properties of Bn2 are perhaps worth being mentioned explicitly. (a) For each n 2 N + , 1  = 1 ,  for all n 2 N + and all  2 Dn . (b) For each " > 0 there exists n 2 N + P such that j'(x) , b(x)j < " holds uniformly for all x 2 f0; 1gn, here b(x) = ni=1 xi 2,i is the real number in I de ned when taking x as its binary decomposition and setting xi = 0 for all i > n. The only we need is to take n such that 2,n < " holds, so that n > lg2 (1=") will do. The reason for which we shall use '(x) instead of b(x) is to ensure that 1 2 Dn , i. e., that '(1n ) = 1. On the other side, when using P ,i in nite binary sequences from f0; 1g1 and setting '(x) = 1 i=1 xi 2 for x = 1 hx1 ; x2 ; : : :i 2 f0; 1g as usual, we would arrive at serious obstacles in what follows, caused by the ambiguity of in nite binary decompositions of some real numbers from I (h1; 0; 0; : : : ; i $ h0; 1; 1; : : :i).

250

Ivan Kramosil

Let : P ([n]) ! Dn be the 1 , 1 mapping de ned by the composition of the mappings  and ', namely (A) = '((A)) = (1 , 2,n),1

Xn

i=1

(A)i 2,i

(9)

for each A  [n] = f1; 2; : : :; ng  N + . Let us de ne a partial operation of addition in Dn in this way: Let 1 ; 2 ; : : : ; m be a sequence of numbers from Dn such that the subsets ,1 (i )  [n] are mutually disjoint (itPfollows imm  = mediately, that if all  's are nonzero, m  n must hold). Then i i df i =1 Wm  (= 1 P _2 2 _2 3 _2


_2 n ). If the subsets ,1 (i ) are not mu2;i=1 i m tually P m disjoint, Pm i=1 i is not de ned. It follows immediately that, if de ned, i=1 i = i=1 i = 1 + 2 +


+ m , where + denotes the standard addition in I . A Boolean algebra B = hB; _B ; ^B ; :B i is called complete, if for all ; 6= D  B the supremum WB; x2D x (abbreviated to WB D) and the in mum VB; x2D x (abbreviated to VB D) are de ned. E. g., all the Boolean algebras Bn0 ; Bn1 ; Bn2 introduced above are complete.

4 Boolean{Valued Sugeno Integrals De nition 4.1. Let be a nonempty set, let B = hB; _; ^; :i be a complete Boolean algebra. B-valued possibilistic measure on is a mapping  : P ( ) ! B such that (;) = 0B ; ( ) = 1B , and (E [ F ) = (E ) _ (FW) for all E; F  . The possibilistic measure  is called distributive, if (A) = !2A (f!g) for all A  . 2

The identity mapping i : P ( ) ! P ( ), i. e., i(E ) = E for each E  , is also a (trivial) Boolean-valued possibilistic measure taking its values in the Boolean algebra hP ( ); [; \; ,
i of all subsets of . De nition 4.2. Let and B be as in De nition 4.1, let  : P ( ) ! B be a Bvalued possibilistic measure on , let f : ! B be a total mapping (function) ascribing a value f (!) 2 B to each ! 2 . Then the B-valued Sugeno integral (B-S. integral, abbreviately) of the function f over the space and with respect to the B-valued possibilistic measure  is de ned by I



f d = W2B [ ^ (f! 2 : f (!) B g)] :

(10)

2

As in the case of real-valued possibilistic measures, if is nite, every Bvalued possibilistic measure on is distributive by de nition, but if is in nite, this need not be the case in general. Or, take (A) = 1B for all in nite subsets of of (A) = 1B for all in nite subsets of and (W A) = 0B for all nite A  including the empty set ;. Then 1B = (A) 6= !2A (f!g) = 0B for

Boolean-Like Interpretation of Sugeno Integral

251

each in nite A  (the supposed completeness of the Boolean algebra B assures that the supremum W!2A (f!g) exists, i. e., is de ned, but not that it equals to (A)). Theorem 4.1. Let be a nonempty set, let B = hB; _B ; ^B ; :B i be a complete Boolean algebra, let  : P ( ) ! B be a distributive B-valued possibilistic measure on , let f : ! B be a mapping (function). Then the relation I



f d = WB;!2 [f (!) ^B (f!g)]

(11)

holds. The index B is purposedly locally explicited here in order to pick out the dierences between Theorems 2.2 and 4.2. 2 Proof. The proof copies that of Theorem 2.2 for the real-valued case above, but now with a careful and detailed veri cation, whether all the steps will pass also for supremum and in mum operations in Boolean algebras. Having performed such a veri cation we can see that the proof of Theorem 2.2 passes well just with operations and relations over unit interval of reals replaced by those de ned in the Boolean algebra B. 2

In what follows, we shall consider, in more detail, Bn0 n -valued possibilistic measures and Sugeno integrals, where Bn0 n is the Boolean algebra of all subsets of the Cartesian product [n]  [n] with respect to the usual set-theoretic operations. We shall describe the subsets of [n]  [n], i. e., the subsets of fhi; j i : i; j 2 N + ; 1  i; j  ng, as binary matrices, in order to facilitate our de nitions of particular classes of such matrices. So, Bn0 n = hBnn ; _ ; ^ ; Bnn ,
i, where  (i) Bnn = hxij ini;j=1 : xij 2 f0; 1g , (ii) hxij ini;j=1 _ hyij ini;j=1 = hxij _ yij ini;j=1 , (iii) hxij ini;j=1 ^ hyij ini;j=1 = hxij ^ yij ini;j=1 , n (iv) Bnn , hxij im i;j =1 = h1 , xij ii;j =1 , here xij _ yij (xij ^ yij , resp.) is the standard supremum (in mum) in f0; 1g, so that xij _ yij = 1 i xij or yij = 1, and xij ^ yij = 1 i xij = yij = 1. A matrix X 2 Bnn is called horizontal, if each of its rows is either 0n (= h0; 0; : : : ; 0i 2 Bn = f0; 1gn) or 1n (= h1; 1; : : : ; 1i)
X is called vertical, if each of its columns is either 0n or 1n . So, denoting by Bnn;h (Bnn;v , resp.) the set of all horizontal (vertical, resp.) matrices from Bnn , we obtain that

hxij ini;j 2 Bnn;h i, for all 1  i; j  n; xij = xi ; (12) n hxij ii;j 2 Bnn;v i, for all 1  i; j  n; xij = x j : Obviously, the zero matrix 0nn 2 Bnn . containing only occurrences of 0, and the unit matrix 1nn 2 Bnn , containing only occurrences of 1, are simulta=1

=1

1

1

neously horizontal and vertical, and there are the only two matrices possesing this property, so that Bnn;h \ Bnn;v = f0nn ; 1nng. Every horizontal matrix X = hxij ini;j=1 is uniquely de ned by the vector hx11 ; x21 ; : : : ; xn1 i 2 Bn ,

252

Ivan Kramosil

denoted by r(X ), every vertical matrix X = hxij ini;j=1 is uniquely de ned by the vector hx11 ; x12 ; : : : ; x1n i 2 Bn , denoted by c(X ), both the mappings r : Bnn;h $ Bn and c : Bnn;v $ Bn being 1 , 1. A matrix X 2 Bnn;h is denoted by X (n; h; );  2 Dn  I , i

'(r(X )) = '(hx11 ; x21 ; : : : ; xn1 i) = (1 , 2,n ),1

Xn

x 2,i = ; i=1 i1

(13)

a matrix X 2 Bnn;v is denoted by X (n; v; );  2 Dn  I , i

'(c(X )) = '(hx11 ; x12 ; : : : ; x1n i) = (1 , 2,n ),1

Xn

j =1

x1j 2,j = :

(14)

As can be easily seen, for each  2 Dn both the matrices X (n; h; ) and X (n; v; ) are de ned uniquely. A matrix X 2 Bnn is called product, if there are Y 2 Bnn;h and Z 2 Bnn;v such that X = Y ^ Z . The subset of all product matrices in Bnn is denoted by Bnn;p . The inclusions Bnn;h  Bnn;p and Bnn;v  Bnn;p are obvious, as X ^ 1nn = X for each X 2 Bnn;h and 1nn ^ X = X for each X 2 Bnn;v , on the other side, Bnn;p 6= Bnn (i. e., not every n  n binary matrix is product), at least if n > 2 holds. Or, given Y 2 Bnn;h and Z 2 Bnn;v , its product Y ^ Z 2 Bnn;p is uniquely determined, and each such Y (Z , resp.) is uniquely determined by r(X ) 2 f0; 1gn (by c(Z ) 2 f0; 1gn, resp.). Hence, the inequality card(Bnn;p )  card(Bnn;h )
card(Bnn;v ) = = (card(f0; 1gn))2 = (2n )2 = 22n holds, but

(15)

card(Bnn ) = card(P ([n]  [n])) = 2(n ) : (16) (2) Let us de ne the mapping ' : P ([n]  [n]) ! I = h0; 1i, which ascribes to each set X of pairs of positive integers not greater than n (identi ed with the matrix hxij ini;j=1 2 Bnn ) the real number '(2) (X ) as follows 2

,




'(2) hxij ini;j=1 = (1 , 2,n ),2

Xn

i=1

Xn

x 2,(i+j) : j =1 ij

(17)

Theorem 4.2. (i) Let X = hxij ini;j=1 ; Y = hyij ini;j=1 be disjoint in the sense that the inequality xij + yij  1 holds for each 1  i; j  n. Then '(2) (X _ Y ) = '(2) (X ) + '(2) (Y ). (ii) For all  2 Dn  I; '(2) (X (n; h; )) = '(2) (X (n; v; )) = , in particular, '(2) (0nn ) = 0; '(2) (1nn ) = 1. (iii) '(2) (X (n; h; ) ^ X (n; v; )) =  (the standard product in I ) for all ; 2 Dn . In other terms, '(2) (X ^ Y ) = '(r(X ))
'(c(Y )) for all X 2 Bnn;h ; Y 2 Bnn;v . 2

Boolean-Like Interpretation of Sugeno Integral

253

Proof. (i) Denote, for each X = hxij ini;j=1 , by (X ) the corresponding subset of [n]  [n], hence, (X ) = fhi; j i : 1  i; j  n; xij = 1g. If xij + yij  1 holds for each 1  i; j  n, then the sets (X ) and (Y ) are disjoint, moreover, (X _ Y ) = (X ) [ (Y ) holds in general. So, denoting by Kn the value (1 , 2,n ),2 , we obtain that

'(2) (X _ Y ) = Kn

,(i+j) =

hi;ji2[n][n] (xij _ yij ) 2

,(i+j) = Kn X ,(i+j) = hi;ji2(X [Y ) 2 hi;ji2(X )[(Y ) 2 X X = Kn hi;ji2(X ) 2,(i+j) + Kn hi;ji2(Y ) 2,(i+j) = Xn Xn Xn Xn = Kn xij 2,(i+j) + Kn yij 2,(i+j) =

= Kn

X

X

i=1

j =1

i=1

= '(2) (X ) + '(2) (Y );

(18)

j =1

and (i) is proved. (ii) Let  2 Dn , let X (n; h; ) = hxij ini;j=1 . Then

Xn xij 2,(i+j) = (19) i =1 j =1 i Xn h Xn = (1 , 2,n ),1 i=1 (1 , 2,n ),1 j=1 xij 2,j 2,i : P As X (n; h; ) is horizontal, (1 , 2,n ),1 nj=1 xij 2,j = 1, if xi1 (= xi2 =


= xin ) =Pn1, this sum being zero, if xi1 = 0. Hence, '(2) (X (n; h; )) = (1 , 2,n ),1 i=1 xi1 2,i , but this expression equals to  just according to the

'(2) (X (n; h; )) = Kn

Xn

de nition of X (n; h; ). For X (n; v; ) the proof is quite analogous and for the particular cases of 0nn and 1nn the equalities also easily follow. (iii) First of all, let us consider the particular case X (n; h; i0 ) = hxij ini;j=1 such that i0 = (1 , 2,n),1 2,i0 for some 1  i0  n. Consequently, r(X (n; h; i0 )) = h0; : : : ; 0; 1; 0; : : :; 0i with the only unit occupying the i0 -th position, so that xi0 j = 1 for all 1  j  n; xij = 0 for all i 6= i0 and all 1  j  n. Let X (n; v; ); 2 Dn , be any vertical matrix. Then

'(2) (X (n; h; i0 ) ^ X (n; v; )) = Kn = (1 , 2,n ),2

Xn

Xn

i=1

Xn

(x ^ y ) 2,(i+j) =(20) j =1 ij ij

(xi0 j ^ yi0 j ) 2,(i0 +j) = j =1 h i Xn = (1 , 2,n ),1 2,i0 (1 , 2,n ),1 j=1 yi0 j 2,j = h ih i Xn Xn = (1 , 2,n),1 i=1 xi1 2,i (1 , 2,n),1 j=1 y1j 2,j = = '(r(X (n; h; i0 ))) '(c(X (n; v; ))) = i0
: Let X (n; h; ) = hxij ini;j=1 be such that  2 Dn . Then

X (n; h; ) = W fX (n; h; k ) : xk1 = 1g

(21)

254

Ivan Kramosil

and X (n; h; k ) are mutually disjoint for dierent k's. So, given 2 Dn , matrices X (n; h; k ) ^ X (n; v; ) are also mutually disjoint for dierent k's and W (22)  1kn;x 1 =1 [X (n; h; k ) ^ X (n; v; )] = = X (n; h; ) ^ X (n; v; ): Hence, due to the already proved part (i) of this Theorem, '(2) (X (n; h; ) h^ X (n; v; )) = (23) i X = 1in; x =1 '(2) (X (n; h; i ) ^ X (n; v; )) = k

i1

X

X





[i ] = 1in; x =1 (1 , 2,n),1 2,i = 1in; x 1 =1 1 h i Xn , n , 1 , i = (1 , 2 ) x 2 =  : i=1 i1 The theorem is proved. 2 The binary n  n matrices introduced and investigated above will play the role of elementary stones in the following construction. De nition 4.3. Let N 2 N + be a positive integer. Binary diagonal hypermatrix of order hN; ni is an N  N -matrix W = hbk` i, k; ` = 1; : : : ; N such that bk` 2 Bnn for each k; ` = 1; 2; : : :; N , i. e., each bk` is a binary n  n matrix, and bk` = 0nn for all 1  k 6= `  N . The space of all binary diagonal hypermatrices of order hN; ni will be denoted by HN;n. The mappingP' = HN;n ! h0; 1) is PN PN  N de ned by ' (hbk` ik;`=1 ) = k=1 `=1 '(2) (bk` ) (= Ni=1 '(2) (bii ), as can be easily seen), where '(2) is the mapping taking Bnn into h0; 1i and de ned above. Finally, we have at hand everything necessary to state and prove the assertion showing that there is a close connection between Boolean-valued Sugeno integrals and the classical integrals. Namely, given a function that takes a probability space into the unit interval of real numbers, there exists a Boolean-valued interpretation of the notions under consideration such that the value of the classical integral of the given function can be obtained as the value of the above de ned projection of the corresponding Boolean-valued Sugeno integral into the unit interval of real numbers. Theorem 4.3. Let = f!1 ; !2 ; : : : ; !N g be a nite nonempty space, let n 2 N + PN be xed, let p : ! Dn be a probability distribution on , i. e., i=1 p(!i ) = 1, let f : ! Dn be a function. For each 1  i  N , let Wp;i = hbk` iNk;`=1 2 HN;n be such that bii = X (n; h; p(!i )); bk` = 0nn for every hk; `i 6= hi; ii, and let Wf;i = hdk` iNk;`=1 2 HN;n be such that bii = X (n; h; p(!i)); bk` = 0nn for every hk; `i 6= hi; ii, and let Wf;i = hdk` iNk;`=1 2 HN;n be such that dii = X (n; v; f (!i )); dk` = 0nn for every hk; `i 6= hi; ii. Then =

i

'

i

N  N (Wp;i ^ Wf;i ) = X p(!i ) f (!i ): ;i=1 i=1

W

(24)

2

Boolean-Like Interpretation of Sugeno Integral

255

Proof. An easy calculation yields that

' =

W

XN



N (Wp;i ^ Wf;i ) = XN ' (Wp;i ^ Wf;i ) = ;i=1 i=1

(25)

'(2) (X (n; h; p(!i)) ^ X (n; v; f (!i ))) =

i=1 XN = i=1 p(!i ) f (!i );

applying Theorem 4.2 (iii) and recalling the obvious fact that the hypermatrices Wp;i ^ Wf;i are disjoint for dierent i's. 2 Obviously, the expression WN;i=1 (Wp;i ^ Wf;i ) can be taken as the value of the Boolean-valued Sugeno integral of the function f : ! I (with the values f (!i ) encoded by the matrices Wf;i ) with respect to the possibilistic measure  on nite = f!1; !2 ; : : : ; !N g (the particular values (f!i g) being de ned by the matrices Wp;i ). This Boolean-valued Sugeno integral takes its values in the Boolean algebra de ned over HN;n by the operations extending _ ; ^ , and 1nn ,
in the natural way to HN;n. Applying the mapping ' to this value PN from HN;n, the value i=1 p(!i ) f (!i ), i. e., that of the classical integral of the function f with respect to the probability distribution p on is obtained. The items [5] and [2] below introduce the notions of the possibilistic measure and possibility theory, [1] presents a detailed explanation of the present state of this domain. Besides [4], also [3] can be used as an introduction dealing with Boolean algebras and related notions.

Acknowledements This work has been sponsored by the grant no. A 1030803 of the GA AS CR.

References 1. 2. 3. 4. 5.

G. De Cooman: Possibility Theory | Part I, II, III. Int. J. General Systems 25 (1997), no. 4, pp. 291{323 (Part I), 325{351 (Part II), 353{371 (Part III). D. Dubois, H. Prade: Theorie des Possibilites. Masson, Paris, 1985. R. Faure, E. Heurgon: Structures Ordonnees et Algebres de Boole. Gauthier{ Villars, Paris, 1971. R. Sikorski: Boolean Algebras | second edition. Springer Verlag, Berlin { Gottingen { Heidelberg { New York, 1964. L. A. Zadeh: Fuzzy Sets as a Basis for a Theory of Possibility. Fuzzy Sets and Systems 1 (1978), no. 1, pp. 3{28.

An Alternative to Outward Propagation for Dempster-Shafer Belief Functions Norbert Lehmann and Rolf Haenni Institute of Informatics, University of Fribourg 1700 Fribourg, Switzerland [email protected] Abstract. Given several Dempster-Shafer belief functions, the framework of valuation networks describes an efficient method for computing the marginal of the combined belief function. The computation is based on a message passing scheme in a Markov tree where after the selection of a root node an inward and an outward propagation can be distinguished. In this paper it will be shown that outward propagation can be replaced by another partial inward propagation. In addition it will also be shown how the efficiency of inward propagation can be improved.

1

Introduction

Dempster-Shafer belief function theory (see [3] and [10]) can be used to represent uncertain knowledge. A piece of evidence is encoded by a belief function. Given many pieces of evidence represented by ϑ1 , . . . , ϑm and one (or many) hypotheses H, the problem to solve is to marginalize the combined belief function ϑ1 ⊗ . . . ⊗ ϑm in an efficient way to the set of variables of the hypothesis H. The framework of valuation networks [12] turns out to be useful for solving this problem because it allows to distribute the belief functions on a Markov tree. The computation is then based on a message passing scheme where the two main operations combination and marginalization are always performed locally on relatively small domains. Depending on the messages which are sent between neighboring nodes of the Markov tree, at least four different architectures can be distinguished: • • • •

the the the the

Shenoy-Shafer architecture (see [14]), Lauritzen-Spiegelhalter architecture (see [7] and [8]), HUGIN architecture (see [4] and [5]), Fast-Division architecture (see [2]).

In [16] a comparision of the former three architectures is given for propagating probability functions. The most popular architecture for propagating belief functions is the Shenoy-Shafer architecture. The Lauritzen-Spiegelhalter architecture and especially the HUGIN architecture are more interesting in the fleld of Bayesian Networks [9]. Finally, the Fast-Division architecture is an attempt to apply ideas contained in the HUGIN architecture and the LauritzenA. Hunter and S. Parsons (Eds.): ECSQARU’99, LNAI 1638, pp. 256-267, 1999.  Springer-Verlag Berlin Heidelberg 1999

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Spiegelhalter architecture for propagating belief functions efficiently. In this paper the Shenoy-Shafer architecture will be used. However, the ideas presented here do not depend on the architecture. For each of these architectures an inward propagation phase and an outward propagation phase can be distinguished. The main contribution of this paper is to show that it is always possible to replace outward propagation by another partial inward propagation. Furthermore, since only inward propagation is required, it is important to speed up inward propagation. The second contribution of this paper is to develop a heuristic to improve the performance of the inward propagation phase. Intermediate results are stored during the flrst inward propagation phase such that the number of combinations needed is minimized. The paper is divided as follows: in Section 2, multivariate Dempster-Shafer belief functions are introduced. Section 3 describes the Shenoy-Shafer architecture. In Section 4, it is shown that instead of an inward propagation followed by an outward propagation, it is sufficient to perform an inward propagation followed by another partial inward propagation. Finally, some improvements to speed up inward propagation are presented in Section 5.

2

Dempster-Shafer Belief Functions

In this section, some basic notions of multivariate Dempster-Shafer belief functions are recalled. More information on the Dempster-Shafer theory of evidence is given in [3], [10], and [15]. Variables and Conflgurations. We deflne £x as the state space of a variable x, i.e. the set of values of x. It is assumed that all variables have flnite state spaces. Upper-case italic letters such as D,E,F ,... denote sets of variables. Given a set D of variables, let £D denote the Cartesian product £D = £{£x : x ∈ D}. £D is called state space for D. The elements of £D are conflgurations of D. Upper-case italic letters from the beginning of the alphabet such as A,B,... are used to denote sets of conflgurations. Projection of Sets of Conflgurations. If D and D0 are sets of variables, 0 D0 µ D and x is a conflguration of D, then x↓D denotes the projection of x to 0 D0 . If A is a subset of £D , then the projection of A to D0 , denoted as A↓D , is 0 0 obtained by projecting each element of A to D0 , i.e. A↓D = {x↓D : x ∈ A}. Extension of Sets of Conflgurations. If D and D0 are sets of variables, D0 µ D, and B is a subset of £D0 , then B ↑D denotes the extension of B to D, i.e. B ↑D = B £ £D\D0 . 2.1

Different Representations

A piece of evidence can be encoded by a belief function. Similar to complex numbers where c ∈ C can be represented in polar or rectangular form, there are

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also different ways to represent the information contained in a belief function. It can be represented as a mass function, as a belief function, or as a commonality function. In this paper commonality functions are not considered. We therefore focus on (unnormalized) mass functions and belief functions. The unusual notation [ϕ]m and [ϕ]b is used instead of mϕ and belϕ , because in our opinion it is more appropriate and convenient. One of the advantages of this notation is its ability to distinguish easily between normalized and unnormalized mass or belief functions. In accordance with Shafer [11] we speak of potentials when no representation specifled. We then write only ϕ, thus without enclosing brackets. Mass Function. A mass function [ϕ]m on D assigns to every subset A of £D a value in [0, 1], that is [ϕ]m : 2ΘD → [0, 1]. The following condition must be satisfled: X [ϕ(A)]m = 1. (1) AµΘD

Intuitively, [ϕ(A)]m is the weight of evidence for A that has not already been assigned to some proper subset of A. Sometimes, a second condition, [ϕ(∅)]m = 0, is imposed. A mass function for which this additional condition holds is called normalized, otherwise it is called unnormalized. Belief Function. A belief function [ϕ]b on D, [ϕ]b : 2ΘD → [0, 1], can be obtained in terms of a mass function: [ϕ(A)]b =

X

[ϕ(B)]m .

(2)

B:BµA

Again, if [ϕ(∅)]b = 0, then the belief function is called normalized. Note that a normalized mass function always leads to a normalized belief function and vice versa. An unnormalized mass or belief function can always be normalized. We write [ϕ]M and [ϕ]B when ϕ is normalized. The transformation is given as follows: ( [ϕ(A)]M = [ϕ(A)]B =

0 [ϕ(A)]m 1¡[ϕ(∅)]m

if A = ∅, otherwise.

[ϕ(A)]b ¡ [ϕ(∅)]b 1 ¡ [ϕ(∅)]b

(3) (4)

Given a potential ϕ on D, D is called the domain of ϕ. The sets A µ £D for which [ϕ(A)]m 6= 0 are called focal sets. We use FS(ϕ) to denote the focal sets of ϕ.

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259

Basic Operations

The basic operations for potentials are combination, marginalization, and extension. For each of these operations there are representations which are more appropriate than others. One advantage of the mass function representation is that every basic operation can be performed more or less easily. In this section, we will therefore focus on mass functions. Combination. Suppose ϕ1 and ϕ2 are potentials on D1 and D2 , respectively. The combination of these two potentials produces an unnormalized potential on D = D1 ∪ D2 : X {[ϕ1 (B1 )]m · [ϕ2 (B2 )]m : B1↑D ∩ B2↑D = A}. (5) [ϕ1 ⊗ ϕ2 (A)]m = Bi µΘDi

Marginalization. Suppose ϕ is a potential on D and D0 µ D. The marginalization of ϕ to D0 produces a potential on D0 : X 0 [ϕ↓D (B)]m = [ϕ(A)]m . (6) A:A↓D0 =B

Extension. Suppose ϕ is a potential on D0 and D0 µ D. The extension of ϕ to D produces a potential on D: ½ [ϕ(B)]m if A = B ↑D , ↑D [ϕ (A)]m = (7) 0 otherwise.

3

The Shenoy-Shafer Architecture

Initially, several potentials ϑ1 , . . . , ϑm are given. Their domains form a hypergraph. For this hypergraph a covering hypertree [6] has to be computed flrst. Then a Markov tree [1] can be constructed where ϑ1 , . . . , ϑm are distributed on the nodes. In such a way every node Ni , 1 • i • `, contains a potential ϕi on the domain Di . Each ϕi is equal to the combination of some ϑj , and ϑ1 ⊗ . . . ⊗ ϑm is always equal to ϕ1 ⊗ . . . ⊗ ϕ` . A Markov tree is the underlying computational structure of the ShenoyShafer architecture. Computation is then organized as a message passing scheme where nodes receive and send messages. 3.1

Local Computations

The Dempster-Shafer theory flts perfectly into the framework of valuation networks [12]. This framework allows to distribute potentials on a Markov tree and to use the Markov tree afterwards as underlying computational structure. The framework of valuation networks involves the two operations marginalization and combination, and three axioms which enable local computation. Potentials satisfy these axioms:

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Axiom A1 (Transitivity of marginalization): Suppose ϕ is a potential on D, and suppose F µ E µ D. Then ¡ ¢↓F ϕ↓F = ϕ↓E . (8) Axiom A2 (Commutativity and associativity): Suppose ϕ1 , ϕ2 , and ϕ3 are potentials on D1 ,D2 , and D3 , respectively. Then ϕ1 ⊗ ϕ2 = ϕ2 ⊗ ϕ1 , ϕ1 ⊗ (ϕ2 ⊗ ϕ3 ) = (ϕ1 ⊗ ϕ2 ) ⊗ ϕ3 .

(9) (10)

Axiom A3 (Distributivity of marg. over combination): Suppose ϕ1 and ϕ2 are potentials on D1 and D2 , respectively. Then ↓(D1 ∩D2 )

(ϕ1 ⊗ ϕ2 )↓D1 = ϕ1 ⊗ ϕ2

.

(11)

Note above that ”=” really means equality and is not only an equivalence relation. 3.2

Message Passing Scheme

Given a Markov tree, computation can be organized as a message passing scheme where nodes receive and send messages to neighbor nodes. A node can send a message to a neighbor node as soon as it has received a message from every other neighbor node. In such a way, a propagation can be started where the leave nodes can compute and send messages flrst. At the end of the propagation, every node will have received and sent a message from and to each of its neighbor nodes. If a node Nk0 has neighbor nodes Nk1 , . . . , Nkn , then the message from Nk0 to a neighbor node Nki is computed as ¡ ¢↑Dki (12) ϕk0 ki = (ϕk0 ⊗ (⊗{ϕkj k0 : 1 • j • n, j 6= i}))↓Dk0 ∩Dki As soon as Nk0 has received a message from each of its neighbor nodes, it can compute a potential ϕ0k0 which is equal to the global potential ϕ := ϕ1 ⊗ . . . ⊗ ϕ` marginalized to the domain Dk0 : ϕ0k0 = ϕk0 ⊗ (⊗{ϕkj k0 : 1 • j • n}).

(13)

Although the Shenoy-Shafer architecture does not impose to designate a root node, we will nevertheless designate a node Nr as root node. In practice, every node could be selected, but it is better to select a node on which some of the queries can be answered. In such a way an inward propagation phase towards the root node and an outward propagation phase can be distinguished. After inward propagation, the root node is able to compute ϕ0r = ϕ↓Dr . If then an outward propagation is performed, then every other node Nk0 can compute ϕ0k0 = ϕ↓Dk0 . Often, only a partial outward propagation depending on the query is performed. For a set H µ £Dh with Dh µ Di for a node Ni it is [ϕ↓Dh (H)]B = [ϕ↓Di (H ↑Di )]B .

(14) ↓Dh

Therefore, a node Ni such that Dh µ Di must be chosen if [ϕ be computed for a set H µ £Dh .

(H)]B has to

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261

Binary Join Trees

The Shenoy-Shafer architecture as it is presented above is not very efficient for Markov trees where nodes have more than three neighbors. During outward propagation unnecessary combinations would be recomputed. This problem can be solved by using binary join trees [13] where each node has at most 3 neighbor nodes. Binary join trees minimize the number of combinations needed by saving intermediate results during propagation. If for example a node has n neighbor nodes, it would need n · (n ¡ 1) combinations to generate all messages of this node. In contrast, a corresponding binary join tree would only need 3n ¡ 4 combinations. Fortunately, every Markov tree can be transformed into a binary join tree by introducing new nodes. For this purpose, for every node with n > 3 neighbor nodes it is sufficient to create n ¡ 3 new nodes.

4

An Alternative to Outward Propagation

Inward propagation followed by an outward propagation corresponds in a way to a complete compilation of the given knowledge allowing then to answer queries of the user very quickly. In the following we propose a method which corresponds only to a partial compilation of the given knowledge. This partial compilation results from the inward propagation phase where intermediate results are stored. Later, instead of performing a complete outward propagation phase, a partial inward propagation is performed for every query. 4.1

The Basic Idea

To understand the basic idea of the method presented in this paper, it is better to forget for a while the Shenoy-Shafer architecture. Suppose the potential ϕ has the domain d(ϕ) = D and suppose H µ £D is given. Then, according to Equation 4 [ϕ(H)]B =

[ϕ(H)]b ¡ [ϕ(∅)]b . 1 ¡ [ϕ(∅)]b

(15)

As shown by the following theorem, there is another way to compute the same result: Theorem 1. Suppose ϕ has domain d(ϕ) = D and suppose H µ £D is given. If υ is such that [υ(H c )]m = 1 and d(υ) = D, then [ϕ(H)]B =

[(ϕ ⊗ υ)(∅)]m ¡ [ϕ(∅)]m . 1 ¡ [ϕ(∅)]m

(16)

Proof. It is always [ϕ(∅)]b = [ϕ(∅)]m . In addition X X [ϕ(A)]m = [ϕ(A)]m [ϕ(H)]b = AµH

=

X

A∩B=∅

A∩H c =∅

([ϕ(A)]m · [υ(B)]m ) = [(ϕ ⊗ υ)(∅)]m .

(17)

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Therefore, the values c1 = [ϕ(∅)]m and c2 = [(ϕ ⊗ υ)(∅)]m are sufficient for calculating [ϕ(H)]B . Our method tries to compute these two values as fast as possible, that’s why we use the Shenoy-Shafer architecture. c1 is computed after the flrst inward propagation phase. If during inward propagation intermediate results are stored on the nodes, then only a partial inward propagation is needed for c2 . 4.2

Restrictions on Queries

There is a price we have to pay when the Shenoy-Shafer architecture is used: [ϕ(H ↑D )]B (which is equal to [ϕ↓Dh (H)]B ) can only be computed for sets H µ £Dh when the underlying Markov tree contains a node Ni with domain Di such that Dh µ Di . If this is not the case, then a new Markov tree has to be built. There is a second restriction which may be more severe: Our method is not able to compute queries of the form [ϕ↓Dh (H)]m and [ϕ↓Dh (H)]M . For a given set H µ Dh , our method computes only the value [ϕ↓Dh (H)]B whereas traditional methods compute the entire mass function [ϕ↓Dh ]m . 4.3

Inward Propagation Phase

Prior to the inward propagation phase a root node Nr has to be selected. In practice, every node could be selected, but it is better to select a node on which some of the queries can be answered. Often, the flrst query is used to determine the root node. After a root node Nr is designated, an inward propagation towards Nr is started. Every node Nk0 with neighbor nodes Nk1 , . . . , Nkn where Nkn denotes the node on the path to the root node then computes ¡ ¢ ϕ˜k0 = ϕk0 ⊗ ϕk1 k0 ⊗ . . . ⊗ ϕkn¡1 k0 . (18) The message ϕk0 kn is computed by an additional marginalization followed by an extension as ³ ´↑Dkn ↓D ∩D . (19) ϕk0 kn = ϕ˜k0 k0 kn The inward propagation phase is flnished as soon as Nr has combined his potential with every incoming message yielding ϕ0r which is equal to ϕ↓Dr . The value c1 = [ϕ↓Dr (∅)]m is of particular interest since it is used afterwards for normalization. Therefore, this value has to be stored somewhere. 4.4

Partial Inward Propagation

Suppose for H µ £Dh the query [ϕ↓Dh (H)]B has to be answered. For this purpose, flrst a node Ni with domain Di such that Dh µ Di has to be selected. If there is no node with this property, then a new Markov tree has to be built. But often there may be several nodes Ni such that Dh µ Di . Then it is best to choose the one which is closest to the root node Nr .

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Now, a new potential υ with d(υ) = Dh and [υ(H c )]m = 1 has to be constructed and adjoined to the node Ni . The idea behind υ is derived from propositional logic where for a set of sentences Σ and another sentence h Σ |= h

⇐⇒

Σ, ¬h |= ⊥.

(20)

A partial inward propagation is then initiated. Note that only the nodes on the path from Ni to Nr have to perform computations if intermediate results are stored during the flrst inward propagation phase. The partial inward propagation is flnished as soon as the root node Nr has received a message and has computed ϕ00r . It is then ϕ00r = (ϕ ⊗ υ)

↓Dr

.

(21)

Of particular interest is the value [(ϕ ⊗ υ)↓Dr (∅)]m because it is used to compute [ϕ↓Dh (H)]B . If we let c1 denote [ϕ↓Dr (∅)]m obtained from the flrst inward propagation phase and c2 = [(ϕ ⊗ υ)↓Dr (∅)]m from the partial inward propagation, then the following theorem holds: Theorem 2. Suppose H µ £Dh with Dh µ Di for some node Ni . In addition, suppose that c1 and c2 are deflned as explained above. Then [ϕ↓Dh (H)]B =

c2 ¡ c1 . 1 ¡ c1

(22)

Proof. According to Subsection 4.1 and if υ is such that [υ(H c )]m = 1 and d(υ) = Dh , then [ϕ↓Dh (H)]B = = = = =

[(ϕ↓Dh ⊗ υ)(∅)]m ¡ [ϕ↓Dh (∅)]m 1 ¡ [ϕ↓Dh (∅)]m [(ϕ ⊗ υ)↓Dh (∅)]m ¡ [ϕ↓Dh (∅)]m 1 ¡ [ϕ↓Dh (∅)]m [(ϕ ⊗ υ)(∅)]m ¡ [ϕ(∅)]m 1 ¡ [ϕ(∅)]m [(ϕ ⊗ υ)↓Dr (∅)]m ¡ [ϕ↓Dr (∅)]m 1 ¡ [ϕ↓Dr (∅)]m c2 ¡ c1 1 ¡ c1

(23) (24) (25) (26) (27)

Partial inward propagation is needed to be able to compute [(ϕ ⊗ υ)↓Dr (∅)]m on the root node Nr . If during the flrst inward propagation phase no intermediate results would have been stored, then a complete inward propagation would be necessary to compute [(ϕ ⊗ υ)↓Dr (∅)]m . Note that partial inward propagation is quite fast since the newly constructed potential υ contains only one focal set. Therefore, adjoining υ to the node Ni generally simplifles computations.

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The Algorithm

Suppose a Markov tree with nodes N1 , . . . , N` is given where each node Ni contains a potential ϕi and let ϕ := ϕ1 ⊗ . . . ⊗ ϕ` denote the global potential. For each Hk in {H1 , . . . , Hm }, Hi µ £Dhk , Dhk µ Di , the query [ϕ↓Dhk (Hk )]B has to be answered. The complete algorithm is then given as follows: Select root node Nr Inward Propagation : ϕ0r = ϕ↓Dr Deflne c1 := [ϕ↓Dr (∅)]m If c1 = 1 then Exit for each Hk in {H1 , . . . , Hm } do Select node Ni such that Dhk µ Di Build υ such that d(υ) = Dhk and [υ(Hk c )]m = 1 Adjoin υ on node Ni Partial Inward Propagation : ϕ00r = (ϕ ⊗ υ)↓Dr Deflne c2 := [(ϕ ⊗ υ)↓Dr (∅)]m 2 ¡c1 Output [ϕ↓Dhk (Hk )]B = c1¡c 1 Next Hk Note that if c1 = 1 is obtained then the given knowledge is completely contradictory. In this case there is nothing to do since c2 would also be equal to 1. Otherwise each query can be answered by a partial inward propagation. Note that when a node Ni has to be selected for a query it is best to choose the node which is closest to the root node Nr .

5 5.1

Some Improvements Improving the Inward Propagation

In the previous section it has been shown that an inward propagation phase followed by another partial inward propagation is sufficient to answer a query. In order to answer queries as fast as possible it is therefore important that inward propagation does not need too much time. In this subsection an algorithm is presented which minimizes the number of combinations needed by storing intermediate results during the flrst inward propagation phase. In addition the algorithm tries to determine a good sequence of combinations in order to speed up inward propagation. Suppose Nk0 has neighbor nodes Nk1 , . . . , Nkn and suppose Nkn is the node towards the root node (if Nk0 is itself the root node, then a new root node connected only to Nk0 has to be virtually inserted). During inward propagation Nk0 has to compute ϕ˜k0 = ϕk0 ⊗ ϕk1 k0 ⊗ . . . ⊗ ϕkn¡1 k0 . Because of associativity and commutativity there are many ways to compute this. Although the flnal result will always be the same there may be big differences in the time needed. Because the time needed to combine two potentials is correlated to the number of focal sets it seems natural to combine flrst potentials possessing fewer focal sets. This heuristic is used in the following algorithm:

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Ψ := {ϕk0 } ∪ {ϕki k0 : 1 • i < n} loop choose ϑk ∈ Ψ and ϑl ∈ Ψ such that |FS(ϑk )| • |FS(ξ)| for all ξ ∈ Ψ and |FS(ϑl )| • |FS(ξ)| for all ξ ∈ Ψ \ {ϑk } Ψ = Ψ ∪ {ϑk ⊗ ϑl } \ {ϑk , ϑl } until |Ψ | = 1 Every node uses this algorithm to combine its incoming messages. At the end the set Ψ contains only the potential ϕ˜k0 . If Nk0 is the root node, then ϕ˜k0 is equal to ϕ↓Dk0 , otherwise the message ϕk0 kn to the inward neighbor Nkn can easily be computed using Equation 19 by performing an additional marginalization followed by an extension. The algorithm can be visualized when for every combination a new node is created. If Nk0 has n > 1 neighbor nodes, then there are exactly n ¡ 1 new nodes with domain Dk0 created. Each of these newly created nodes serves to store the corresponding intermediate result. As an example in Figure 1 two trees are shown for a given node with 4 neighbor nodes. The tree on the left corresponds to the computation of (((ϕk0 ⊗ ϕk1 k0 ) ⊗ ϕk2 k0 ) ⊗ ϕk3 k0 ) whereas the one on the right corresponds to ((ϕk0 ⊗ ϕk1 k0 ) ⊗ (ϕk2 k0 ⊗ ϕk3 k0 )).

Fig. 1. Visualizing the Algorithm.

The method presented here corresponds to the technique of binary join trees because both methods minimize the number of combinations needed by storing intermediate results during inward propagation. But here the nodes of the Markov tree are not restricted to at most 3 neighbor nodes. But because combination is a binary operation the visualization of the algorithm leads always to a binary join tree.

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In [17] another method is presented which minimizes the number of combinations by saving intermediate results during propagation. Every message from the outward neighbors of Nk0 are combined accumulatively with the potential of Nk0 itself. In Figure 1 the binary join tree on the left where the newly created nodes are on a line corresponds to this method. If n denotes the number of neighbor Qn+1 ¡ ¢ nodes of Nk0 then there are k=2 k2 = (n+1)!·n! possible connections. Only one 2n of them corresponds to the method presented in [17] whereas our method tries to select a good connection depending on ϕk0 and the incoming messages ϕki k0 . 5.2

Selection of nodes

In the algorithm presented in Subsection 4.5 a node Ni has to be selected on which a newly constructed potential υ has to be adjoint. As already mentioned there may be many nodes Ni which could be used for this purpose but it is best to select the one which is closest to the root node Nr . When the heuristic presented in the previous subsection is used during the flrst inward propagation phase then node Ni has stored some intermediate results, each of them with the domain Di . To adjoin υ it is once again best to select the intermediate result which is closest to the root node. In such a manner the way to the root node is as short as possible.

6

Conclusion

We have shown that given a Markov tree a query can be answered by an inward propagation followed by another partial inward propagation instead of an inward propagation followed by an outward propagation. In order to answer queries as fast as possible it is therefore important to speed up inward propagation. By storing intermediate results during the flrst inward propagation phase and the use of a heuristic which combines potentials possessing fewer focal sets inward propagation can be improved.

7

Acknowledgements

This work was partially supported by grant No. 21–42927.95 of the Swiss National Foundation for Research.

References 1. R.G. Almond. Graphical Belief Modeling. Chapman & Hall, London, UK, 1995. 2. R. Bissig, J. Kohlas, and N. Lehmann. Fast-division architecture for dempstershafer belief functions. In Dov M.Gabbay, R. Kruse, A. Nonnengart, and H.J. Ohlbach, editors, First International Joint Conference on Qualitative and Quantitative Practical Reasoning ECSQARU-FAPR’97, Bad Honnef, pages 198–209. Springer-Verlag, 1997.

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3. A.P. Dempster and A.Kong. Uncertain evidence and artiflcial analysis. Journal of Statistical Planning and Inference, 20:355–368, 1988. 4. F.V. Jensen, K.G. Olesen, and S.K. Andersen. An algebra of bayesian belief universes for knowledge-based systems. Networks, 20(5):637–659, 1990. 5. F.V. Jensen, S.L. Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, 4:269– 282, 1990. 6. J. Kohlas and P.A. Monney. A Mathematical Theory of Hints. An Approach to the Dempster-Shafer Theory of Evidence, volume 425 of Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, 1995. 7. S.L. Lauritzen and F.V. Jensen. Local computation with valuations from a commutative semigroup. Annals of Mathematics and Artificial Intelligence, 21(1):51–70, 1997. 8. S.L. Lauritzen and D.J. Spiegelhalter. Local computations with probabilities on graphical structures and their application to expert systems. Journal of Royal Statistical Society, 50(2):157–224, 1988. 9. J. Pearl. Fusion, propagation and structuring in belief networks. Artificial Intelligence, 29:241–288, 1986. 10. G. Shafer. The Mathematical Theory of Evidence. Princeton University Press, 1976. 11. G. Shafer. An axiomatic study of computation in hypertrees. Working Paper 232, School of Business, The University of Kansas, 1991. 12. P.P. Shenoy. A valuation-based language for expert systems. International Journal of Approximate Reasoning, 3:383–411, 1989. 13. P.P. Shenoy. Binary join trees for computing marginals in the Shenoy-Shafer architecture. International Journal of Approximate Reasoning, 17:239–263, 1997. 14. P.P. Shenoy and G. Shafer. Axioms for probability and belief functions propagation. In R.D. Shachter and al., editors, Proceedings of the 4th Conference on Uncertainty in Artificial Intelligence, pages 169–198. North Holland, 1990. 15. Ph. Smets. Belief functions. In D. Dubois Ph. Smets, A. Mamdani and H. Prade, editors, Nonstandard logics for automated reasoning, pages 253–286. Academic Press, 1988. 16. V. Lepar and P.P. Shenoy. A comparison of Lauritzen-Spiegelhalter, Hugin and Shenoy-Shafer architectures for computing marginals of probability distributions. Uncertainty in Artificial Intelligence, 14:328–337, 1998. 17. H. Xu. An efficient implementation of belief function propagation. In D’Ambrosio B.D., Smets Ph., and Bonissone P.P., editors, Proceedings of the 7th Conference on Uncertainty in Artificial Intelligence, pages 425–432. Morgan Kaufmann, San Mateo, CA, 1997.

On bottom-up pre-processing techniques for automated default reasoning Thomas Linke and Torsten Schaub Institut fur Informatik, Universitat Potsdam Postfach 60 15 53, D{14415 Potsdam, Germany flinke,[email protected]

Abstract. In default logic, possible sets of conclusions from a default

theory are given in terms of extensions of that theory. Each such extension is generated through a set of defaults rules. In this paper, we are concerned with identifying default rules belonging to all sets of default rules generating dierent extensions. This is interesting from several perspectives. First, it allows for approximating the set of so-called skeptical conclusions of a default theory, that is, those conclusions belonging to all extensions. Second, it provides a technique usable for pre-processing default theories, because such default rules are applicable without knowing nor altering the extensions of the initial theory. The fact that our technique leaves the resulting conclusions unaected makes it thus applicable as a universal pre-processing tool to all sorts of computational tasks.

1 Introduction Default logic [6] is one of the best known and most widely studied formalizations of default reasoning due to its very expressive and lucid language. In default logic, knowledge is represented as a default theory, which consists of a set of formulas and a set of default rules for representing default information. Possible sets of conclusions from a default theory are given in terms of extensions of that theory. A default theory can possess zero, one, or multiple extensions because dierent ways of resolving con icts among default rules lead to dierent alternative extensions. In fact, any such extension is uniquely characterizable through its so-called set of generating default rules. In this paper, we are concerned with identifying default rules belonging to all sets of generating default rules. This is interesting from several perspectives. First, it allows for approximating the set of so-called skeptical conclusions of a default theory, that is, those conclusions belonging to all extensions. Second, it provides a technique usable for pre-processing default theories, because such default rules are applicable without knowing nor altering the extensions of the initial theory. The fact that our technique leaves the resulting conclusions unaffected makes it thus applicable as a universal pre-processing tool to all sorts of computational tasks, starting from testing existence of extensions, over any sort of query-answering, up to the computation of entire extensions. A. Hunter and S. Parsons (Eds.): ECSQARU’99, LNAI 1638, pp. 268-278, 1999.  Springer-Verlag Berlin Heidelberg 1999

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For accomplishing our task, we draw on the notions of blocking sets and block graphs introduced in [4]. There, these concepts were used for characterizing default theories guaranteeing the existence of extensions and for supporting query-answering. Here, we exploit the information gathered in block graphs for the aforementioned pre-processing task.

2 Background A default rule is an expression of the form  : . We sometimes denote the prerequisite  of a default rule  by p(), its justi cation by j () and its consequent

by c().1 A set of default rules D and a set of formulas W form a default theory2
= (D; W ), that may induce one, multiple or no extensions : De nition 1. [6] Let
= (D; W ) be a default theory. For any set of formulas S , let ,
(S ) be the smallest set of formulas S 0 such that 1. W S 0 ; 2. Th(S 0) = S 0 ; 3. For any  : D; if  S 0 and S then S 0 : A set of formulas E is an extension of
i ,
(E ) = E: Observe that E is a xed-point of ,
. Any such extension represents a possible set of beliefs about the world. For simplicity, we assume for the rest of the paper that default theories (D; W ) comprise nite sets only. Additionally, we assume that for each default rule  in D, we have that W j () is consistent. This can be done without loss of generality, because we can clearly eliminate all rules  from D for which W j () is inconsistent, without altering the set of extensions. We call two default theories extension equivalent, if they have the same extensions. Consider the standard example where birds y, birds have wings, penguins are birds, and penguins don't y along with a formalization through default theory (D1 ; W1 ) where 

2

2

:

62

2

[

[



D1 = b : f:ab b ; b w: w ; p b: b ; p : ::f ab p



(1)

and W1 = f abb ; f abp ; p . We let f , w , b , :f abbreviate the previous default rules by appeal to their consequents. Our example yields two extensions, viz. E1 = Th(W1 b ; w ; f ) and E2 = Th(W1 b ; w ; f ), while theory (D1 p : ab :x :x ; W1 ) has no extension. Adding ab to (1) eliminates extension E2 . Further, de ne for a set of formulas S and a set of defaults D, the set of generating default rules as GD(D; S ) =  D S p() and S j () : We f:

!

[ f

f

g

!

g

: g

[ f

2

[

b

f

1

g

b

2

j

`

6` :

g

This generalizes to sets of default rules in the obvious way. If clear from the context, we sometimes refer with
to (D; W ) or its components D and W (and vice versa) without mention.

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Thomas Linke and Torsten Schaub

see that the two extensions of (1) are generated by GD(D; E1 ) = w ; b ; :f and GD(D; E2 ) = f ; w ; b , respectively. Following [7], we call a set of default rules D grounded in a set of formulas S i there exists an enumeration i i2I of D such that we have for all i I that S c( 0 ; : : : ; i,1 ) p(i ). Note that E = Th(W c(GD(D; E ))) forms an extension of (D; W ) if GD(D; E ) is grounded in W . As proposed by [3, 1], we call a set of default rules D weakly regular wrt a set of formulas S i we have for each  D that S c(D) j (  )3 . A default logic is said to enjoy one of these properties according to its treatment of default rules that generate extensions. Reiter's default logic leads to grounded and weakly regular sets of generating default rules. For capturing the interaction between default rules under weak regularity, [4] introduced the concept of blocking sets : f

f

g

g

h

2

[

f

g

i

`

[

2

:

[

6`

f g

De nition 2. [4] Let
= (D; W ) be a default theory. For  D and B D, 2

we de ne B as a potential blocking set of , written B 7!
, i { W [ c(B ) ` :j () and { B is grounded in W .



B is an essential blocking set of , written B 
, i { B
 and { (B 0 )
00 for no 0 B and no 00 B  . Let
() = B B 
 be the set of all essential blocking sets of . These blocking sets provide candidate sets for denying the application of . We give 7!

7!

nf

B

f

g

7!

j

2

7!

2

[ f g

g

the sets of blocking sets obtained in our example at the end of this section. In what follows, we let the term blocking set refer to essential blocking sets. This is justi ed by a result in [5], showing that essential blocking sets are indeed sucient for characterizing the notion of consistency used in Reiter's default logic:4 For a set D0 D grounded in W , we have that D0 is weakly regular wrt W i we have for each 0 D0 and each B 0 D0 that B 0
(0 ). The problem with blocking sets is that there may be exponentially many in the worst case. This is why [4] put forward the notion of a block graph, as a compact abstraction of actual blocking sets. 

2



62 B

De nition 3. [4] Let
= (D; W ) be a default theory. The block graph G(
) = (V
; A
) of
is a directed graph with vertices V
= D and arcs

A
= (0 ; ) 0 B for some B f

j

2


(  )g :

2 B

(Recall that a directed graph G is a pair G = (V; A) such that V is a nite, non-empty set of vertices and A V V is a set of arcs.) We observe that the space complexity of block graphs is quadratic in the number of default rules; 3 Opposed to this, strongly regular stipulates S [ c(D) [ j (D) 6` ?. 4 We let B  B stand for B  B and B = 6 B. 

0

0



0

On Bottom-Up Pre-processing Techniques for Automated Default Reasoning

271

its construction5 faces the same time complexity as the extension-membershipproblem [2]; they are both in 2P . Note that the eorts put into constructing a block graph are however meant to amortize over subsequent tasks; notably its construction (and reduction, see operation \ " below) are both incremental. A crucial notion is that of a cycle in a block graph: A directed cycle in G = (V; A) is a nite subset C V such that C = v1 ; ; vn and (vi ; vi+1 ) A for each 1 i < n and (vn ; v1 ) A. The length of a directed cycle in a graph is the total number of arcs occurring in the cycle. Additionally, we call a cycle even (odd ) if its length is even (odd). Accordingly, a default theory is said to be non-con icting, well-ordered or even, depending on whether its block graph has no arcs, no cycles or only even cycles, respectively. [4] show that these three classes guarantee the existence of extensions. Default theory (1) yields the following blocking sets:
(f ) = :f
(w ) =
(b ) =
(:f ) = b ; f We get a block graph with vertice set D1 (indicated by white nodes) and (solid) arcs (:f ; f ), (f ; :f ) and (b ; :f ) : j





f






g

2

2

B

ff

B

;

B

;

B

ff

f ab

gg

gg

w :x

b

:f

b

The addition of ab = p ab: ab to (1) augments
(f ) by ab . We get additionally
(ab ) = , re ecting the fact that ab is unblockable, that is, applicable without consistency check. The addition to the block graph is indicated by (lightgray) node ab and (dashed) arc (ab ; f ). The further addition of :x = :: xx to (1) leaves the above blocking sets unaected and yields additionally
(:x ) = :x re ecting self blockage. This leads to an additional (lightgray) node :x and a (dotted) odd loop (:x ; :x ) in the augmented block graph. Finally, we need the following technical instrument for (what we call) shifting : ,Let
= (D; W ) be a default theory and D0 D. We de ne

D0 as D (D0 D0 ) ; W c(D0 ) where D0 =  D W c(D0 ) j () . Intuitively, the shift operation
D0 results in a default theory, simulating the application of the rule set D0 to theory
. The purpose of D0 is to eliminate defaults whose justi cation is inconsistent with the facts of
D0 . In fact, [5] shows that a set E of formulas is an extension of some default theory
i it is an extension of
D0 , provided that D0 GD(D; E ) is grounded in W . b

b

B

b

B

b

;

ff

bg

b

b

B

f

b

gg



j

:

n

[

[

g

f

2

j

[

`

j

j

j

5

That is, a corresponding decision problem.



272

Thomas Linke and Torsten Schaub

3 Pre-processing Our aim is now to exploit the information gathered in a block graph for identifying default rules belonging to all sets of generating default rules. We begin with the formal foundations for this endeavor. De nition 4. Let
= (D; W ) be a default theory and D0 D. We de ne D0 D as active in
i D0 is grounded in W and for all 0 D0 there is no  D such that (; 0 ) A
. Active rules, say, are thus applicable without consistency check since they possess no blocking sets. The block graph of our exemplary theory (D1 ; W1 ) indicates two active sets: b and b ; w . While the addition of :x results in the same (putatively) active sets, the addition of ab yields additionally ab , b ; ab , and b ; w ; ab . It should be clear that the union of all active sets is also an active set. Activeness provides us with the following two results: Theorem 1. Let
= (D; W ) be a default theory and D0 D. If D0 is an active set in
, then D0 GD(D; E ) for each extension E of
. Provided that D0 is active in
, we have that all consequences of W c(D0 ) do belong to all extensions of
; they form a subset of the so-called skeptical theorems of
(given by the intersection of all extensions). Now we may consider a bottom-up approach shifting all active default knowledge to the set of facts. This approach is backed up by the next theorem: Theorem 2. Let
= (D; W ) be default theory and D0 D. If D0 is an active set in
, then
is extension equivalent to
D0 . The applications of this result are manifold. In particular, it leads also to a reduction of the block graph: When computing the block graph G(
), we may keep track of all active blocking sets. If we let B denote their union, we may then build
B , reduce G(
) to G(
B ) and perform all subsequent tasks, like extension-construction or query-answering, relative to
B and G(
B ). We make this intuition precise in the sequel. First, we need the following de nition. Let G = (V; A) be a directed graph. A node v V is a source of G if there is no v0 V such that (v0 ; v) A, that is, there are no arcs pointing on sources. We denote the set of all sources of a graph G by SG . The idea is now to repeatedly apply the default rules indicated by the sources of the block graph. We capture the single application of all such rules in the following way. De nition 5. Let
be a default theory and G(
) its block graph. We de ne  (
) as the maximal set  (
) SG(
) being grounded in W . Observe that  (
) is unique and, according to De nition 4, it is an active set of default rules. With De nition 5 we are able to identify default rules common to all sets of generating default rules: 



2

2

2

f

g

f

g

f

b

f

bg f

bg

bg





[



j

[

j[

j[

j[

2

2



j[

2

On Bottom-Up Pre-processing Techniques for Automated Default Reasoning

273

De nition 6. Let
= (D; W ) be a default theory. De ne
0 =
, C0 =  (
0 ) and for i > 0


i+1 =
i Ci

Ci+1 =  (
i+1 ) :

and

j

S

Then, CD(D; W ) = i0 Ci is de ned as the set of core default rules of
. We note that CD(D; W ) is nite and well-de ned, since
is supposed to be nite. Also, observe the dierence between CD(D; W ) and GD(D; E ): While the latter depends on an existing extension, the former relies \merely" on the initial theory (through its block graph). We put `merely' in quotes, since the computation of a block graph is in the worst case as expensive as that of an extension. The dierence is however that an extension is an object of \on-line" computations, while a block graph (as well as the set of core default rules) is meant to be computed \o-line", so that the eorts put into its construction amortize over subsequent tasks.6 The idea is now to repeatedly extract the default rules indicated by the sources of the block graph and to shift the consequents of the \groundable" ones to the facts of the theory in focus. This process is repeated until all applicable rules at the sources have been applied. For example, we get CD(D1 ; W1 ) = fb ; w g and CD(D1 [ fabb g; W1 ) = fb ; w ; abb g :

We now elaborate upon the theories resulting from this process. To begin with, we obtain the following corollary to theorems 1 and 2: Corollary 1. Let
= (D; W ) be default theory. Then,
is extension equivalent to
CD(D; W ). Shifting the core rules in our rst two exemplary theories yields the following ones: j

(D1

ab

[ f

(D1 ; W1 ) CD(D1 ; W1 ) = ( f ; :f ; W1 b ; w ) ; W1 ) CD(D1 ab ; W1 ) = ( :f ; W1 b ; w ; ab b ) : j

bg

j

[ f

f

bg

f

g

g

[f

[f

g

g

While the rst theory results in a cyclic block graph, given on the left, the second one yields an arcless, single-noded graph, given on the right:

f

:f 6

:f

Note also that there may be an exponential number of extensions, while there is always a single block graph.

274

Thomas Linke and Torsten Schaub

According to the type of (initial) default theory (well-ordered, even or general) CD(D; W ) plays a dierent role in skeptical default reasoning. Theorem 3. Let
be a well-ordered default theory. Then, CD(D; W ) = GD(D; E ) for the (single) extension E of
. Clearly, this applies also to non-con icting theories. Otherwise, we have: Theorem 4. Let
be a default theory. Then, we have CD(D; W ) GD(D; E ) for each extension E of
. Observe that this result integrates the case where
has no extension; extensions are guaranteed for even theories (cf. Section 2). In order to actually use CD(D; W ) for skeptical reasoning, however, one has to ensure that
CD(D; W ) has an extension because there may be remaining (destructive) odd cycles in block graph G(
CD(D; W )) which kill all possible extensions of
CD(D; W ) (and with it all extensions of
). For instance, (D1 :x ; W1 ) results in CD(D1 :x ; W1 ) = b ; w , although it has no extenions. Next, we therefore elaborate upon the structure of
CD(D; W ). Theorem 5. Let
be a default theory. If G(
CD(D; W )) is acyclic, then G(
CD(D; W )) arcless. That is, shifting the set of core rules yields block graphs having either no arcs at all or at least one cycle. While the rst case amounts to computing the single extension of the underlying theory, the cycle(s) obtained in the second case indicate the dierent choices that lead to multiple putative extensions. The former can be made precise as follows. Corollary 2. Let
be a default theory. If G(
CD(D; W )) is acyclic, then
has a single extension. Existing cycles in G(
CD(D; W )) may however still lead to none, one or multiple extensions. For this take W = and in turn D = :: xx , D = :: xx ; :xx ; :::xx , and D = :xx ; :::xx , respectively. Clearly, if
is well-ordered, then G(
CD(D; W )) is arcless. The next theorem shows what happens, if
is even. Theorem 6. If
is an even default theory, then G(
CD(D; W )) is either even or arcless. In both cases,
CD(D; W ) is thus also guaranteed to have some extension. Finally, we obtain the following recursive procedure for computing the set of core default rules: Procedure core(
= (D; W ) : Default theory ) 1. construct the block graph G(
) of
2. compute  (
) from G(
) 3. if  (
) = then return else return core(
 (
))  (
). 

j

j

j

[ f

[ f

g

f

g

g

j

j

j

j

j

;

f

f

g

j

j

j

;

;

j

[

g

f

g

On Bottom-Up Pre-processing Techniques for Automated Default Reasoning

275

The next theorem shows that procedure core computes CD(D; W ): Theorem 7. If
is a default theory, then CD(D; W ) = core(
). Although the computation of CD(D; W ) is meant as an o-line compilation process, one may be reluctant to repeatedly compute the block graph at each recursive level. There are two constructive answers to this objection: First, G(
 ) is computable from G(
) by deleting arcs starting at defaults belonging to the same blocking sets as . These are indicated by the arcs in the block graph. Second, one may approximate at each level the exact block graph G(
D0 ) by simply deleting vertices D0 D0 (recall that D0 =  D W c(D0 ) j () ) and their connecting arcs in the block graph of the original theory. Denoting this modi ed procedure by core0 (
), it is easy to show that core0 (
) core(
) because G(
D0 ) has the same vertices but less arcs and thus more sources than the graph obtained by simply deleting vertices D0 D0 . jf g

j

[

f

2

j

[

` :

g



j

[

4 Breaking even cycles The idea put forward in the last section was to pre-process a default theory by starting at the outer nodes, namely the sources, of the block graph and by subsequently working inwards until we encounter either a cycle or no further arcs. While the latter case results in an extension, the former one may leave back rather large parts of the block graph that remain unexplored. This section furnishes a method for breaking up certain even cycles in order to make their successor nodes accessible to the pre-processing approach given in the previous section. We need the following graph theoretical concepts. For a directed graph G = (V; A) and vertex v V , de ne the reachable predecessors of v as (v) = S i0 i (v) where 0(v) = v and i (v) = uS(u; w) A and w i,1 (v) for i > 1. Then, for subset S V , de ne (S ) = v2S (v). Furthermore, we need: De nition 7. Let G = (V; A) be a directed graph and C V a cycle in G. C is a dominating cycle in G i C (C ) is the only cycle in the subgraph in G which is induced by nodes (C ). That is, there is no other cycle between the predecessors of C ; C is thus a cycle placed at the \outskirts" of the graph. For instance, :f ; f is a dominating cycle in the block graph of (D1 ; W1 ). Observe that such cycles do not necessarily exist, since cycles may themselves depend upon each other in a cyclic way. Even (dominating) cycles are decomposable in a natural way. De nition 8. Let G = (V; A) be a directed graph and C = v1; : : : ; vk an even cycle in G. We de ne the canonical partition of C as C1 ; C2 where C1 = v1 ; v3 ; : : : ; v2n,1 and C2 = v2 ; v4 ; : : : ; v2n for 2n = k. By de nition, we have C1 C2 = because C1 contains all nodes with even indices and C2 contains all nodes with odd indices. The canonical partition of the even cycle in the block graph of G((D1 ; W1 )) is :f ; f . 2

f g

f

j

2

2

g







f

g

f

h

f

g

f

\

g

;

hf

g f

g

i

gi

276

Thomas Linke and Torsten Schaub

Interestingly, each part of such a partition induces a dierent extension of the underlying theory:

Theorem 8. Let
be an even default theory such that G(
CD(D; W )) conj

tains a dominating cycle C and let hC1 ; C2 i be the canonical partition of C . Then C1  GD(D; E1 ) and C2  GD(D; E2 ) for dierent extensions E1 and E2 of
.

In our example, :f belongs to GD(D; E1 ) = w ; b ; :f , while f gives rise to GD(D; E2 ) = f ; w ; b . This can be put together to the following procedure: f

f

g

g

Procedure core (
= (D; W ) : Default theory ) +

1. Let CD = core(
) and
0 =
jCD /*
0 =
jCD(D; W ) */ 0 2. if G(
) has dominating cycle C then for the canonical partition hC1 ; C2 i of C return CD [ (core+ (
0 jC1 ) \ core+ (
0 jC2 )) else return CD.

We have the following result.

Theorem 9. If
is a default theory, then core(
) core (
) GD(D; E ) 

for each extension E of
.

+



For illustration, let us extent our running example (1) by defaults, expressing that \birds that can be assumed to be ight-less are ratites" and \ ying birds and ratites are animals" (leaving facts W1 = f abb ; f abp ; p unchanged): f:



b : :ab b ; b : w ; p : b ; p : :ab p f w b :f



!

 [

!

g



b : :f ; (b ^ f ) _ r : a ; ; W 1 r a



where r and a abbreviate the new default rules. We get two extensions E1 = Th(W1 b ; w ; f ; r ; a ) and E2 = Th(W1 b ; w ; f ; a ). Note that a contributes to both extensions. It is easy to verify that the default theory has the following block graph: [ f

:

g

[ f

r

f

w

a

b

:f

g

Let us now step through procedure core+ : As with our initial default theory, we obtain core rules CD = b ; w resulting in
0 = ( f ; :f ; r ; a ; W1 b ; w ). This yields the following reduced blockgraph G(
0 ): f

g

f

g

[f

g

On Bottom-Up Pre-processing Techniques for Automated Default Reasoning

277

f

r

:f

a

We get thus an even cycle C = :f ; f that dominates rules r and a , so that they are outside the scope of our pre-processing techniques. While r is still blockable, a is not grounded. Now, the canonical partition of C is :f ; f . From this, we get (i)
0 C1 = ( r ; a ; W1 b ; w f ) along with core+ (
0 C1 ) = r ; a and (ii)
0 C2 = ( a ; W1 b ; w f ) along with core+ (
0 C1 ) = a . Intersecting the results from both recursions gives a , so that our original call to core+ returns b ; w a . Although this technique allows pre-processing beyond dominating even cycles, as exempli ed by a , it should be applied with care since there may be an exponential number of such cycles in the worst case. f

hf

g f

gi

j

j

j

f

g

f

f

g

f

g

j

f

g

[f

g [ f: g

[ f

g [ f g

g

g

f

g[f

g

5 Concluding remarks We have exploited the notion of a block graph for identifying default rules belonging to all sets of default rules generating dierent extensions. This has provided us with a technique that is applicable as a universal pre-processing tool to all sorts of computational tasks. Although this proceeding faces the same computational complexity as the computation of an extension, it should be seen as an investment that amortizes over subsequent computations. In the future, it will be interesting to see in how far more general classes of even cycles or yet odd loops can be broken up, so that one might eventually end up with a method for computing entire extensions.

References 1. C. Froidevaux and J. Mengin. Default logic: A uni ed view. Computational Intelligence, 10(3):331{369, 1994. 2. G. Gottlob. Complexity results for nonmonotonic logics. Journal of Logic and Computation, 2(3):397{425, June 1992. 3. F. Levy. Computing extensions of default theories. In R. Kruse and P. Siegel, editors, Proceedings of the European Conference on Symbolic and Quantitative Approaches for Uncertainty, volume 548 of Lecture Notes in Computer Science, pages 219{226. Springer Verlag, 1991. 4. T. Linke and T. Schaub. An approach to query-answering in reiter's default logic and the underlying existence of extensions problem. In J. Dix, L. Fari~nas del Cerro, and U. Furbach, editors, Logics in Arti cial Intelligence, Proceedings of the Sixth European Workshop on Logics in Arti cial Intelligence, volume 1489 of Lecture Notes in Arti cial Intelligence, pages 233{247. Springer Verlag, 1998.

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Thomas Linke and Torsten Schaub

5. Th. Linke. New foundations for automation of default reasoning. Dissertation, University of Bielefeld, 1999. 6. R. Reiter. A logic for default reasoning. Arti cial Intelligence, 13(1-2):81{132, 1980. 7. C. Schwind. A tableaux-based theorem prover for a decidable subset of default logic. In M. Stickel, editor, Proceedings of the Conference on Automated Deduction. Springer Verlag, 1990.

Probabilistic Logic Programming under Maximum Entropy Thomas Lukasiewicz1 and Gabriele Kern-Isberner2 1

Institut fur Informatik, Universitat Gieen Arndtstrae 2, D-35392 Gieen, Germany

[email protected]

2

Fachbereich Informatik, FernUniversitat Hagen P.O. Box 940, D-58084 Hagen, Germany [email protected]

Abstract. In this paper, we focus on the combination of probabilistic

logic programming with the principle of maximum entropy. We start by de ning probabilistic queries to probabilistic logic programs and their answer substitutions under maximum entropy. We then present an ef cient linear programming characterization for the problem of deciding whether a probabilistic logic program is satis able. Finally, and as a central contribution of this paper, we introduce an ecient technique for approximative probabilistic logic programming under maximum entropy. This technique reduces the original entropy maximization task to solving a modi ed and relatively small optimization problem.

1 Introduction Probabilistic propositional logics and their various dialects are thoroughly studied in the literature (see especially [19] and [5]; see also [15] and [16]). Their extensions to probabilistic rst-order logics can be classi ed into rst-order logics in which probabilities are de ned over the domain and those in which probabilities are given over a set of possible worlds (see especially [2] and [9]). The rst ones are suitable for describing statistical knowledge, while the latter are appropriate for representing degrees of belief. The same classi cation holds for existing approaches to probabilistic logic programming: Ng [17] concentrates on probabilities over the domain. Subrahmanian and his group (see especially [18] and [4]) focus on annotation-based approaches to degrees of belief. Poole [22], Haddawy [8], and Jaeger [10] discuss approaches to degrees of belief close to Bayesian networks [21]. Finally, another approach to probabilistic logic programming with degrees of belief, which is especially directed towards ecient implementations, has recently been introduced in [14]. Usually, the available probabilistic knowledge does not suce to specify completely a distribution. In this case, applying the principle of maximum entropy is a well-appreciated means of probabilistic inference, both from a statistical and A. Hunter and S. Parsons (Eds.): ECSQARU’99, LNAI 1638, pp. 279-292, 1999.  Springer-Verlag Berlin Heidelberg 1999

280

Thomas Lukasiewicz and Gabriele Kern-Isberner

from a logical point of view. Entropy is an information-theoretical measure [26] re ecting the indeterminateness inherent to a distribution. Given some consistent probabilistic knowledge, the principle of maximum entropy chooses as the most appropriate representation the one distribution among all distributions satisfying that knowledge which has maximum entropy (ME). Within a rich statistical rst-order language, Grove et al. [7] show that this ME-distribution may be taken to compute degrees of belief of formulas. Paris and Vencovska [20] investigate the foundations of consistent probabilistic inference and set up postulates that characterize ME-inference uniquely within that framework. A similar result was stated in [27], based on optimization theory. Jaynes [11] regarded the ME-principle as a special case of a more general principle for translating information into a probability assignment. Recently, the principle of maximum entropy has been proved to be the most appropriate principle for dealing with conditionals [13] (that is, using the notions of the present paper, ground probabilistic clauses of the form (H jB )[c1 ; c2 ] with c1 = c2 ). The main idea of this paper is to combine probabilistic logic programming with the principle of maximum entropy. We thus follow an old idea already stated in the work by Nilsson [19], however, lifted to the rst-order framework of probabilistic logic programs. At rst sight, this project might seem an intractable task, since already probabilistic propositional logics under maximum entropy suer from eciency problems (which are due to an exponential number of possible worlds in the number of propositional variables). In this paper, however, we will see that this is not the case. More precisely, we will show that the ecient approach to probabilistic logic programming in [14], combined with new ideas, can be extended to an ecient approach to probabilistic logic programming under maximum entropy. Roughly speaking, the probabilistic logic programs presented in [14] generally carry an additional structure that can successfully be exploited in both classical probabilistic query processing and probabilistic query processing under maximum entropy. The main contributions of this paper can be summarized as follows:  We de ne probabilistic queries to probabilistic logic programs and their correct and tight answer substitutions under maximum entropy.  We present an ecient linear programming characterization for the problem of deciding whether a probabilistic logic program is satis able.  We introduce an ecient technique for approximative probabilistic logic programming under maximum entropy. More precisely, this technique reduces the original entropy maximizations to relatively small optimization problems, which can easily be solved by existing ME-technology. The rest of this paper is organized as follows. Section 2 introduces the technical background. In Section 3, we give an example. Section 4 concentrates on deciding the satis ability of probabilistic logic programs. In Section 5, we discuss probabilistic logic programming under maximum entropy itself. Section 6 nally summarizes the main results and gives an outlook on future research. All proofs are given in full detail in the appendix.

Probabilistic Logic Programming under Maximum Entropy

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2 Probabilistic Logic Programs and Maximum Entropy Let  be a rst-order vocabulary that contains a nite and nonempty set of predicate symbols and a nite and nonempty set of constant symbols (that is, we do not consider function symbols in this framework). Let X be a set of object variables and bound variables. Object variables represent elements of a certain domain, while bound variables describe real numbers in the interval [0; 1]. An object term is a constant symbol from  or an object variable from X . An atomic formula is an expression of the kind p(t1 ; : : : ; tk ) with a predicate symbol p of arity k  0 from  and object terms t1 ; : : : ; tk . A conjunctive formula is the false formula ?, the true formula >, or the conjunction A1 ^


^ Al of atomic formulas A1 ; : : : ; Al with l > 0. A probabilistic clause is an expression of the form (H jB )[c1 ; c2 ] with real numbers c1 ; c2 2 [0; 1] and conjunctive formulas H and B dierent from ?. A probabilistic program clause is a probabilistic clause (H jB )[c1 ; c2 ] with c1  c2 . We call H its head and B its body. A probabilistic logic program P is a nite set of probabilistic program clauses. Probabilistic program clauses can be classi ed into facts, rules, and constraints as follows: facts are probabilistic program clauses of the form (H j>)[c1; c2 ] with c2 > 0, rules are of the form (H jB )[c1 ; c2 ] with B 6= > and c2 > 0, and constraints are of the kind (H jB )[0; 0]. Probabilistic program clauses can also be divided into logical and purely probabilistic program clauses: logical program clauses are probabilistic program clauses of the kind (H jB )[1; 1] or (H jB )[0; 0], while purely probabilistic program clauses are of the form (H jB )[c1 ; c2 ] with c1 < 1 and c2 > 0. We abbreviate the logical program clauses (H jB )[1; 1] and (H jB )[0; 0] by H B and ? H ^ B , respectively. The semantics of probabilistic clauses is de ned by a possible worlds semantics in which each possible world is identi ed with a Herbrand interpretation of the classical rst-order language for  and X (that is, with a subset of the Herbrand base over ). Hence, the set of possible worlds I is the set of all subsets of the Herbrand base HB  . A variable assignment maps each object variable to an element of the Herbrand universe HU  and each bound variable to a real number from [0; 1]. For Herbrand interpretations I , conjunctive formulas C , and variable assignments , we write I j= C to denote that C is true in I under . A probabilistic interpretation Pr is a mapping from I to [0; 1] such that all Pr (I ) with I 2 I sum up to 1. The probability of a conjunctive formula C in the interpretation Pr under a variable assignment  , denoted Pr  (C ), is de ned as follows (we write Pr (C ) if C is variable-free): P Pr (I ) : Pr  (C ) = I 2I ; I j= C

A probabilistic clause (H jB )[c1 ; c2 ] is true in the interpretation Pr under a variable assignment , denoted Pr j= (H jB )[c1 ; c2 ], i c1
Pr  (B )  Pr  (H ^ B )  c2
Pr  (B ). A probabilistic clause (H jB )[c1 ; c2 ] is true in Pr , denoted Pr j= (H jB )[c1 ; c2 ], i Pr j= (H jB )[c1 ; c2 ] for all variable assignments . A probabilistic interpretation Pr is called a model of a probabilistic clause F i Pr j= F . It is a model of a set of probabilistic clauses F , denoted Pr j= F , i

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Pr is a model of all probabilistic clauses in F . A set of probabilistic clauses F is satis able i a model of F exists. Object terms, conjunctive formulas, and probabilistic clauses are ground i they do not contain any variables. The notions of substitutions, ground substitutions, and ground instances of probabilistic clauses are canonically de ned. Given a probabilistic logic program P , we use ground (P ) to denote the set of all ground instances of probabilistic program clauses in P . Moreover, we identify  with the vocabulary of all predicate and constant symbols that occur in P . The maximum entropy model (ME-model) of a satis able set of probabilistic clauses F , denoted ME [F ], is the unique probabilistic interpretation Pr that is a model of F and that has the greatest entropy among all the models of F , where the entropy of Pr , denoted H (Pr ), is de ned by: P H (Pr ) = , Pr (I )
log Pr (I ) : I 2I

A probabilistic clause F is a maximum entropy consequence (ME-consequence) of a set of probabilistic clauses F , denoted F j=? F , i ME [F ] j= F . A probabilistic clause (H jB )[c1 ; c2 ] is a tight maximum entropy consequence (tight MEconsequence) of a set of probabilistic clauses F , denoted F j=?tight (H jB )[c1 ; c2 ], i c1 is the minimum and c2 is the maximum of all ME  [F ](H ^ B ) = ME  [F ](B ) with ME  [F ](B ) > 0 and variable assignments . A probabilistic query is an expression of the form 9(H jB )[c1 ; c2 ] or of the form 9(H jB )[x1 ; x2 ] with real numbers c1 ; c2 2 [0; 1] such that c1  c2 , two dierent bound variables x1 ; x2 2 X , and conjunctive formulas H and B dierent from ?. A probabilistic query 9(H jB )[t1 ; t2 ] is object-ground i H and B are ground. Given a probabilistic query 9(H jB )[c1 ; c2 ] with c1 ; c2 2 [0; 1] to a probabilistic logic program P , we are interested in its correct maximum entropy answer substitutions (correct ME-answer substitutions), which are substitutions  such that P j=? (HjB)[c1 ; c2 ] and that  acts only on variables in 9(H jB )[c1 ; c2 ]. Its ME-answer is Yes if a correct ME-answer substitution exists and No otherwise. Whereas, given a probabilistic query 9(H jB )[x1 ; x2 ] with x1 ; x2 2 X to a probabilistic logic program P , we are interested in its tight maximum entropy answer substitutions (tight ME-answer substitutions), which are substitutions  such that P j=?tight (HjB)[x1 ; x2 ], that  acts only on variables in 9(H jB )[x1 ; x2 ], and that x1 ; x2  2 [0; 1]. Note that for probabilistic queries 9(H jB )[x1 ; x2 ] with x1 ; x2 2 X , there always exist tight ME-answer substitutions (in particular, object-ground probabilistic queries 9(H jB )[x1 ; x2 ] with x1 ; x2 2 X always have a unique tight ME-answer substitution).

3 Example We give an example adapted from [14]. Let us assume that John wants to pick up Mary after she stopped working. To do so, he must drive from his home to her oce. However, he left quite late. So, he is wondering if he can still reach her in time. Unfortunately, since it is rush hour, it is very probable that he runs into

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a trac jam. Now, John has the following knowledge at hand: given a road (ro) in the south (so) of the town, he knows that the probability that he can reach (re) S through R without running into a trac jam is 90% (1). A friend just called him and gave him advice (ad) about some roads without any signi cant trac (2). He also clearly knows that if he can reach S through T and T through R, both without running into a trac jam, then he can also reach S through R without running into a trac jam (3). This knowledge can be expressed by the following probabilistic rules (R, S , and T are object variables): (re (R; S ) j ro (R; S ) ^ so (R; S ))[0:9; 0:9] (re (R; S ) j ro (R; S ) ^ ad (R; S ))[1; 1] (re (R; S ) j re (R; T ) ^ re (T; S ))[1; 1] :

(1) (2) (3)

Some self-explaining probabilistic facts are given as follows (h, a, b, and o are constant symbols; the fourth clause describes the fact that John is not sure anymore whether or not his friend was talking about the road from a to b): (ro (h ; a) j >)[1; 1]; (ad (h ; a) j >)[1; 1] (ro (a; b) j >)[1; 1]; (ad (a; b) j >)[0:8; 0:8] (ro (b; o ) j >)[1; 1]; (so (b; o ) j >)[1; 1] : John is wondering whether he can reach Mary's oce from his home, such that the probability of him running into a trac jam is smaller than 1%. This can be expressed by the probabilistic query 9(re (h ; o ))[:99; 1]. His wondering about the probability of reaching the oce, without running into a trac jam, can be expressed by 9(re (h ; o ))[X1 ; X2 ], where X1 and X2 are bound variables.

4 Satis ability In this section, we concentrate on the problem of deciding whether a probabilistic logic program is satis able. Note that while classical logic programs without negation and logical constraints (see especially [1]) are always satis able, probabilistic logic programs may become unsatis able, just for logical inconsistencies through logical constraints or, more generally, for probabilistic inconsistencies in the assumed probability ranges.

4.1 Naive Linear Programming Characterization The satis ability of a probabilistic logic program P can be characterized in a straightforward way by the solvability of a system of linear constraints as follows. Let LC  be the least set of linear constraints over yI  0 (I 2 I ) containing:

P

(1) I 2I yI = 1 P P P (2) c1
I 2I ; I j= B yI  I 2I ; I j= H ^B yI  c2
I 2I; I j= B yI for all (H jB )[c1 ; c2 ] 2 ground (P ):

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It is now easy to see that P is satis able i LC  is solvable. The crux with this naive characterization is that the number of variables and of linear constraints is linear in the cardinality of I and of ground (P ), respectively. Thus, especially the number of variables is generally quite large, as the following example shows.

Example 4.1 Let us take the probabilistic logic program P that comprises all the probabilistic program clauses given in Section 3. If we characterize the satis ability of P in the described naive way, then we get a system of linear constraints that has 264  18
1018 (!) variables and 205 linear constraints. 4.2 Reduced Linear Programming Characterization We now present a new system of linear constraints to characterize the satis ability of a probabilistic logic program P . This new system generally has a much lower size than LC  . In detail, we combine some ideas from [14] with the idea of partitioning ground (P ) into active and inactive ground instances, which yields another substantial increase of eciency. We need some preparations: Let P denote the set of all logical program clauses in P . Let P denote the least set of logical program clauses that contains H B if the program P contains a probabilistic program clause (H jB )[c1 ; c2 ] with c2 > 0. We de ne a mapping R that maps each ground conjunctive formula C to a subset of HB  [ f?g as follows. If C = ?, then R(C ) is HB  [ f?g. If C 6= ?, then R(C ) is the set of all ground atomic formulas that occur in C . For a set L of logical program clauses, we de ne the operator TL " ! on the set of all subsets of HB  [ f?g as usual. For this task, we need the immediate consequence operator TL , which is de ned as follows. For all I  HB  [ f?g:

TL (I ) = S fR(H ) j H B 2 ground (L) with R(B )  I g : For all I  HB  [ f?g, we de ne TL " !(I ) as the union of all TL " n(I ) with n < !, where TL" 0(I ) = I and TL " (n + 1)(I ) = TL(TL " n(I )) for all n < !. We adopt the usual convention to abbreviate TL " (;) by TL " . The set ground (P ) is now partitioned into active and inactive ground instances as follows. A ground instance (H jB )[c1 ; c2 ] 2 ground (P ) is active if R(H ) [ R(B )  TP " ! and inactive otherwise. We use active (P ) to denote the set of all active ground instances of ground (P ). We are now ready to de ne the index set I P of the variables in the new system of linear constraints. It is de ned by I P = I P0 \I , where I P0 is the least set of subsets of HB  [ f?g with: () TP " ! 2 I P0 , ( ) TP " !(R(B )); TP " !(R(H ) [ R(B )) 2 I P0 for all purely probabilistic program clauses (H jB )[c1 ; c2 ] 2 active (P ), ( ) TP " !(I1 [ I2 ) 2 I P0 for all I1 ; I2 2 I P0 . The index set I P just involves atomic formulas from TP " !:

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Lemma 4.2 It holds I  TP " ! for all I 2 IP . The new system of linear constraints LC P itself is de ned as follows: LC P is the least set of linear constraints over yI  0 (I 2 I P ) that contains:

P

(1) I 2I P yI = 1 P P P (2) c1
I 2I P ; I j= B yI  I 2I P ; I j= H ^B yI  c2
I 2I P ; I j= B yI for all purely probabilistic program clauses (H jB )[c1 ; c2 ] 2 active (P ). We now roughly describe the ideas that carry us to the new linear constraints. The rst idea is to just introduce a variable for each I  TP " !, and not for each I  HB  anymore. This also means to introduce a linear constraint only for each member of active (P ), and not for each member of ground (P ) anymore. The second idea is to exploit all logical program clauses in P . That is, to just introduce a variable for each TP " !(I ) with I  TP " !. This also means to introduce a linear constraint only for each purely probabilistic member of active (P ). Finally, the third idea is to exploit the structure of all purely probabilistic members of active (P ). That is, to just introduce a variable for each I 2 I P . The following important theorem shows the correctness of these ideas.

Theorem 4.3 P is satis able i LC P is solvable. We give an example to illustrate the new system of linear constraints LC P .

Example 4.4 Let us take again the probabilistic logic program P that comprises all the probabilistic program clauses given in Section 3. The system LC P then consists of ve linear constraints over four variables yi  0 (i 2 [0:3]): y0 + y1 + y2 + y3 = 1 0:9
(y0 + y1 + y2 + y3 )  y1 + y3 0:9
(y0 + y1 + y2 + y3 )  y1 + y3 0:8
(y0 + y1 + y2 + y3 )  y2 + y3 0:8
(y0 + y1 + y2 + y3 )  y2 + y3 More precisely, the variables yi (i 2 [0:3]) correspond as follows to the members of I P (written in binary as subsets of TP " ! = fro (h; a); ro (a; b); ro (b; o), ad (h; a); ad (a; b); so (b; o); re (h; a); re (a; b); re (b; o); re (h; b); re (a; o); re (h; o)g):

y0 =^ 111101100000; y1 =^ 111101101000 y2 =^ 111111110100; y3 =^ 111111111111 : Moreover, the four linear inequalities correspond to the following two active ground instances of purely probabilistic program clauses in P : (re (b; o) j ro (b; o) ^ so (b; o))[0:9; 0:9]; (ad (a; b) j >)[0:8; 0:8] :

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5 Probabilistic Logic Programming under ME In this section, we concentrate on the problem of computing tight ME-answer substitutions for probabilistic queries to probabilistic logic programs. Since every general probabilistic query can be reduced to a nite number of object-ground probabilistic queries, we restrict our attention to object-ground queries. In the sequel, let P be a satis able probabilistic logic program and let Q = 9(GjA)[x1 ; x2 ] be an object-ground query with x1 ; x2 2 X . To provide the tight ME-answer substitution for Q, we now need ME [P ](A) and ME [P ](G ^ A).

5.1 Exact ME-Models The ME-model of P can be computed in a straightforward way by solving the following entropy maximization problem over the variables yI  0 (I 2 I ):

P yI log yI

max ,

I 2I

subject to LC  .

(4)

However, as we already know from Section 4.1, the set of linear constraints LC  has a number of variables and of linear constraints that is linear in the cardinality of I and of ground (P ), respectively. That is, especially the number of variables of the entropy maximization (4) is generally quite large. Example 5.1 Let us take again the probabilistic logic program P that comprises all the clauses given in Section 3. The entropy maximization (4) is done subject to a system of 205 linear constraints over 264  18
1018 variables.

5.2 Approximative ME-Models

We now introduce approximative ME-models, which are characterized by optimization problems that generally have a much smaller size than (4). Like the linear programs in Section 4.1, the optimization problems (4) suer especially from a large number of variables. It is thus natural to wonder whether the reduction technique of Section 4.2 also applies to (4). This is indeed the case, if we make the following two assumptions: (1) All ground atomic formulas in Q belong to TP " !. (2) Instead of computing the ME-model of P , we compute the ME-model of active (P ) (that is, we approximate ME [P ] by ME [active (P )]). Note that P can be considered a logical approximation of the probabilistic logic program P . This logical approximation of P does not logically entail any other ground atomic formulas than those in TP " !. Hence, both assumptions (1) and (2) are just small restrictions (from a logic programming point of view). To compute ME [active (P )](A) and ME [active (P )](G ^ A), we now have to adapt the technical notions of Section 4.2 as follows. The index set I P must be adapted by also incorporating the structure of the query Q into its de nition. More precisely, the new index set I P;Q is de ned by I P;Q = I P0 ;Q \ I , where I P0 ;Q is the least set of subsets of HB  [ f?g with:

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TP " !; TP " !(R(A)); TP " !(R(G) [ R(A)) 2 I P0 ;Q , TP " !(R(B )); TP " !(R(H ) [ R(B )) 2 I P0 ;Q for all purely probabilistic program clauses (H jB )[c1 ; c2 ] 2 active (P ), ( ) TP " !(I1 [ I2 ) 2 I P0 ;Q for all I1 ; I2 2 I P0 ;Q . Also the new index set I P;Q just involves atomic formulas from TP " !: Lemma 5.2 It holds I  TP " ! for all I 2 IP;Q. The system of linear constraints LC P must be adapted to LC P;Q, which is the least set of linear constraints over yI  0 (I 2 I P;Q ) that contains: P (1) I 2I P;Q yI = 1 P P P (2) c1
I 2I P;Q ; I j= B yI  I 2I P;Q ; I j= H ^B yI  c2
I 2I P;Q ; I j= B yI for all purely probabilistic program clauses (H jB )[c1 ; c2 ] 2 active (P ). Finally, we need the following de nitions. Let IP = fL j L  TP " !g and let aI (I 2 I P;Q ) be the number of all possible worlds J 2 TP " !(IP ) \ IP that are a superset of I and that are not a superset of any K 2 I P;Q that properly includes I . Roughly speaking, I P;Q de nes a partition fSI j I 2 I P;Q g of TP " !(IP ) \ IP and each aI with I 2 I P;Q denotes the cardinality of SI . Note especially that aI > 0 for all I 2 I P;Q , since I P;Q  TP " !(IP ) \ IP . We are now ready to characterize the ME-model of active (P ) by the optimal

() ( )

solution of a reduced optimization problem. Theorem 5.3 For all ground conjunctive formulas C with TP " !(R(C )) 2 IP;Q: ME [active (P )](C ) =

PI2I

P;Q ; I j=C

yI? ;

where yI? with I 2 I P;Q is the optimal solution of the following optimization problem over the variables yI  0 with I 2 I P;Q :

max ,

P

I 2I P;Q

yI (log yI , log aI ) subject to LC P;Q .

(5)

The tight ME-answer substitution for the probabilistic query Q to the ground probabilistic logic program active (P ) is more precisely given as follows.

Corollary 5.4 Let yI? with I 2 IP;Q be the optimal solution of (5). a) If yI? = 0 for all I 2 I P;Q with I j= A, then the tight ME-answer substitution for the query 9(GjA)[x1 ; x2 ] to active (P ) is given by fx1 =1; x2=0g. b) If yI? > 0 for some I 2 I P;Q with I j= A, then the tight ME-answer substitution for the query 9(GjA)[x1 ; x2 ] to active (P ) is given by fx1 =d; x2 =dg, where d =

PI2I

P;Q ; I j=G^A

P

yI? = I 2I P;Q ; I j=A yI? .

We give an example to illustrate the optimization problem (5).

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Example 5.5 Let us take again the probabilistic logic program P from Section 3. The tight ME-answer substitution for the query 9(re (h ; o ))[X1 ; X2 ] to active (P ) is given by fX1 =0:9353; X2=0:9353g, since ME [active (P )](re (h ; o )) = y3? + y4? + y5? + y6? = 0:9353, where yi? (i 2 [0 : 6]) is the optimal solution of the following optimization problem over the variables yi  0 (i 2 [0:6]): max ,

P6 yi (log yi , log ai) subject to LCP;Q ,

i=0

where (a0 ; a1 ; a2 ; a3 ; a4 ; a5 ; a6 ) is given by (3; 1; 1; 1; 6; 5; 2) and LC P;Q consists of the following ve linear constraints over the seven variables yi  0 (i 2 [0:6]):

y0 + y1 + y2 + y3 + y4 + y5 + y6 = 1 0:9
(y0 + y1 + y2 + y3 + y4 + y5 + y6 )  y1 + y3 + y5 0:9
(y0 + y1 + y2 + y3 + y4 + y5 + y6 )  y1 + y3 + y5 0:8
(y0 + y1 + y2 + y3 + y4 + y5 + y6 )  y2 + y3 + y6 0:8
(y0 + y1 + y2 + y3 + y4 + y5 + y6 )  y2 + y3 + y6 More precisely, the variables yi (i 2 [0:6]) correspond as follows to the members of I P;Q (written in binary as subsets of TP " ! = fro (h; a); ro (a; b); ro (b; o), ad (h; a); ad (a; b); so (b; o); re (h; a); re (a; b); re (b; o); re (h; b); re (a; o); re (h; o)g):

y0 =^ 111101100000; y1 =^ 111101101000; y2 =^ 111111110100; y3 =^ 111111111111 y4 =^ 111101100001; y5 =^ 111101101001; y6 =^ 111111110101 : Furthermore, the variables yi (i 2 [0 : 6]) correspond as follows to the members of TP " !(IP ) \ IP (written in binary as subsets of TP " !). Note that ai with i 2 [0:6] is given by the number of members associated with yi .

y0 =^ h111101100000; 111101100100; 111101110100i y1 =^ h111101101000i; y2 =^ h111111110100i; y3 =^ h111111111111i y4 =^ h111101100001; 111101100011; 111101100101; 111101100111; 111101110101; 111101110111i y5 =^ h111101101001; 111101101011; 111101101101; 111101101111; 111101111111i y6 =^ h111111110101; 111111110111i : Finally, the four linear inequalities correspond to the following two active ground instances of purely probabilistic program clauses in P : (re (b; o) j ro (b; o) ^ so (b; o))[0:9; 0:9]; (ad (a; b) j >)[0:8; 0:8] : Note that we used the ME-system shell SPIRIT (see especially [23] and [24]) to compute the ME-model of active (P ).

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5.3 Computing Approximative ME-Models

We now brie y discuss the problem of computing the numbers aI with I 2 I P;Q and the problem of solving the optimization problem (5). As far as the numbers aI are concerned, we just have to solve two linear equations. For this purpose, we need the new index set I P+;Q de ned by I P+;Q = I P00;Q \ I , where I P00;Q is the least set of subsets of HB  [ f?g with: () ;; R(A); R(G) [ R(A) 2 I P00;Q , ( ) R(B ); R(H ) [ R(B ) 2 I P00;Q for all (H jB )[c1 ; c2 ] 2 active (P ), ( ) I1 [ I2 2 I P00;Q for all I1 ; I2 2 I P00;Q . We start by computing the numbers sJ with J 2 I P+;Q , which are the unique solution of the following system of linear equations:

P

jT " !j,jI j for all I J 2I +P;Q ; J I sJ = 2 P

2 I P+;Q .

We are now ready to compute the numbers aJ with J 2 I P;Q , which are the unique solution of the following system of linear equations:

PJ2I

P;Q ; J I

aJ =

PJ2I+

P;Q ; J I; J j=P

sJ for all I 2 I P;Q .

As far as the optimization problem (5) is concerned, we can build on existing ME-technology. For example, the ME-system PIT (see [6] and [25]) solves entropy maximization problems subject to indierent possible worlds (which are known to have the same probability in ME-models). It can thus directly be used to solve the optimization problem (5). Note also that if the probabilistic logic program P contains just probabilistic program clauses of the form (H jB )[c1 ; c2 ] with c1 = c2 , then the optimization problem (5) can easily be solved by standard Lagrangean techniques (as described in [24] and [25] for entropy maximization).

6 Summary and Outlook In this paper, we discussed the combination of probabilistic logic programming with the principle of maximum entropy. We presented an ecient linear programming characterization for the problem of deciding whether a probabilistic logic program is satis able. Furthermore, we especially introduced an ecient technique for approximative query processing under maximum entropy. A very interesting topic of future research is to analyze the relationship between the ideas of this paper and the characterization of the principle of maximum entropy in the framework of conditionals given in [13].

Appendix

Proof of Lemma 4.2: The claim is proved by induction on the de nition of I P0 as follows. Let I 2 I P0 with I 6= HB  [ f?g.

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() If I = TP " !, then I  TP " !, since TP " !  TP " !. ( ) If I = TP " !(R(B )) or I = TP " !(R(H )[R(B )) for some purely probabilistic (H jB )[c1 ; c2 ] 2 active (P ), then I  TP " !, since R(H ) [ R(B )  TP " !. ( ) If I = TP " !(I1 [ I2 ) for some I1 ; I2 2 I P0 , then I  TP " !, since I1 ; I2 6= HB  [ f?g and thus I1 [ I2  TP " ! by the induction hypothesis. 2

Proof of Theorem 4.3: We rst need some preparation as follows. We show that all purely probabilistic program clauses from active (P ) can be interpreted by probability functions over a partition fSI j I 2 I P g of TP " !(I ) \ I . That is, as far as active (P ) is concerned, we do not need the ne granulation of I . For all I 2 I P let SI be the set of all possible worlds J 2 TP " !(I ) \ I that are a superset of I and that are not a superset of any K 2 I P that properly includes I . We now show that fSI j I 2 I P g is a partition of TP " !(I) \ I . Assume rst that there are two dierent I1 ; I2 2 I P and some J 2 TP " !(I ) \ I with J 2 SI1 \ SI2 . Then J  I1 [ I2 and thus J  TP " !(I1 [ I2 ). Moreover, it holds TP " !(I1 [ I2 ) 2 I P by ( ) and TP " !(I1 [ I2 )  I1 or TP " !(I1 [ I2 )  I2 . But this contradicts the assumption J 2 SI1 \ SI2S. Assume next that there are some J 2 TP " !(I) \ I that do not belong to fSI j I 2 I P g. We now construct an in nite chain I0  I1 


of elements of I P as follows. Let us de ne I0 = TP " !. It then holds I0 2 I P by () and also J  I0 . But, since J 62 SI0 , there must be some I1 2 I P with J  I1 and I1  I0 . This argumentation can now be continued in an in nite way. However, the number of subsets of HB  is nite and we have thus arrived at a contradiction. We next show that for all I 2 I P , all possible worlds J 2 TP " !(I) \ I , and all ground conjunctive formulas C with TP " !(R(C )) 2 I P , it holds J j= C for some J 2 SI i J j= C for all J 2 SI . Let J j= C for some J 2 SI . It then holds J  I , J  R(C ), and thus J  TP " !(R(C )). We now show that I  TP " !(R(C )). Assume rst I  TP " !(R(C )). But this contradicts J 2 SI . Suppose next that I 6 TP " !(R(C )) and I 6 TP " !(R(C )). Since J  I [ TP " !(R(C )), we get J  TP " !(I [ TP " !(R(C ))). Moreover, it holds TP " !(I [ TP " !(R(C ))) 2 I P by ( ) and TP " !(I [ TP " !(R(C )))  I . But this contradicts J 2 SI . Hence, we get I  TP " !(R(C )). Since J  I for all J 2 SI , we thus get J  R(C ) for all J 2 SI . That is, J j= C for all J 2 SI . The converse trivially holds. We are now ready to prove the theorem as follows. Let Pr be a model of P . Let yI (I 2 I P ) be de ned as the sum of all Pr (J ) with J 2 SI . It is now easy to see that yI (I 2 I P ) is a solution of LC P . Conversely, let yI (I 2 I P ) be a solution of LC P . Let the probabilistic interpretation Pr be de ned by Pr (I ) = yI if I 2 I P and Pr (I ) = 0 otherwise. It is easy to see that Pr is a model of all logical program clauses in P and of all purely probabilistic program clauses in active (P ). Let us now take a purely probabilistic program clause (H jB )[c1 ; c2 ] from ground (P ) n active (P ). Assume that R(B ) contains some Bi 62 TP " !. By Lemma 4.2, we then get Bi 62 I for all I 2 I P . Hence, it holds Pr (B ) = 0 and thus Pr j= (H jB )[c1 ; c2 ]. Suppose now that R(B )  TP " ! and that R(H ) contains some Hi 62 TP " !. But this contradicts the assumption c2 > 0. That is, Pr is a model of P . 2

Probabilistic Logic Programming under Maximum Entropy

291

Proof of Lemma 5.2: The claim can be proved like Lemma 4.2 (by induction on the de nition of I P;Q ). The proof makes use of R(G) [ R(A)  TP " !. 2 Proof of Theorem 5.3: Since active (P ) does not involve any other atomic formulas than those in TP " !, we can restrict our attention to probability functions over the set of possible worlds IP = fL j L  TP " !g. Like in the proof of Theorem 4.3, we need some preparations as follows. We show that all purely probabilistic program clauses from active (P ) can be interpreted by probability functions over a partition fSI j I 2 I P;Q g of IP : For all I 2 I P;Q let SI be the set of all possible worlds J 2 TP " !(IP ) \ IP that are a superset of I and that are not a superset of any K 2 I P;Q that properly includes I . By an argumentation like in the proof of Theorem 4.3, it can easily be shown that fSI j I 2 I P;Q g is a partition of TP " !(IP ) \ IP and that for all I 2 I P;Q , all possible worlds J 2 TP " !(IP ) \ IP , and all ground conjunctive formulas C with TP " !(R(C )) 2 I P;Q , it holds J j= C for some J 2 SI i J j= C for all J 2 SI . Given a model Pr of active (P ), we can thus de ne a model Pr ? of active (P ) P ? by Pr (L) = 1=aI
J 2SIPr (J )if L 2 TP " !(IP ) \IP , where I 2 I P;Q such that L 2 SI , and Pr ? (L) = 0 otherwise. Hence, it immediately follows that for all I 2 I P;Q and all J1 ; J2 2 SI : ME [active (P )](J1 ) = ME [active (P )](J2 ). Hence, for all ground conjunctive formulas C with TP " !(R(C )) 2 I P;Q : P ME [active (P )](C ) = I 2I P;Q ; I j=C aI x?I ; where x?I (I 2 I P;Q ) is the optimal solution of the following optimization problem over the variables xI  0 (I 2 I P;Q ): P aI xI log xI subject to LC0P;Q , max , I 2I P;Q

where LC 0P;Q is the least set of constraints over xI  0 (I 2 I P;Q ) containing: P (1) I 2I P;Q aI xI = 1 P P P (2) c1
I 2I P;Q ; I j= B aI xI  I 2I P;Q ; I j= H ^B aI xI  c2
I 2I P;Q ; I j= B aI xI for all purely probabilistic program clauses (H jB )[c1 ; c2 ] 2 active (P ). Thus, we nally just have to perform the variable substitution xI = yI =aI . 2

References 1. K. R. Apt. Logic programming. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, chapter 10, pages 493{574. MIT Press, 1990. 2. F. Bacchus, A. Grove, J. Y. Halpern, and D. Koller. From statistical knowledge bases to degrees of beliefs. Artif. Intell., 87:75{143, 1996. 3. B. de Finetti. Theory of Probability. J. Wiley, New York, 1974. 4. A. Dekhtyar and V. S. Subrahmanian. Hybrid probabilistic programs. In Proc. of the 14th International Conference on Logic Programming, pages 391{405, 1997.

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5. R. Fagin, J. Y. Halpern, and N. Megiddo. A logic for reasoning about probabilities. Inf. Comput., 87:78{128, 1990. 6. V. Fischer and M. Schramm. tabl | a tool for ecient compilation of probabilistic constraints. Technical Report TUM-I9636, TU Munchen, 1996. 7. A. J. Grove, J. H. Halpern, and D. Koller. Random worlds and maximum entropy. J. Artif. Intell. Res., 2:33{88, 1994. 8. P. Haddawy. Generating Bayesian networks from probability logic knowledge bases. In Proceedings of the 10th Conference on Uncertainty in Arti cial Intelligence, pages 262{269. Morgan Kaufmann, 1994. 9. J. Y. Halpern. An analysis of rst-order logics of probability. Artif. Intell., 46:311{ 350, 1990. 10. M. Jaeger. Relational Bayesian networks. In Proceedings of the 13th Conference on Uncertainty in Arti cial Intelligence, pages 266{273. Morgan Kaufmann, 1997. 11. E. T. Jaynes. Papers on Probability, Statistics and Statistical Physics. D. Reidel, Dordrecht, Holland, 1983. 12. R. W. Johnson and J. E. Shore. Comments on and corrections to \Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy". IEEE Trans. Inf. Theory, IT-29(6):942{943, 1983. 13. G. Kern-Isberner. Characterizing the principle of minimum cross-entropy within a conditional-logical framework. Artif. Intell., 98:169{208, 1998. 14. T. Lukasiewicz. Probabilistic logic programming. In Proc. of the 13th European Conference on Arti cial Intelligence, pages 388{392. J. Wiley & Sons, 1998. 15. T. Lukasiewicz. Local probabilistic deduction from taxonomic and probabilistic knowledge-bases over conjunctive events. Int. J. Approx. Reason., 1999. To appear. 16. T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 1999. To appear. 17. R. T. Ng. Semantics, consistency, and query processing of empirical deductive databases. IEEE Trans. Knowl. Data Eng., 9(1):32{49, 1997. 18. R. T. Ng and V. S. Subrahmanian. A semantical framework for supporting subjective and conditional probabilities in deductive databases. J. Autom. Reasoning, 10(2):191{235, 1993. 19. N. J. Nilsson. Probabilistic logic. Artif. Intell., 28:71{88, 1986. 20. J. B. Paris and A. Vencovska. A note on the inevitability of maximum entropy. Int. J. Approx. Reasoning, 14:183{223, 1990. 21. J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA, 1988. 22. D. Poole. Probabilistic Horn abduction and Bayesian networks. Artif. Intell., 64:81{129, 1993. 23. W. Rodder and G. Kern-Isberner. Representation and extraction of information by probabilistic logic. Inf. Syst., 21(8):637{652, 1997. 24. W. Rodder and C.-H. Meyer. Coherent knowledge processing at maximum entropy by SPIRIT. In Proceedings of the 12th Conference on Uncertainty in Arti cial Intelligence, pages 470{476. Morgan Kaufmann, 1996. 25. M. Schramm, V. Fischer, and P. Trunk. Probabilistic knowledge representation and reasoning with maximum entropy - the system PIT, Technical report (forthcoming), 1999. 26. C. E. Shannon and W. Weaver. A Mathematical Theory of Communication. University of Illinois Press, Urbana, Illinois, 1949. 27. J. E. Shore and R. W. Johnson. Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Trans. Inf. Theory, IT-26:26{37, 1980.

Lazy Propagation and Independence of Causal In uence Anders L. Madsen1 and Bruce D'Ambrosio2 1

2

Department of Computer Science, Aalborg University Fredrik Bajers Vej 7C, DK-9220 Aalborg , Denmark [email protected]

Department of Computer Science, Oregon State University 303 Dearborn Hall, Corvallis, OR 97331, USA [email protected]

The eÆciency of algorithms for probabilistic inference in Bayesian networks can be improved by exploiting independence of causal in uence. In this paper we propose a method to exploit independence of causal in uence based on on-line construction of decomposition trees. The eÆciency of inference is improved by exploiting independence relations induced by evidence during decomposition tree construction. We also show how a factorized representation of independence of causal in uence can be derived from a local expression language. The factorized representation is shown to t ideally with the lazy propagation framework. Empirical results indicate that considerable eÆciency improvements can be expected if either the decomposition trees are constructed on-line or the factorized representation is used. Abstract.

1

Introduction

Bayesian networks is an increasingly popular knowledge representation framework for reasoning under uncertainty. A Bayesian network consists of a directed, acyclic graph and a set of conditional probability distributions. For each variable in the graph there is a conditional probability distribution of the variable given its parents. The most common task performed on a Bayesian network is calculation of the posterior marginal distribution for all non-evidence variables given a set of evidence. The complexity of probabilistic inference in Bayesian networks is, however, known to be NP -hard in general [2]. In this paper we consider how a special kind of structure within the conditional probability distributions can be exploited to reduce the complexity of inference in Bayesian networks. The structure we want to exploit is present when the parents of a common child interact on the child independently. This is referred to as independence of causal in uence (ICI), see eg. [4{6, 11, 12, 14]. There exists basically two dierent approaches to inference in Bayesian networks. One approach is based on building a junction tree representation T of the A. Hunter and S. Parsons (Eds.): ECSQARU’99, LNAI 1638, pp. 293-304, 1999.  Springer-Verlag Berlin Heidelberg 1999

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Bayesian network G and then propagating evidence in T by message passing. T is constructed based on an o-line determined elimination order of the variables in G . The other approach is based on direct computation. Direct computation is based on determining the elimination order or order of pairwise combination of potentials on-line. G is pruned to remove all variables not relevant for the query of interest and an elimination order of the variables or a combination order of the potentials of the pruned graph is determined. Lazy propagation is a recently proposed junction tree based inference architecture [8]. A junction tree representation T of G is constructed based on moralization and triangulation of G . The nodes of T are cliques (maximal, complete subgraphs) of the triangulation of G . The distributions of G are associated with cliques of T such that the domain of a distribution is a subset of the clique with which it is associated. In the lazy propagation architecture distributions associated with a clique C are not combined to form the initial clique potential. Instead, a multiplicative decomposition of each clique and separator potential is maintained and potentials are only combined during inference when necessary.

Inference in T is based on message passing between adjacent cliques of T . Each message consists of a set of potentials with domains that are subsets of the separator between the two adjacent cliques. A message from Ci to Cj over S is computed from a subset, S , of the potentials associated with Ci . S consists of the potentials relevant for calculating the joint of S . All variables in domains of potentials in S , but not in S are eliminated one at a time by marginalization. The factorized decomposition of clique and separator potentials and the use of direct computation to calculate messages make it possible to take advantage of barren variables and independence relations induced by evidence during inference. The eÆciency of the lazy propagation algorithm can also be improved by exploiting ICI.

De nition 1. The parent cause variables C1 ; : : :; C of a common child eect variable E are causally independent wrt. E if there exists a set of contribution variables E 1 ; : : :; E with the same domain as E such that: n

C

{ 8 =1 { E=E i;j

Cn

? fC ; E g and 


E where  is a commutative and associative binary operator. ^ 6= : E

;::: ;n C1

i

j

Ci

j

Cj

Cn

With de nition 1, P (E j C1 ; : : : ; Cn ) can be expressed as a sum over the product of a set of usually much simpler probability distributions:

P (E j pa(E )) =

X f

j

E C1 ;::: ;E Cn E =E C1

YP E n






E Cn

g

(

Ci

j C ): i

i=1

For each cause variable in an ICI model some state is designated to be distinguished. For most real-world models this will be the state bearing no eect on

Lazy Propagation and Independence of Causal Influence

295

E [6]. In general, ICI is exploited when cause variables are eliminated before the eect variable E as eliminating E rst will reconstruct the entire conditional probability distribution P (E j C1 ; : : : ; C ). n

2

Lazy Parent Divorcing

Parent divorcing [9] is a method to exploit ICI based on changing the structure of the Bayesian network before the junction tree representation is constructed. Let G be a Bayesian network and let M be an ICI model in G with variables E , C1 ; : : : ; Cn and conditional probability distribution PM = P (E j C1 ; : : : ; Cn ). With parent divorcing a number of variables, say Y1 ; : : : ; Ym , are introduced as intermediate variables between C1 ; : : : ; Cn and E . The variables Y1 ; : : : ; Ym are introduced to cache the intermediate results of combining cause variables pairwise. The cause, eect, and intermediate variables are arranged in a decomposition tree T D constructed once and o-line (in the case of n = 4 and computation order E = ((C1  C2 )  (C3  C4 ))):

P (E j C1 ; C2 ; C3 ; C4 ) =

f

X

j

X g j X 

Y1 ;Y2 E =Y1 Y2

f

E C1 ;E C2 Y1 =E C1

f

j

E C3 ;E C4 Y2 =E C3



P (E j C1 )P (E j C2 ) C1

E C2

g

C2

P (E j C3 )P (E j C4 ): C3

E C4

g

C4

Knowledge about the possible structure of the evidence, say E , and the impact of the structure of T D on the remaining part of G should be considered when T D is constructed. There exist, however, examples where it o-line is impossible to construct T D such that the space requirements are minimal given dierent E . j

P (E C1 ; C2 ; C3 ) A

B

C

C1

C2

C3

E

EC1 C2 C3

C1 C2

C1 C3

C2 C3

AC1 C2

BC1 C3

CC2 C3

P (A C1 ; C2 ) P (C1 )

P (B C1 ; C3 ) P (C2 )

P (C C2 ; C3 ) P (C3 )

j

(A)

j

(B)

j

Fig. 1: A Bayesian network G and corresponding junction tree T .

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Anders L. Madsen and Bruce D’Ambrosio

Example 1. Consider the Bayesian network G shown in gure 1(A). Let

M be the ICI model describing the relationship between E and its parents. Three dierent decomposition trees TAD , TBD , and TCD can be constructed using parent divorcing on M, see gure 2. By considering three dierent evidence scenarios E1 , E2 , and E3 it will become clear that it is impossible o-line to construct a decomposition tree with minimal space requirements for all three scenarios. That is, for each of TAD , TBD , and TCD there exists a set of evidence that makes one of the other trees a better choice. As an example gure 3 shows a junction tree T corresponding to constructing TBD o-line. If E1 = fA = a; B = bg, then the potential with largest domain size is constructed when a message is passed from the root to EC1 Y . The set of potentials associated with the root of T after it has received messages from all adjacent cliques is  = fP (a j C1 ; C2 ), P (C1 ), P (C2 ), P (C3 ), P (b j C1 ; C3 ), P (Y j C2 ; C3 )g. Variables C2 and C3 have to be eliminated, but neither C2 nor C3 can be eliminated without creating a potential over variables: Y , C1 , C2 , C3 . If either TAD or TCD is constructed, then the largest domain size is three instead of four variables. On the other hand, if E2 = fB = b; C = cg, then TBD and TCD are optimal. Similarly, if E3 = fA = a; C = cg, then TAD and TBD are optimal. A

B

C

A

B

C

A

B

C

C1

C2

C3

C1

C2

C3

C2

C1

C3

Y

TAD

Y

E

(A)

TBD

E

(B)

Y

TCD

E

(C)

Fig. 2: The three possible ways to divorce C1 , C2 , and C3 .

Example 1 shows that no matter what decomposition tree is constructed o-line there exists a set of evidence such that one of the other trees is a better choice. We propose a method based on performing the decomposition on-line and only when needed. The method is referred to as lazy parent divorcing. Instead of constructing a speci c decomposition tree T D for M o-line, dierent decomposition trees are constructed on-line. First, a junction tree representation T of G is constructed without recognizing M. Next, the distributions of G are associated with cliques of T as usual except that only a speci cation of PM similar to de nition 1 is used. Probabilistic inference in a junction tree based inference architecture consists of two phases. The rst phase is message passing between adjacent cliques and the

Lazy Propagation and Independence of Causal Influence P (Y

297

j C2 ; C3 )

Y C1 C2 C3

C1 C2

C1 C3

AC1 C2

BC1 C3

j

C2 C3

j

C1 Y

CC2 C3

EC1 Y

P (C C2 ; C3 ) P (C3 )

P (E C1 ; Y )

j

P (A C1 ; C2 ) P (B C1 ; C3 ) P (C1 ) P (C2 )

j

Fig. 3: A junction tree for the Bayesian network shown in gure 2(B).

second phase is calculation of posterior marginals. In either phase an ICI model can be relevant for a calculation. If  is the set of potentials relevant for some calculation and PM 2  represents an ICI model M, then a decomposition tree for M is constructed on-line before the calculation is performed.

T shown in gure 1(B). Assume E1 = fA = a; B = bg. The set of potentials associated with the root of T after it has received messages from all adjacent cliques is  = fP (a j C1 ; C2 ), P (b j C1 ; C3 ), P (C1 ), P (C2 ), P (C3 ), P (E j C1 ; C2 ; C3 )g. P (E j C1 ; C2 ; C3 ) is not stored explicitly, only a speci cation similar to de nition 1 is stored. The potential with the largest domain size is constructed when the posterior marginal P (E ) is calculated. Based on the rst ve potentials of  and the structure of E1 either TAD or TCD is constructed. Either of TAD or TCD gives a largest potential domain size of three variables. The decomposition tree TAD , for instance, corresponds to the following decomposition: Example 2. Continuing example 1 consider the junction tree

P (E j C1 ; C2 ; C3 ) =

f

X j  X

Y;E C3 E =Y

f

j

E C1 ;E C2 Y

P (E j C3 ) C3

E C3

g

P (E j C1 )P (E j C2 ): C1

=E C1



E C2

g

C2

The evidence scenarios E2 and E3 are handled similarly. The problem of nding an optimal decomposition tree for M is expected to be NP -hard. The heuristic used in the current implementation is based on a score s of making two cause variables Ci and Cj parents of a common intermediate child variable. The score is equal to the sum of the weights of the ll-ins needed to complete the neighborhood of Ci and Cj in the domain graph of :

s(C ; C ) = i

j

V;W

2

[

adj (Ci )

X

nf

adj (Cj )

Ci ;Cj

jV j
jW j:

g 62 ;V

adj (W )

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Anders L. Madsen and Bruce D’Ambrosio

It is clear that for a given ICI model the decomposition tree constructed on-line is one of the decomposition trees that could have been constructed o-line. As shown above it is, however, not always possible to construct a decomposition tree T D o-line such that T D is optimal for all possible sets of evidence. Even if it is possible to construct T D o-line, it is not easy to determine the structure of T D o-line. In the case of on-line construction of decomposition trees all the relevant information is represented in the set of potentials relevant for a calculation. A new decomposition tree T is constructed each time an ICI model M is found relevant for a calculation. Note that it might be unnecessary to construct T at all. This happens when the eect variable E of M is marginalized out and none of its descendants have received evidence, when certain kinds of evidence on E is present, or when none of the variables in M are relevant for the calculation.

3

The Factorized Representation

The main idea of the factorized representation proposed by [13] is to introduce a set of intermediate variables representing P (E  e) for each state e of E except one. The intermediate variable E e corresponds to the product of the contributions from the cause variables to E  e. This is used to calculate P (E = ei ) as P (E = ei ) = P (E  ei ) P (E  ei 1 ) where ei 1 < ei are states of E .

3.1 Factorized Representation of Noisy-OR Consider a noisy-OR interaction model M consisting of binary variables E , C1 , and C2 . We will use M to describe how the factorized representation can be derived from the local expression language introduced by [3]. A local expression describing the dependence of E on C1 and C2 can be speci ed as below (subscripts denote the domain on which the expression is de ned):

exp(E j C1 ; C2 ) =1

j

E =t C1 ;C2

(1E =tjC1 qE =tjC1 )(1E =tjC2 qE =tjC2 ) + (1E =f jC1 qE =f jC1 )(1E =f jC2 qE =f jC2 );

where qE =tjC1 and qE =f jC1 both represent P (E = t j C1 = t; C2 = f ) and P (E = t j C1 = f; C2 = f ), but de ned on dierent domains. As M is a noisy-OR interaction model P (E = t j C1 = f; C2 = f ) = 0.

The expression exp(E j C1 ; C2 ) has the disadvantage that it is only available to the expression for the entire network exp(E; C1 ; C2 ) if we distribute over exp(E j C1 ; C2 ). This operation is, in general, exponential in the number of causes of M and is therefore an expensive operation. By introducing an intermediate variable E f with two states v and I as an intermediate variable between C1 , C2 and E it is possible to represent M with two expressions:

Lazy Propagation and Independence of Causal Influence

exp(E  j C1 ; C2 ) =1  f

E

j

f =I C ;C 1 2

+ (1E  =vjC1 qE  =vjC1 )(1E  =vjC2 qE   f exp(E j E ) =1E=tjE =I 1E=tjE =v + 1E=f jE =v : f

f

f

f

f

299

j );

f =v C 2

f

The expression exp(E j C1 ; C2 ) can be reconstructed by combining the expressions above and marginalizing E f out:

exp(E j C1 ; C2 ) = =

=

X exp E j E X j

E

f

E

f

X E

f

(

f

)  exp(E f j C1 ; C2 )

(1E =t E 

1E =tjE 

f =I

 (1

E

1E =t;E 

q 

E

)

j )

f =v C 1

E

q 

f =vjC2

j ))

f =v C 2

E

j

f =I C ;C 1 2

(1E =t;E 

j

f =v C 1

E =t;E

q

E =t;E

j

f =v C 1

E =f;E

f =vjC2

f =vjC1 )

q

f =vjC2

+ (1E =f;E 

 (1

f =v

f =I jC1 ;C2 + (1E f =vjC1

 (1

 (1

+ 1E =f jE 

f =v

E =t;E

q

E =f;E

q

f =vjC2 )

f =vjC1 )

E =f;E

f =vjC2 )

= 1E =tjC1 ;C2 (1E =tjC1 qE =tjC1 )(1E =tjC2 + (1E =f jC1 qE =f jC1 )(1E =f jC2

q q

j ) = j 2 ):

E =t C2

E

f C

Based on the above derivation of exp(E f j C1 ; C2 ) and exp(E j E f ) potentials H (E j E f ), G(E f j C1 ), and G(E f j C2 ) are constructed, see table 1. Each G(E f j Ci ) describes the contribution from Ci to E  f . The potential H (E j E f ) describes how to calculate the distribution of E from the distributions of the intermediate variables. P (E j C1 ; C2 ) can be obtained by marginalizing out E f of the product of H and the G's:

P (E j C1 ; C2 ) =

X H E j E Y G E j C :

E f

(

f

)

2

i=1

(

f

i

)

(1)

300

Anders L. Madsen and Bruce D’Ambrosio

Table 1. C1

Potentials for G(E f j C1 ) (A), G(E f j C2 ) (B), and H (E j E f ) (C).



E

f

C2

v f t

1

I

1

j

qE =t C =t;C =f 1 2



E

f

E

v

1 1

f t

I

1

1 qE =tjC =f;C =t 1 1 2

1

(A)



(B)

f

E f

v I

t

1 1 0 1

(C)

Note that equation 1 only uses marginalization of variables and multiplication of potentials, no special operator is required. Therefore, no constraints are imposed on the elimination order or the order in which potentials are combined. The H and G potentials do not have the usual properties of probability potentials. The H potential includes negative numbers and for some parent con gurations the distribution of E consists of all zeros. Both the H and G potentials have the property that for a given parent con guration the entries in the potential do not necessarily sum to one. This implies that marginalizing a head variable out of one of these potentials does not result in a unity potential.

3.2 Factorized Representation of Noisy-MAX The factorized representation of noisy-OR interaction models can be generalized to capture noisy-MAX interaction models. Let M be a noisy-MAX interaction model with variables E , C1 ; C2 ; : : : ; Cn . One intermediate variable E e is introduced for each state e of E except the highest state. If E has m + 1 states, then a graphical interpretation of the factorized representation of M can be depicted as shown in gure 4. C1

C2

E

e1

E

Cn

e2

E

em

E

Fig. 4: A graphical interpretation of the factorized representation of M.

The complexity of ICI model representation is reduced from exponential in n to exponential in jE j. As jE j is usually much smaller than n considerable eÆciency improvements can be expected.

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Example 3. Let M be a noisy-MAX interaction model where each variable has ordered domain l < m < h. The potentials required to represent M in factorized form are shown in table 2 (where qE>ejC =c is shorthand for qE>ejC =c;8 6= C =l ). The potential P (E j C1 ; : : : ; Cn ) can be obtained by eliminating E l and E m from the factorized representation of M: i

P (E j pa(E )) =

Table 2.

H (E j E  ; E  ) l

E l ;E m

1 m 1 h 1

E



v

l

2

i

j

G(E  j C )G(E  j C ): l

m

C pa(E )

j j j

qE>l C =l qE>l C =m qE>l C =h

C I

1 1 1

(A)

4

Y

m

j

Potentials for G(E l j C ) (A), G(E m j C ) (B), and H (E j E l ; E m ) (C).

C

l

X

i

1 m 1 h 1 l

E



v

m

j j j

qE>m C =l qE>m C =m qE>m C =h

(B)

E



I

1 1 1

l

E



m

E l

v v I I

0 I 1 v 0 I 0 (C) v

m

0 1 1 0

h

0 0 1 1

Empirical Results

A series of tests were performed in order to determine the eÆciency of exploiting ICI with lazy parent divorcing and factorized representation during inference in the lazy propagation architecture. The eÆciency of the approaches is compared with the parent divorcing and temporal transformation [6] approaches. The tests were performed on six dierent versions of the CPCS-network [10] with 40, 145, 245, 360, 364, and 422 nodes, respectively. All the ICI models in the networks are noisy-MAX interaction models. For each network evidence sets with sizes varying from 0 to 25 instantiated variables are chosen randomly. For each size, 25 dierent sets were generated. The average size (in terms of numbers) of the largest potential calculated during inference in the 145 and 245-nodes networks are shown in gure 5. See [7] for additional test results. The test results indicate that lazy parent divorcing and the factorized representation oer considerable improvements in space eÆciency compared to parent divorcing and temporal transformation. The eÆciency of lazy parent divorcing and the factorized representation seems to be of the same level. The factorized representation is, however, more eÆcient that lazy parent divorcing for small sets of evidence in the 422-nodes network.

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40000 Parent Divorcing Temporal Transformation Factorized Representation Lazy Parent Divorcing

Parent Divorcing Temporal Transformation Factorized Representation Lazy Parent Divorcing

35000

250

Largest potential (average)

Largest potential (average)

30000 200

150

100

25000

20000

15000

10000 50 5000

0

0 0

5

10 15 Number of evidence variables

(A) 145-nodes network. Fig. 5.

5

20

25

0

5

10 15 Number of evidence variables

20

25

(B) 245-nodes network.

Plots of the test results for the 145 and 245-nodes subnetworks.

Discussion

The lazy parent divorcing and factorized representation approaches to exploiting ICI t naturally into the lazy propagation framework. Lazy parent divorcing postpones the construction of decomposition trees in the same way as lazy propagation postpones the combination of potentials. The factorized representation oers a multiplicative decomposition of ICI model distributions similar to the multiplicative decomposition of clique and separator potentials already exploited in the lazy propagation architecture. Both lazy parent divorcing and the factorized representation introduce a number of intermediate variables to reduce the complexity of inference. The intermediate variables are not part of the original Bayesian network and therefore not part of the junction tree constructed from the original Bayesian network. This raises the issue of when to eliminate intermediate variables most eÆciently. The heuristic currently used in the implementation is to eliminate an intermediate variable V when the number of ll-ins added by eliminating V is two or less. Lazy propagation exploits a decomposition of the clique and separator potentials while the factorized representation and lazy parent divorcing oers a decomposition of conditional probability distributions of the eect variable in an independence of causal in uence model. The decomposition of conditional probability distributions is only exploited if some cause variables are eliminated before the intermediate variables as eliminating all the intermediate variables before any cause variable will reconstruct the conditional probability distribution of the eect variable given the cause variables. We have described how the factorized representation of noisy-OR interaction models can be generalized to noisy-MAX interaction models. The factorized representation cannot, however, be generalized to represent all ICI models eÆ-

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ciently. An in depth analysis of the limitations of the factorized representation is given in [13]. Examples of the models eÆciently representable with the factorized representation are noisy-OR, noisy-MAX, noisy-MIN, and noisy-AND. Noisy-ADD is an example of an ICI model not eÆciently representable using the ideas of factorized representation [13]. Lazy parent divorcing does not, on the other hand, have any limitations with respect to representation of general ICI models. The ideas of lazy parent divorcing also seems applicable to construction of decomposition trees when the structure of the conditional probability distributions is such that the independence relations only hold in certain contexts. This is known as context speci c independence, see for instance [1]. How to use the ideas of lazy parent divorcing to exploit context speci c independence is a topic of future research. In [15] it is reported that on average 7Mb of memory is required to propagate randomly chosen evidence in the 245-nodes network using the heterogeneous factorization to exploit ICI. The average size of the largest potential calculated using either lazy parent divorcing or the factorized representation is in this network 5K . It has not been possible to compare lazy parent divorcing and the factorized representation with the approach suggested by [11] as they do not oer any empirical results. For a given ICI model M the corresponding factorized representation can be constructed o-line. The lazy parent divorcing approach, on the other hand, requires additional time on-line to construct the decomposition tree for M during inference. The impact of the factorized representation and lazy parent divorcing on the time eÆciency of inference is a topic of future research.

6

Conclusion

In this paper two dierent approaches to exploiting ICI to improve the eÆciency of probabilistic inference in Bayesian networks are presented. Both the factorized representation and lazy parent divorcing t nicely into the lazy propagation framework. Lazy parent divorcing is a method to exploit ICI based on on-line construction of decomposition trees. This makes it possible to exploit structure of the evidence to improve the eÆciency of decomposition tree construction. The lazy propagation architecture is based on a multiplicative decomposition of clique and separator potentials and it is readily extended to exploit the multiplicative decomposition oered by the factorized representation. The reader should note that neither lazy parent divorcing nor the factorized representation imposes any constraints on the order in which variables can be eliminated or the order in which potentials can be combined. The empirical results indicate that considerable improvements in eÆciency of probabilistic inference can be expected by the use of either lazy parent divorcing or the factorized representation to exploit ICI.

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References 1. C Boutilier, N Friedman, M. Coldszmidt, and D. Koller. Context-speci c independence in bayesian networks. In Proceedings of the Twelth Conference on Uncertainty in Arti cial Intelligence, pages 115{123, 1996. 2. G. F. Cooper. The computational complexity of probabilistic inference using Bayesian belief networks. Arti cial Intelligence, 42:393{405, 1990. 3. B. D'Ambrosio. Local Expression Language for Probabilistic Dependence. International Journal of Approximate Reasoning, 13(1):61{81, 1995. 4. F. J. Diez. Local Conditioning in Bayesian Networks. Arti cial Intelligence, 87:1{ 20, 1994. 5. D. Heckerman. Causal Independence for Knowledge Acquisition and Inference. In Proceedings of the Ninth Conference on Uncertainty in Arti cial Intelligence, pages 122{127, 1993. 6. D. Heckerman and J. S. Breese. A New Look at Causal Independence. In Proceedings of the Tenth Conference on Uncertainty in Arti cial Intelligence, pages 286{292, 1994. 7. A. L. Madsen and B. D'Ambrosio. Lazy Propagation and Independence of Causal In uence. Technical report, Department of Computer Science, Oregon State University, 1998. 8. A. L. Madsen and F. V. Jensen. Lazy Propagation in Junction Trees. In Proceedings of the Fourteenth Conference on Uncertainty in Arti cial Intelligence, pages 362{ 369, 1998. 9. K. G. Olesen, U. Kjrul, F. Jensen, F. V. Jensen, B. Falck, S. Andreassen, and S.K. Andersen. A MUNIN network for the median nerve | a case study on loops. Applied Arti cial Intelligence, 3, 1989. Special issue: Towards Causal AI Models in Practice. 10. M. Pradhan, G. Provan, B. Middleton, and M. Henrion. Optimal Knowledge Engineering for Large Belief Networks. In Proceedings of the Tenth Conference on Uncertainty in Arti cial Intelligence, pages 484{490, 1994. 11. I. Rish and R. Dechter. On the impact of causal independence. In Stanford Spring Symposium on Interactive and Mixed-Initiative Decision-Theoretic Systems, pages 101{108, 1998. 12. S. Srinivas. A Generalization of the Noisy-Or Model. In Proceedings of the Ninth Conference on Uncertainty in Arti cial Intelligence, pages 208{218, 1993. 13. M. Takikawa. Representations and Algorithms for EÆcient Inference in Bayesian Networks. PhD Thesis, Department of Computer Science, Oregon State University, 1998. 14. N. L. Zhang and D. Poole. Exploiting Causal Independence in Bayesian Network Inference. Journal of Arti cial Intelligence Research, 5:301{328, 1996. 15. N. L. Zhang and L. Yan. Independence of Causal In uence and Clique Tree Propagation. In Proceedings of the Thirteenth Conference on Uncertainty in Arti cial Intelligence, pages 472{280, 1997.

A Monte Carlo Algorithm for Combining Dempster-Shafer Belief Based on Approximate Pre-computation? Serafn Moral1 and Antonio Salmeron2 1

Dpt. Computer Science and Arti cial Intelligence University of Granada, 18071 Granada, Spain [email protected]

Dpt. Statistics and Applied Mathematics University of Almera, 04120 Almera, Spain 2

[email protected]

Abstract. In his paper, a new Monte Carlo algorithm for combining

Dempster-Shafer belief functions is introduced. It is based on the idea of approximate pre-computation, which allows to obtain more accurate estimations by means of carrying out a compilation phase previously to the simulation. Some versions of the new algorithm are experimentally compared to the previous methods.

1 Introduction From a computational point of view, one important problem of Dempster-Shafer theory of evidence [2, 10] is the complexity of the computation of the combination of independent pieces of evidences by means of Dempster's rule. It is known that the exact computation of the combination is a NP-hard problem [7]. This fact motivates the development of approximate schemes for performing Dempster's rule. Moral and Wilson have studied a wide class of approximate algorithms based on Monte Carlo methods. Those algorithms can be classi ed into two groups: Markov Chain Monte Carlo algorithms [5] and Importance Sampling algorithms [6]. In this paper we propose a method that corresponds to the importance sampling group. The new feature we introduce is the approximate pre-computation phase, earlier utilized in probabilistic reasoning [3, 4]. The paper starts with some basic concepts and notation in Sec. 2. Known algorithms are reviewed in Sec. 3, and the fundamentals of importance sampling in Evidence Theory are given in Sec. 3.3. The new scheme we have developed is presented in Section 4. Section 5 is devoted to the description of the experiments carried out to check the performance of the new algorithms, and the paper ends with the conclusions in Sec. 6. ?

This work has been supported by CICYT under projects TIC97-1135-C04-01 and TIC97-1135-C04-02.

A. Hunter and S. Parsons (Eds.): ECSQARU’99, LNAI 1638, pp. 305-315, 1999.  Springer-Verlag Berlin Heidelberg 1999

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2 Basic Concepts and Notation Let  be a nite sample space. By 2 we denote the set of all the subsets of . A mass function over  is a function m : 2 ! [0; 1] verifying that m(;) = 0 P and A22 m(A) = 1. Those sets for which m is strictly positive are called focal sets. A function Bel : 2 ! [0; 1] is said to be a beliefPfunction over  if there exists a mass function m over  verifying Bel(X ) = AX m(A) 8X 2 2 : This de nition establishes a one-to-one correspondence between mass functions and belief functions [10]. Other functions can be de ned from a mass function: namely, the plausibility and the commonality. P A plausibility function over  is a function Pl : 2 ! [0; 1] de ned as Pl(X ) = A\X 6=; m(A) for all X 2 2 . A commonality function over  is a mapping Q : 2 ! [0; 1] de ned as Q(X ) = P m(A) for all X 2 2 . AX Any of the functions de ned above contains the same information, and it is a matter of convenience which one to use. However, mass functions may be more compact if only the masses of the focal sets are given. Now assume we have n mass functions m1 ; : : : ; mn over . The information they contain can be collected in a new mass function m = m1


mn , where

denotes the combination by Dempster's rule: m(X ) = k,1 where k =

P

X A1 ;:::;An  A1 \


\An =X

m1 (A1 )


mn (An ) ; 8X   ;

A1 ;:::;An  m1 (A1 )


mn (An ). A1 \


\An 6=;

(1)

P

The combined measure of belief over  is given by Bel(X ) = AX m(A). The aim of this paper will be the estimation of Bel(X ) for a given X  . Next, we show how the calculation of Bel(X ) can be formulated as the estimation of a probability value by sampling from a multivariate probability distribution. We shall denote by Pi a probability distribution over 2 de ned as Pi (X ) = mi (X ) for all X 2 2 . An ordered collection of sets (C1 ; : : : ; Cn ), Ci 2 2 will be called a con guration, and denoted by letter C . Let 0 be the set of all possible n-dimensional con gurations, and = f(C1 ; : : : ; Cn ) 2 0 j \ni=1 Ci 6= ;g. From now on, the intersection of all the sets in a con guration C , \ni=1 Ci , will be abbreviated by IC . Q The following distribution can be de ned over 0 : P0 (C ) = ni=1 Pi (Ci ) for 0 all C 2 . Let PDS be a probability distribution over such that for all C 2 , PDS(C ) = P0 (C )=P0 ( ). From this notation, P P it follows that, for all X  , m(X ) = C 2 PDS (C ), and Bel(X ) = C 2 PDS (C ). IC =X

IC X

So Bel(X ) can be estimated by sampling con gurations, C , with probability distribution PDS and calculating the relative frequencies with which IC  X . The

A Monte Carlo Algorithm for Combining Dempster-Shafer Belief

307

main problem of Monte Carlo algorithms is to sample from this distribution PDS. P0 is simple to manage and simulate, because it is the product of n probability distributions, but PDS makes a global restriction by conditioning to , so that the selections of the dierent components are not independent.

3 Previous Simulation Algorithms 3.1 The Simple Monte Carlo Algorithm (MC)

One simple way of obtaining con gurations C with probability PDS was proposed by Wilson [11] and consists of selecting sets Ci 2 2 , i = 1; : : : ; n with probability Pi (Ci ) until we get a con guration of sets C = (C1 ; : : : ; Cn ) such that IC 6= ;. In this way, we obtain a con guration C with probability proportional to PDS(C ). The main drawback of this method is that it may be very dicult to get a sequence with nonempty intersection. More precisely, the probability of obtaining such a sequence is P0 ( ). The value 1 , P0 ( ) is known as the con ict among the evidences. So, we can say that this algorithm does not perform well if the con ict is high (i.e. if P0 ( ) is close to 0). See [6, 11] for more details.

3.2 The Markov Chain Monte Carlo Algorithm (MCMC)

Unlike the simple Monte Carlo, in this case the trials are not independent. Instead they form a Markov chain. It means that the result of each trial depends only in the result of the previous one. Moral and Wilson [5, 6] designed a method for generating a sample from PDS that starts with an initial con guration C 0 = (C10 ; : : : ; Cn0 ). Then, the next element in the sample is selected by changing only one of the coordinates in the current con guration and so on. More precisely, one con guration C j is constructed from C j,1 by changing the i-th coordinate in C j,1 . The new set Cij in the i-th coordinate is selected verifying IC j 6= ; and with chance proportional to Pi (Cij ). For each con guration C i , the corresponding trial will be successful if IC i  X . Bel(X ) is again estimated by the proportion of successes.

3.3 Importance Sampling Algorithms

Importance sampling is a popular technique in Statistics, used to reduce the variance in Monte Carlo simulation [8]. It has been successfully applied to approximate probabilistic reasoning [3, 4] and to belief functions calculation [6]. The idea of this technique is to try to obtain points in those regions of higher importance (probability), using during the sampling process a distribution easier to handle than the original one (PDS ), in the sense that it must be easy to obtain a sample from it. Consider a probability distribution P over such that P (C ) > 0 for all C 2 with PDS(C ) > 0. Then, we can write

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Bel(X ) = =

X C 2

IC X

PDS(C ) =

X C 2

X PDS(C )X (IC )

C 2

P (C )

PDS(C )X (IC )



P (C ) = E PDS(PZ)(ZX) (IZ )



;

where X (IC ) = 1 if IC  X and 0 otherwise, and Z is a random vector with distribution P. Thus, w = PDS (Z)X (IZ )=P(Z) is an unbiased estimator of Bel(X ) with variance Var(w) = E [w2 ] , E [w]2 X P2DS(C )X2 (IC )  2 =  (C ))2 P (C ) , Bel (X ) (P C 2

X P2DS(C )X2 (IC ) = , Bel2 (X ) :  (C ) P C 2

Now let (C 1 ; : : : ; C N ) be a sample of elements of obtained by sampling from P , and wi the value of the estimator for C i , also called weight of P C i . Then c Bel(X ) can be approximated by the sample mean as Bel(X ) = (1=N ) Ni=1 wi . Note that

8 P (C i ) < DS wi = : P (C i ) 0 c X ) = (1=N ) P which implies that Bel(

if IC i  X ; otherwise,

(2)

ICi X wi .

The variance of this sample mean estimator is N ! X 1 c X )) = Var Var(Bel( N i=1 wi = 1 (N Var(w))

N2

= N1

X P2DS(C )X2 (IC )

0C2

BX = N1 B B@ C 2

IC X

P(C )

!

, Bel2 (X )

1 P2DS (C ) , Bel2 (X )C CC ; P(C ) A

(3)

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where we have used that X2 (IC ) = 1 for IC  X . The minimum variance is equal to 0, and is achieved when P = PDS(
j X ), where X = fC 2 j IC  X g. For each C 2 , PDS (C j X ) = PDS(C \

X )=PDS( X ), and PDS ( X ) = Bel(X ). It means that we need the value of Bel(X ) to construct a P which minimizes the variance of the estimation, and this value is unknown: it is precisely what we are trying to calculate. Furthermore, P depends on X and dierent simulations are necessary for dierent sets of interest X . To obtain a small variance, we must select P close to PDS. P must be easy to handle, in the sense that it must be easy to simulate a sample from it. In this case the minimum variance we can reach is (1=N )Bel(X )(1 , Bel(X )), which is always less than or equal to 1=(4N ). Once P is selected, the problem is how to compute the weights of the con gurations. For a given con guration C i , its weight is wi = PDS (C i )=P(C i ). P (C i ) is easy to compute if we select P appropriately.P But it is not that easy to calculate PDS(C i ) = P0 (C i )=P0 ( ), because P0 ( ) = C 2 P0 (C ). Moral and Wilson [6] propose to use this weight instead: Wi = P0 (C i )=P(C i ). In this case, 1 X W = 1 P0 ( ) X w ; i N I i X i N IC i X C P is an estimator of Bel(X )
P0 ( ). Since (1=N ) Ni=1 Wi is an unbiased estimator of P0( ), then we can estimate Bel(X ) with the value

P c X ) = PICNi X Wi : Bel( W i=1 i

(4)

Moral and Wilson [6] propose two algorithms based on the importance sampling technique.

Importance Sampling with Restriction of Coherence (IS1). This scheme

is very similar to the simple Monte Carlo algorithm (see Sec. 3.1). The dierence is that, instead of generating a con guration C k and then checking the condition C k 2 , whenever a new coordinate Cik is going to be generated, it is chosen with probability proportional to Pi (Cik ) from those sets verifying that Cik \ T i , 1 ( j=1 Cjk ) 6= ;. In this way, only con gurations in are selected. Following this procedure, the probability of selecting a given con guration C = (C1 ; : : : ; Cn ) is n P (C ) Y i i

0

= Qn P (PC ) (i) ; (5) P ( i ) i=1 CO i=1 CO P ,1 Cj )6=; Pi (Ci ). where PCO is the probability of coherence: PCO (i) = Ci \(Tij=1 P (C ) =

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Q

Then, the weight corresponding to C is W = P0 (C )=P (C ) = ni=1 PCO (i). Therefore, for a sample (C 1 ; : : : ; C N ) generated from P , the estimation of Bel(X ) is

P Qn P (j ) I  X i C c X ) = PN Qn j=1 CO : Bel( P (j ) i=1 j=1 CO

(6)

Selecting Con gurations Containing an Element (IS2). This procedure is based on selecting an element  2  and then selecting a con guration such

that each one of its coordinates contains . In this way, every con guration has nonempty intersection: at least  belongs to all the sets. P Each  is selected with probability P() = Q0()= 2 Q0 ( ), where Q0 is an approximation of the commonality function Q of the combined belief. Let be  = fC 2 j  2 IC g. Then a con guration C is selected from  with probability proportional to PDS(C ). This selection is easy since for C 2 , P0 (C j ) = P0 (C j \ ), which is equal to PDS(C j ) = PDS (C )=Q(). Then, choosing a con guration with probability proportional to PDS is the same as choosing it with probability proportional to P0 . Thus, we have that P (C ) = .

X

2IC

P()PDS (C )=Q() /

X Q0()

PDS(C ) 2IC Q()

Therefore, the weight W associated with C is proportional to 1=

X

2IC

Q0 ()=Q() ;

which is equal to 1=jIC j in the case that Q0 = Q. The performance of this algorithm in experimental evaluations is quite good [6], especially if the focal elements have a small number of elements. However with big focal elements the algorithm may have problems: more diculty in the approximation of Q0 and more variation in the weights even if Q0 = Q. Imagine for example, a belief function with two focal elements: one has only one element, , and the other the rest of elements of . Both have a mass of 0.5. If we select an element with a probability proportional to its commonality, then since all elements have the same commonality, the distribution is uniform. The focal element with only one element will be selected with a probability 1=jj, and weight 1, and the other focal with probability (jj , 1)=jj, and weight 1=(jj , 1). If the number of elements of  is high, this will produce a great variance of the weights (most of the elements in the sample will have a small relative weight), which will produce poor estimations.

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4 Importance Sampling Based on Approximate Pre-computation Here we propose a new importance sampling method. It is based on the algorithm for computing marginals in Bayesian networks developed by Hernandez, Moral and Salmeron [3, 4]. The new algorithm is similar to the importance sampling with the restriction of coherence, but here, a previous step is carried out in which some information is collected in order to obtain a better sample. Note that each coordinate Ci in a con guration C is selected taking into account just mass mi and the intersection of the coordinates already simulated. We propose to use also some information coming from the masses corresponding to the coordinates that still have not been simulated. In this way we try to avoid rare con gurations with low weight. Let us describe the algorithm in more detail. We assume we are not able to compute Dempster's rule. It means that the resources are limited and the number of focal sets in m is too big to be handled by the computer. Let s be the maximum number of focal sets that the system can handle. Given two masses m1 and m2 , their approximate combination with limit s, will be a mass m1 ~ s m2 , calculated in the usual way (adding the non-empty intersections of focal elements of m1 and m2 ), but performing the following reduction step each time the limit s is surpassed: 1. Select two focal sets B1 and B2 and remove them from the list of focal sets of m1 ~ s m2 . Let be B = B1 [ B2 . 2. If B is already a focal set of m1 ~ s m2 , then (m1 ~ s m2 )(B ) := (m1 ~ s m2 )(B ) + (m1 ~ s m2 )(B1 ) + (m1 ~ s m2 )(B2 ) ;

else, add B as a new focal set of the combination with (m1 ~ s m2 )(B ) := (m1 ~ s m2 )(B1 ) + (m1 ~ s m2 )(B2 ) : The idea is to carry out an approximate and feasible computation by replacing two focal sets B1 and B2 by their union, each time the maximum number of focal sets s is surpassed. Using this procedure, n new masses m1 ; : : : ; mn can be de ned according to the following recursive formula: m1 = m1 , mk = mk ~ s mk,1 , where k = 2; : : : ; n. Note that these masses can always be computed since they do not have more than s focal sets. Now we de ne the following sampling procedure. To obtain a con guration C = (C1 ; : : : ; Cn ) the sets are calculated from n to 1 according to the following distributions



 ,1 Ci \ Ci+1 \


\ Cn 6= ; ; Pi (Ci jCi+1 ; : : : ; Cn ) = k0i Pi (Ci )Pli,1 (Ci ) ifotherwise ;

(7)

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where Pli,1 is the plausibility corresponding to mass mi,1 , Pl0  1 and ki,1 is P a normalization constant given by ki = A\(Tnj=i+1 Cj )6=; Pi (A)Pli,1 (A), with k1 = 1. We have to check that P is valid as sampling distribution, that is, it must be positive whenever PDS is. The only thing we have to check is that the product of all Pli is positive when PDS is, but this is true since in the approximate combination, removed sets are replaced by the union of both. Then, if a set has positive mass, then it has also positive plausibility after the deletion. The intuition behind this sampling procedure is the following: If Pli were the exact plausibility associated with m1


mi , for each i, then the sampling distribution P would be equal to the desired distribution PDS (this is simple to see from the de nition of this probability distribution). With approximate combinations mi we are trying to simulate as close as possible to the distribution PDS. One remarkable property of the approximate combination is that it can be detected when the exact calculation is feasible: it would be the case in which the size threshold is not surpassed in any of the combinations performed to compute m1 ; : : : ; mn . If this happens, in mn we obtain the exact combination by Dempster's rule, and then there is no need to simulate. The weight assigned to a con guration C will be

Q

n n P (C ) 0 Y ki i=1 i i W = PP((CC)) = = : Q  n  , 1 P1 (C1 ) i=2 ki Pi (Ci )Pli,1 (Ci ) i=2 Pli,1 (Ci )

(8)

4.1 The Main Algorithm The pseudo-code for performing the tasks concerning this algorithm can be as follows. Let MASSi (Y ) be a function returning the sum of Pi (A)Pli,1 (A) subject to A\Y 6= ;, and SELECTi (Y ) returning one element A 2 2 subject to A\Y 6= ; with probability Pi (A). Let PLAUi (Y ) be a function returning Pli,1 (Y ). The following algorithm computes an estimation of Bel(X ) with X   from a sample of size N :

FUNCTION ISAPNs m1 = m1 for v = 2 to n mv = mv ~ s mv,1

next v

S1 := 0:0 ; S2 := 0:0 for v = 1 to N Y :=  ; W := 1:0 for j = n downto 2 W := W  MASSj (Y )=PLAUj (Y ) Cj := SELECTj (Y ) Y := Y \ Cj

A Monte Carlo Algorithm for Combining Dempster-Shafer Belief

313

next j

C1 := SELECT1 (Y ) Y := Y \ C1 Let C = (C1 ; : : : ; Cn ) if IC  X then S1 := S1 + W else S2 := S2 + W

next v S return S1+1S2

4.2 Particular Cases The performance of the ISAP algorithm depends on the way in which the approximate combination is done. More precisely, the critical decision to be made is the selection of the sets to be removed in step 2.a in the approximate combination. Those sets should be selected such that the distance between the mass before and after the elimination be as small as possible. We have considered the following measure of the cost of removing two sets A and B and adding their union A [ B with m(A [ B ) = m(A) + m(B ): d(A; B ) = m(A)jB , Aj + m(B )jA , B j :

(9)

Thus, when performing the approximate combination, a rst option can be to compute d(A; B ) for every focal sets A and B , and select those producing the lowest cost. We will reference the ISAP algorithm with this option as ISAP1. However, if the number of focal elements to be compared is high, it can be too costly to compute all the values of the cost function. In order to skip this problem, we have tested two alternatives. In the rst one, the focal sets of the combination are kept in a list such that whenever a new focal set is inserted, it is placed between those two sets for which the addition of the costs with respect to the new set is the lowest. Then, when two sets must be removed, the cost is computed just for adjacent sets in the list. We call ISAP2 this case of the algorithm. The second option is similar to the former, but now the sets are inserted in the list in lexicographical order. This will be called ISAP3.

5 Experimental Evaluation The performance of all the algorithms described in this paper has been experimentally tested. The experiment consisted of combining 9 belief functions de ned over a sample space  with 50 elements. The number of focal sets in each function was chosen following a Poisson distribution with mean 4.0. Each focal set was determined by rst generating a random number p 2 [0; 1] and then including each  2  with probability pp (so that to obtain focal sets with a high number of elements).

314

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The experiment has been repeated 10 times for dierent belief functions. In each repetition, a set X   was selected at random and its belief Bel(X ) was exactly computed and also approximated by the Monte Carlo algorithms. The number of runs (i.e. the sample size) was set to 5000, and each algorithm was executed 100 times. c X ) was the approximation given by one execution of a Monte Carlo If Bel( algorithm, the error in the estimation was calculated as

c X ) , Bel(X )j Bel( pjBel( ; X )(1 , Bel(X ))

(10)

in order to make the errors independent of the value Bel(X ). The mean of the errors is shown in table 1. According to the obtained results, we can say that the new algorithms in general provide better results than the previous ones. However, there are not clear dierences among the three versions of the ISAP algorithm. We think this is due to the de nition of function d (see Eq. 9). The results of the new methods do not seem to depend on the value of P0 ( ). In fact, we have checked that in other examples with much lower values of P0 ( ), the error levels keep steady. Exp. P0 ( ) MC MCMC IS1 IS2 ISAP1 ISAP2 ISAP3 1 0.210590 0.028638 0.013578 0.015282 0.015407 0.009971 0.012571 0.011800 2 0.218410 0.026885 0.011841 0.014306 0.016761 0.011758 0.010731 0.012193 3 0.221991 0.022345 0.014493 0.013641 0.014716 0.011360 0.011528 0.010813 4 0.269354 0.021218 0.012873 0.015125 0.015278 0.010635 0.010534 0.010434 5 0.323525 0.016665 0.014102 0.014066 0.015277 0.011263 0.010742 0.011107 6 0.383930 0.018580 0.012625 0.015151 0.014154 0.012355 0.011051 0.012406 7 0.439027 0.016712 0.013032 0.015624 0.013595 0.012182 0.011447 0.010541 8 0.455528 0.018179 0.013355 0.013546 0.015219 0.011607 0.011489 0.010506 9 0.559075 0.015046 0.011789 0.012395 0.021883 0.011388 0.011904 0.012044 10 0.568261 0.014193 0.010899 0.011204 0.017189 0.010423 0.011737 0.011192

Table 1. Errors when combining 9 belief functions

6 Conclusions We have presented a new technique for combining belief functions that shows a good performance. This conclusion is supported by the experiments we have carried out, showing a behaviour clearly superior to existing algorithms. One feature of this scheme is that it can be detected when the exact combination can be done. Many aspects are still to be studied; for instance, a good measure of the cost of removing focal sets could provide more accurate estimations. We have only given a simple, but perhaps not very accurate cost function for an approximation step. This is supported by the fact that there are not important dierences between the algorithm (ISAP1) that considers the cost of joining all

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the pairs of focal elements and the algorithms (ISAP2 and ISAP3) considering only some of them. Besides, instead of representing mass functions as lists of focal sets, a more sophisticated representation could be used. The semi-lattice representation [9] would facilitate the realization of the operations among masses. The algorithm we have developed deals with beliefs de ned over the same sample space. Some more works are needed to extend it to the case of multivariate belief functions. Exact algorithms for this case can be found in [1, 9].

References 1. Bissig, R., Kohlas, J., Lehmann, N. (1997). Fast-division architecture for DempsterShafer belief functions. D. Gabbay, R. Kruse, A. Nonnengart and H.J. Ohlbach (eds.), Qualitative and Quantitative Practical Reasoning, First International Joint Conference on Qualitative and Quantitative Practical Reasoning; ECSQARU{ FAPR'97. Lecture Notes in Arti cial Intelligence. Springer. 2. Dempster, A.P. (1967). Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics, (38):325{339. 3. Hernandez, L.D., Moral, S., Salmeron, A. (1996). Importance sampling algorithms for belief networks based on approximate computation. Proceedings of the Sixth International Conference IPMU'96. Vol. II, 859{864. 4. Hernandez, L.D., Moral, S., Salmeron, A. (1998). A Monte Carlo algorithm for probabilistic propagation based on importance sampling and strati ed simulation techniques. International Journal of Approximate Reasoning, (18):53{91. 5. Moral, S., Wilson, N. (1994). Markov-Chain Monte-Carlo algorithms for the calculation of Dempster-Shafer Belief. Proceedings of the Twelfth National Conference on Arti cial Intelligence (AAAI-94) Vol. 1, 269{274. 6. Moral, S., Wilson, N. (1996). Importance sampling algorithms for the calculation of Dempster-Shafer belief. Proceedings of the IPMU-96 Conference, Vol. 3, 1337{1344. 7. Orponen, P. (1990). Dempster's rule is #P-complete. Arti cial Intelligence, (44):245{253. 8. Rubinstein, R.Y. (1981). Simulation and the Monte Carlo Method. Wiley (New York). 9. Salmeron, A., Jensen, F.V. (1997). HUGIN architecture for propagating belief functions. Technical Report #GAD9702. Data Analysis Group. University of Almera. 10. Shafer, G. (1976). A mathematical theory of evidence. Princeton University Press, Princeton, NJ. 11. Wilson, N. (1991). A Monte Carlo algorithm for Dempster-Shafer belief. Proc. of the 7th Conference on Uncertainty in Arti cial Intelligence, B. D'Ambrosio, P. Smets and P. Bonissone (eds.), Morgan Kaufmann, 1991.

An Extension of a Linguistic Negation Model allowing us to Deny Nuanced Property Combinations Daniel Pacholczyk LERIA, University of Angers, 2 Bd Lavoisier, F-49045 Angers Cedex 01, FRANCE [email protected]

Abstract. In this paper, we improve the abilities of a previous model of representation of vague information expressed under an affirmative or negative form. The study more especially insists on information referring to linguistic negation. The extended definition of the linguistic negation that we propose makes it possible to deny nuanced property combinations based upon disjunction and conjunction operators. The properties that this linguistic negation possesses now allow considering it as a generalization of the logical one, and this is in satisfactory agreement with linguistic analysis.

1

Introduction

In this paper, we present a new model dealing with affirmative or negative information within a fuzzy context. We refer to the methodology proposed in ([13], [14], [15], [16]) to represent nuanced information, and more particularly, information referring to linguistic negation. The modelisation has been conceived in such a way that the user deals with statements expressed in natural language, that is to say, referring to a graduation scale containing a finite number of linguistic expressions. The underlying information being generally evaluated in a numerical way, this approach is based upon the fuzzy set theory proposed by Zadeh in [19]. Previous models allow us to represent a rule like “if the wage is not high, then the summer holidays are not very long” and a fact like “the wage is really low”. But, in previous methodology, facts and rules are only based upon nuances of one property. So, they do not accept facts or rules based upon combinations of properties as they may appear in knowledge bases including a rule like, “if the man is not visible in the crowd, he is not medium or tall”, or a fact like “the man is rather small or very small”. Our main idea has been to extend the definition of linguistic negation proposed in ([13], [14], [15], [16]) in order to take into account combinations of nuanced properties based on conjunction or disjunction connectors. Representing nuances in terms of membership functions in the context of the fuzzy set theory ([19]) is nothing less than a difficult task and numerous works tackle different aspects of this problem: modifiers ([1], [2], [3], [5], [6], [12], [20], [7], [13], [14], [15]) or linguistic labels ([18]). Section 2 briefly presents a previous model [7] allowing the representation of A. Hunter and S. Parsons (Eds.): ECSQARU’99, LNAI 1638, pp. 316-327, 1999.  Springer-Verlag Berlin Heidelberg 1999

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nuanced properties. In Section 3, we present the main ideas leading to the previous methodology. More precisely, we point out that it is necessary to leave a representation of linguistic negation in terms of a one-to-one correspondence and to turn towards a one-to-many correspondence, called here a multiset function. We recall previous results leading to the standard forms of negation. The definitions of the linguistic negation and of the intended meaning of a denied property are based upon the ones proposed in ([16], [17]). Some modifications have been made to obtain better accordance with the linguistic analysis proposed in ([11], [4], [9], [8], [10]). Section 4 is devoted to new developments which improve the abilities of the previous model. We extend the definition of linguistic negation in order to be able to deny previous combinations of nuanced properties. Then, we can define the set of intended meanings of “x is not A”. Finally, we study the particular contraposition law. It appears clearly that this more powerful concept of linguistic negation can be viewed as a generalization of the logical one.

2

The Universe Description

We suppose that our discourse universe is characterised by a finite number of concepts Ci. A set of properties1 Pik is associated with each Ci, whose description domain is denoted as Di. The properties Pik are said to be the basic properties connected with concept Ci. For example, the concepts of “height”, “wage” and “appearance” should be understood as qualifying individuals of the human type2. The concept “wage” can be characterised by the basic fuzzy properties “low”, “medium” and “high”. Linguistic modifiers bearing on these basic properties permit us to express nuanced knowledge. This work uses the model proposed in [7] to represent affirmative information expressed in the form « x is fαmβPik » or « x is not fαmβPik » in the case of negation3. In this context, expressing a property like “fαmβPik” called here nuanced property, requires a list of linguistic terms. 1

This approach to linguistic negation is semi-qualitative. Indeed, the knowledge is expressed in natural language, but the underlying information is evaluated in a numerical way. Note that a qualitative approach of linguistic negation based upon similarity relation is under study. 2 The choice of referential types, fixes standard values of basic properties for typical individuals of the chosen referential types. 3 An important difference exists between the variables occurring in linguistic fuzzy assertions and those defining membership degrees of the associated fuzzy sets. Consider, for instance, the linguistic predicate “tall” associated with the concept “height”. Fuzzy set theory tells us that “tall” must be represented by a fuzzy subset “tall” of the real line, whose membership function, denoted as µtall,, associates with any u∈R a quantity α=µtall(u): in other words, any x, having u as “height” value (i.e. h(x)=u), belongs to the fuzzy set “tall” to a degree α. Logically speaking, x satisfies the predicate “tall” to a degree α (i. e. “x is tall” is α–true) if α=µtall(u)=µtall(h(x)). So, x is a linguistic variable for individuals of the human type, and u variable for the corresponding real evaluations. In order to simplify the presentation of our model, if no confusion is possible, for any property A associated with a concept C, and represented by a fuzzy set A, if c(x) denotes the value of C for x, then the notation α=µA(x)

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Two ordered sets of modifiers4 are selected depending on their modifying effects. - The first one groups translation modifiers resulting somehow in both a translation and a possible precision variation of the basic property: For example, the set of translation modifiers could be M7={extremely little, very little, rather little, moderately (∅), rather, very, extremely} totally ordered by the relation: mα); G(:p j>) 6j plldg G(p ^ :p j>) as well as G(p j>); G(:p j>) 6j plldg G(q j>) if q is not logically implied by p or :p. In particular, for any phasing function  and independence relation I the consistency check blocks the second derivation step in the following counterintuitive derivation. In the derivation below, as well as in following ones, the rst blocked step is represented by a dashed line. G(pj>)(ffG(pj>)gg;0) wc G(p _ qj>)(ffG(pj>)gg;(wc)) sa G(p _ q j :p)(ffG(pj>);:pgg;(sa)) G(:pj>)(ffG(:pj>)gg;0) cta G(q ^ :pj>)(ffG(pj>);:p;G(:pj>))gg;(cta)) wc G(qj>)(ffG(pj>);:p;G(:pj>)gg;(wc))

380

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The label of the goal potentially derived by sa contains besides the premise G(pj>) also the strengthening :p. Consequently sa cannot be applied, because it would result in a goal with contradictory ful llment p and strengthening :p. In general, it is obvious that the two premises G(p j>) and G(:p j>) cannot be combined in any single case, because the derived goal would contain the goals with contradictory ful llments p and :p. The second example concerns context sensitivity. Consider the two goals `I prefer coee' G(c j>) and `I prefer tea if there is no coee' G(t j:c). The logic plldg combines strengthening of the antecedent (sa) with weakening of the consequent (wc) without validating the following counterintuitive derivation of `I prefer coee or no tea if there is no coee' G(c _ :tj:c) from the rst premise.

G(cj>)(ffG(cj>)gg;0) wc G(c _ :tj>)(ffG(cj>)gg;(wc))

G(c _ :tj:c)(ffG(cj>);:cgg;(sa))

sa

The sa rule cannot be applied, because the label of the potentially derived goal contains besides the premise G(c j>) also the strengthening :c. In general, we call a goal G( j ) a secondary goal of the primary goal G(1 j 1 ) if and only if ^ 1 is classically inconsistent, and a secondary goal can neither be derived from a primary one, nor can they be combined in a single case. The third example concerns overriding. Consider the goals `Sue prefers to marry John over not' G(j j>) and `Sue prefers to marry Fred over not' G(f j>). The plldg derivation below shows that the goal `Sue prefers to marry John and Fred' G(j ^ f j>) can defeasibly be derived, because the conjunction rule can be derived from sa and cta if (cta)  (sa).

G(j j>)(ffG(jj>)gg;0) sa G(j jf )(fG(jj>);f gg;(sa)) G(f j>)(ffG(f j>)gg;0) cta G(j ^ f j>)(ffG(jj>);f;G(f j>)gg;(cta)))

Now consider the additional goal `Sue prefers not to be married to both John and Fred' G(:(j ^ f ) j>). If the goals are independent, i.e. I contains at least (G(j j >); G(:(j ^ f ) j >)) and (G(f j >); G(:(j ^ f ) j >)), then the derivation above is still valid, because additional independent premises do not in uence valid derivations. Otherwise the derivation is not valid, because the sa rule is blocked. The counterintuitive goal cannot be derived, because the ful llment of G(j ^ f j>) violates the additional premise.

G(j j>)(ffG(jj>)gg;0)

sa

G(j jf )(fG(jj>);f gg;(sa)) G(f j>)(ffG(f j>)gg;0) cta G(j ^ f j>)(ffG(jj>);f;(f j>)gg;(cta)) Note that the fact that the conjunction rule only holds non-monotonically implies that we have to be careful whether we formalize a goal for p ^ q by the

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formula G(p) ^ G(q ) or by G(p ^ q ), where we write G() for G(j>). Although we have G(p); G(q ) j plldg G(p ^ q), G(p ^ q)j plldg G(p) and G(p ^ q)j plldg G(q), they derive dierent consequences when other goals are added to the goal base. For example with only dependent goals we have G(p); G(q ); G(r) j plldg G(rj:(p ^ q)) and G(p ^ q ); G(r) 6  j plldg G(rj:(p ^ q)), and with dependent goals we also have G(p); G(q); G(rj:p) 6  j plldg G(rj:p ^ :q) and G(p ^ q); G(rj:p)j plldg G(rj:p ^ :q). Hence, both representations are incomparable, in the sense that in some circumstances the former is more adventurous, and in other circumstances the latter. Finally, consider dependent goals in B1 = (fG(p); G(q )g; ;) and independent goals in B2 = (fG(p); G(q )g; f(G(p); G(q )); (G(q ); G(p))g). The ful llment of G(p j :q ) violates G(q j >), and the former goal can therefore only be derived when the two goals are independent. We thus have B1  6j plldg G(p j:q) and B2  j lldg G(pj:q). In other words, we have inheritance to sub-ideal subclasses (:q) only for independent goals. Due to lack of space we cannot further discuss the independence relation and the procedure to update it when additional goals are added to the goal base. 4

The relation between

plldg

and

pllg

The following proposition shows that plldg derives a subset of pllg introduced in [16].

Proposition 1. Let pllg be the plldg in which the consistency check R is replaced by the following check RF pllg on the consistency of the ful llments of all elements of F . RF

pllg : G(j )(F;p) may only be derived if the strengthenings together with the ful llments of goals of each Fi 2 F are consistent.

Note that the related inference relation `pllg of pllg is not relative to the goal base, because RF pllg does not refer to it. If we have B j plldg , then we have B `pllg  (but obviously not necessarily vice versa).

Proof We prove that if RF holds, then RF pllg also holds. Assume RF pllg does not hold. This means that the derived goal violates the goal it is derived from. Therefore RF does not hold, because each goal depends on itself (I is irre exive). The following proposition shows how plldg can be de ned as pllg with additional restrictions.

Proposition 2. Let B = (fG(1 j 1 ); : : : ; G(n j n )g; I ) be an unlabelled goal base, and let the materializations of dependent goals of G(i j i ) of B = 8 9 (S; I ) ?? G(j j j ) 2 S; < = not I (G(i j i ); G(j j j )); be MDG(G(i j i ) j B ) = f j ! j g ? ? : ? i ^ i ^ j ^ j is consistent ;

382

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We have B j lldg G( j ) if and only if there is a labelled goal G( j )L that can be derived from the set of goals

fG(i j i )ffG  j ( i

ggMDG(G(i j i)jB) j 1  i  ng

i)

in pllg, i.e. in plldg in which the consistency check RF is replaced by the check RF pllg of Proposition 1.

Proof We prove that RF implies RF pllg and vice versa. ) Assume RF pllg does not hold. This means that there is an Fi 2 F which is inconsistent, and this Fi contains a case with an element of MDG, say fG(i j i ); j ! j g. This represents exactly that the derived goal violates the goal G(j j j ) (that depends on G(i j i )). Consequently RF does not hold. ( If RF pllg holds then, arguing analogously, there is no violated dependent goal. Hence RF holds. In the following section we consider surprising consequences of this relation:

modi ed versions of the important phasing theorems of pllg also hold for plldg. 5

The one-phase logic

lldg

and four-phase logic

4lldg

We are interested in the following one- and four-phase logic, that represent the two systems discussed in the introduction with respectively a global consistency constraint and a local consistency constraint with phasing.

De nition 4 (lldg,4lldg). The logic lldg is the plldg with the phasing function  de ned by (sa) = 1, (cta) = 1, (wc) = 1, (or) = 1. The logic 4lldg is the plldg with (sa) = 1, (cta) = 2, (wc) = 3, (or) = 4. In Theorem 1 below we show that the phasing does not restrict the set of derivable goals: for each lldg derivation there is an equivalent 4lldg derivation. This proof rewriting theorem is analogous to Theorem 1 for llg and 4llg in [16]. We rst prove two propositions. The rst says that for each case the ful llments of the premises of the case together with the strengthenings imply the ful llment of the derived goal.

Proposition 3. For each goal G( j )(F;p) derived in plldg we have for each Fi 2 F that the ful llments of goals in Fi together with the strengthenings in Fi classically imply  ^ . Proof By induction on the structure of the proof tree. The property trivially holds for the premises, and it is easily seen that the proof rules retain the property. The second proposition says that we can rewrite two steps of an lldg proof into a 4lldg proof.

Proposition 4. We can replace two subsequent steps of an lldg derivation by an equivalent 4lldg derivation without violating a consistency constraint.

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Proof In Proposition 3 in [16] this has been proven for llg and 4llg. It can be generalized for lldg and 4lldg, because plldg derivations can be considered as pllg derivations with modi ed labels for premises, see Proposition 2. To illustrate this generalization, we repeat one reversal in Figure 1 below. We have to prove that if the rst derivation does not violate RF , then neither does the second derivation. Call the sets of ful llments of the three premises respectively F1 , F2 and F3 . The rst item of the label of the rst derived conditional is F1  (F2 [ F3 ) = (F1  F2 ) [ (F1  F3 ) and the rst item of the label of the second derived conditional is (F1  f 1 g  F2 ) [ (F1  f 2 g  F3 )

The two sets contain equivalent formulas, because each element of F2 classically implies 1 and each element of F3 classically implies 2 (Proposition 3). The other proofs are analogous.

G( j 1 ) G( j 2 ) or G(j ^ ( 1 _ 2 )) G( j 1 _ 2 ) cta G( ^ j 1 _ 2 ) G(j ^ ( 1 _ 2 )) G(j ^ ( 1 _ 2 )) sa sa G(j ^ 1 ) G( j 1 ) G(j ^ 2 ) G( j 2 ) cta cta G( ^ j 1 ) G( ^ j 2 ) or G( ^ j 1 _ 2 ) Fig. 1.

e.2. Reversing the order of or4 and cta2

From the two previous propositions follows the following generalization of Theorem 1 of [16]. This theorem illustrates the advantage of labelled deductive systems to inductively generalize of the rewriting of two subsequent steps to the rewriting of arbitrary proofs.

Theorem 1 (Equivalence lldg and 4lldg). Let B be a goal base. We have B j lldg G(j ) if and only if Bj 4lldg G(j ). Proof ( Every 4lldg derivation is a lldg derivation. ) We can take any lldg derivation and construct an equivalent 4lldg derivation, by iteratively replacing two subsequent steps in the wrong order by several steps in the right order, see Proposition 4. Moreover, Theorem 2 shows that in 4lldg the global consistency check on the label can be replaced by a local consistency check on the conjunction of

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the antecedent and consequent of the goal. This is dierent from the analogous Theorem 2 for 4llg in [16], because for non-defeasible goals a consistency check on phase-1 goals is suÆcient, whereas for defeasible goals we have to check phase-2 goals too.

Theorem 2. Let RAC be the following condition. RAC : G(j )(F;p) may only be derived from G(1 j 1 ) (and G(2 j2 )) if it does not violate a dependent goal (of the premise used to derive G(1 j 1 ) (and G(2 j2 ))). Consider any potential derivation of 4lldg, satisfying the condition Rp but not necessarily RF . Then the following conditions are equivalent: 1. The derivation satis es RF everywhere, 2. The derivation satis es condition RAC throughout phase 1 and 2.

Proof Through phase 1 and 2, F in the label has a unique element. The conjunction of antecedent and consequent of the derived goal is equivalent to the conjunction of ful llments of goals of its label and the strengthenings. Hence, in phase 1 and 2 RF holds i RAC holds. In phase 3 and 4 no goals can be violated. This proves the completeness of the phased proof theory given in the introduction, because we can simply de ne atomic goals as phase-1 goals, compound goals as phase-2 goals, and combined goals as phase-4 goals. The goal G( j ) is an atomic goal of goal base B if there is a goal G( j 0 ) 2 B such that classically implies 0 , i.e. ` 0 , and  ^ does not violate a goal dependent on G(j 0 ). The goal G(1 ^ : : : ^ n j ) is a compound goal of B if there is a set of atomic goals fG(1 j ); G(2 j ^ 1 ); : : : ; G(n j ^ 1 ^ : : : ^ n 1 )g of B such that ^ 1 ^ : : : ^ i does not violate a goal dependent on the goal used to derive G(i 1 j ^ 1 ^ : : : ^ i 2 ) for i = 2 : : : n. The goal G(j 1 _ : : : _ n ) is a combined goal of B if there is a set of compound goals fG(1 j 1 ); : : : ; (G(n j n )g of B such that i `  for i = 1 : : : n. 6

Conclusions

In this paper we studied a labelled logic of defeasible goals, with several desirable properties. The logic does not have the identity or con ict averse strategies, but it has restricted strengthening of the antecedent, weakening of the consequent, the conjunction rule, cumulative transitivity and the disjunction rule. We introduced phasing in the logic and we proved two phasing theorems. Phasing makes the proof theory more eÆcient, because only a single order of rule application has to be considered. Restricted applicability is of course well-known from logics of defeasible reasoning, where monotony and transitivity only hold in a restricted sense and where restricted alternatives of monotony have been developed, e.g. cautious and rational monotony. However, it is important to note that the logic of defaults is

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quite dierent from the logic of defeasible goals. For example, according to the generally accepted Kraus-Lehmann-Magidor paradigm [7] the identity (called supra-classicallity) holds in the logic of defaults. Moreover, the conjunction rule holds without restrictions, whereas in the logic of goals it has to be restricted to formalize the marriage example. This distinction is not surprising, because the quantitative counterparts { utilities and probabilities { also have completely dierent properties. A systematic study of these two types of non-monotonic reasoning is the subject of present investigations. References 1. F. Bacchus and A.J. Grove. Utility independence in a qualitative decision theory. In Proceedings of KR'96, pages 542{552, 1996. 2. C. Boutilier. Toward a logic for qualitative decision theory. In Proceedings of the KR'94, pages 75{86, 1994. 3. J. Doyle, Y. Shoham, and M.P. Wellman. The logic of relative desires. In Sixth International Symposium on Methodologies for Intelligent Systems, Charlotte, North Carolina, 1991. 4. J. Doyle and M.P. Wellman. Preferential semantics for goals. In Proceedings of the AAAI'91, pages 698{703, Anaheim, 1991. 5. D. Gabbay. Labelled Deductive Systems, volume 1. Oxford University Press, 1996. 6. Z. Huang and J. Bell. Dynamic goal hierarchies. In Intelligent Agent Systems: Theoretical and Practical Issues, LNAI 1027, pages 88{103. Springer, 1997. 7. S. Kraus, D. Lehmann, and M. Magidor. Nonmonotonic reasoning, preferential models and cumulative logics. Arti cial Intelligence, 44:167{207, 1990. 8. J. Lang. Conditional desires and utilities - an alternative approach to qualitative decision theory. In Proceedings of the ECAI'96, pages 318{322, 1996. 9. D. Makinson. On a fundamental problem of deontic logic. In P. McNamara and H. Prakken, editors, Norms, Logics and Information Systems. New Studies on Deontic Logic and Computer Science. IOS Press, 1998. 10. D. Makinson and L. van der Torre. The logic of reusable propositional output. Submitted, 1999. 11. J.D. Mullen. Does the logic of preference rest on a mistake? Metaphilosophy, 10:247{255, 1979. 12. J. Pearl. From conditional oughts to qualitative decision theory. In Proceedings of the UAI'93, pages 12{20, 1993. 13. S.-W. Tan and J. Pearl. Qualitative decision theory. In Proceedings of the AAAI'94, 1994. 14. S.-W. Tan and J. Pearl. Speci cation and evaluation of preferences under uncertainty. In Proceedings of the KR'94, pages 530{539, 1994. 15. L. van der Torre. Labeled logics of conditional goals. In Proceedings of the ECAI'98, pages 368{369. John Wiley & Sons, 1998. 16. L. van der Torre. Phased labeled logics of conditional goals. In Logics in Arti cial Intelligence, LNAI 1489, pages 92{106. Springer, 1998. 17. L. van der Torre and Y. Tan. Cancelling and overshadowing: two types of defeasibility in defeasible deontic logic. In Proceedings of the IJCAI'95, pages 1525{1532. Morgan Kaufman, 1995.

Logical Deduction using the Local Computation Framework Nic Wilson1 and Jer^ome Mengin2 1

2

School of Computing and Mathematical Sciences, Oxford Brookes University, Gipsy Lane, Headington, Oxford OX3 0BP, U.K. Institut de Recherche en Informatique de Toulouse, Universite Paul Sabatier, 118 route de Narbonne, F-31082 Toulouse Cedex [email protected], [email protected]

Keywords: Deduction, Theorem Proving, First-order logic, Circumscription

1 Introduction Computation in a number of uncertainty formalisms has recently been revolutionized by the notion of local computation. [13] and [9] showed how Bayesian probability could be eciently propagated in a network of variables; this has already lead to sizeable successful applications, as well as a large body of literature on these Bayesian networks and related issues (e.g., the majority of papers in the Uncertainty in Arti cial Intelligence conferences over the last ten years). In the late `Eighties, Glenn Shafer and Prakash Shenoy [18] abstracted these ideas, leading to their Local Computation framework. Remarkably, the propagation algorithms of this general framework give rise to ecient computation in a number of spheres of reasoning: as well as Bayesian probability [16], the Local Computation framework can be applied to the calculation of Dempster-Shafer Belief [18,12], in nitesimal probability functions [21], and Zadeh's Possibility functions. This paper shows how the framework can be used for the computation of logical deduction, by embedding model structures in the framework; this is described in section 3, and is an expanded version of work in [22]. This work can be applied to many of the formalisms used to represent and reason with common sense including predicate calculus, modal logics, possibilistic logics, probabilistic logics and non-monotonic logics. Local Computation is based on a structural decomposition of knowledge into a network of variables, in which there are two fundamental operations, combination and marginalization. The combination of two pieces of information is another piece of information which gives the combined eect; it is a little like conjunction in classical logic. Marginalization projects a piece of information relating a set of variables, onto a subset of the variables: it gives the impact of A. Hunter and S. Parsons (Eds.): ECSQARU’99, LNAI 1638, pp. 386-396, 1999.  Springer-Verlag Berlin Heidelberg 1999

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the piece of information on the smaller set of variables. Axioms are given which are sucient for the propagation of these pieces of information in the network. General propagation algorithms have been de ned, which are often ecient, depending, roughly speaking, on topological properties of the network. The reason that Local Computation can be very fast is that the propagation is expressed in terms of much smaller (`local') problems, involving only a small part of the network. Finite sets of possibilities (or constraints) can be propagated with this framework, and so deduction in a nite propositional calculus can be performed by considering sets of possible worlds; this is implemented in, for example, PULCINELLA [15], and described formally in [17]. However, dealing with sets of possible worlds is often not computationally ecient; it is only recently [7,8] that it has been shown how to use Local Computation to directly propagate sets of formulae in a nite propositional calculus. In the next section we introduce the Local Computation framework. We describe in section 3 how model structures can be embedded in the framework. This is applied to rst-order predicate calculus in section 4. The last section discusses further applications and advantages of this approach.

2 Axioms for Local Computation The primitive objects in the Local Computation framework are an index set  (often called the set of variables ) and for each r  a set r , called the set of r-valuation, or, the valuations on r (an r-valuation is intended to represent a piece of information related to the set of variables r). The set of valuations S is de ned to be r r . We assume a function : , called combination, such that if A r and B s then A B r[s , for r; s . If A and B represent pieces of information then A B is intended to represent an aggregation of the two pieces Sof information. We also assume that, for each r , there is a function r : sr s r , called marginalization to r. It # r is intended that A will represent what piece of information A implies about variables r. The framework assumes that the following axioms are veri ed: Axiom LC1 (Commutativity and associativity of combination): Suppose A, B and C are valuations. Then A B = B A and A (B C) = (A B) C. Axiom LC2 (Consonance of marginalization) : Suppose A is a t-valuation and r s t . Then (A#s )#r = A#r . Axiom LC3 (Distributivity of marginalization over combination) 1: Suppose A is an r-valuation and B is an s-valuation and r t r s . Then (A B)#t = A B #s\t . 

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LC3 is slightly stronger than the corresponding axiom A3 given in [18], (their axiom is LC3 but with the restriction that r = t); it turns out to be occasionally useful to have this stronger axiom.

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Let A1 ; : : :; An be valuations with, for i = 1; : : :; n, Ai ri . Many problems can be expressed as calculating (A1 An )#r0 for some subset of variables of Sn particular interest r0 r ; the valuation (A1 An )#r0 will represent i=1 i the impact of all the pieces of information on variables r0; in Bayesian probability this computes the marginal of a joint probability distribution; we will show below how testing the consistency of a set of formulae in propositional or predicate calculus can be expressed in this way. Elements of r will generally be simpler objects when r is small; for example they may be sets of formulae using only a small number of propositional symbols; also combination and marginalization will generally be much easier on the simpler objects (the computational complexity of these operations is typically exponential in r ). Direct computation of (A1 Sn An)#r0 will often be infeasible as it involves a valuation in r where r = i=1 ri . It can be seen that axioms LC1, LC2 and LC3 allow the computation of (A1 An )#r0 to be broken down into a sequence of combinations and marginalizations, each within some ri (i.e., local computations), if = r0 ; : : :; rn is a hyperforest. Brie y, is said to be a hyperforest if its elements can be ordered as s0 ; : : :; sn (known asSa construction sequence) where, for i = 1; : : :; n, there exists ki < i with si j