A descriptive approach to database languages and dynamic complexity 9798208399255

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A descriptive approach to database languages and dynamic complexity Patnaik, Sushant, Ph.D. University of Massachusetts, 1994

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A DESCRIPTIVE APPROACH TO DATABASE LANGUAGES AND DYNAMIC COMPLEXITY

A Dissertation Presented by

SUSHANT PATNAIK

Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY September 1994 Computer Science Department

0 Copyright by Sushant Patnaik 1994 All Rights Reserved

A DESCRIPTIVE APPROACH TO DATABASE LANGUAGES AND DYNAMIC COMPLEXITY

A Dissertation Presented by

SUSHANT PATNAIK

Approved as to style and content by:

il Immerman, Chair /

I

.

\

/

avid W. Stemple, Member

David A. M. Barrington, Member

Paris C. Kanellakis, Member

Nathaniel Whitaker, Member

1 W. Richards Adrion, Departm&^t Chair Computer Science

Dedicated to Ma } Baba and i?667l(Z

ACKNOWLEDGEMENTS I am grateful to Neil Immerman for his constant support, instruction and mentorship throughout the course of my dissertation. I was fortunate to have him as my advisor. I learnt a lot from his insightful thinking, his visionary ideas in Logic and Computer Science and his research philosophy. I am indebted to him and Susan Landau for inspiring and egging me on at a crucial stage in my graduate school career. Their encouraging words led me to complete my doctorate. I thank Dave Stemple for stimulating discussions on matters related to research and otherwise, Dave Barrington for inspiring and supporting me and for being avail­ able all the time for every little problem during my initial years in graduate school, Paris Kanellakis and Nate Whitaker for their concrete thoughtful comments on my research and presentation. I am obliged to Ramesh Sitaraman for showing me the feasibility of a probablistic construction of dynamic expanders, and Nabil Kahale for his proof of an explicit construction of dynamic expanders. Finally, I thank all my colleagues in the theory group, especially James Corbett, Kousha Etessami, Zhi-Li Zhang and Jose Antonio Medina, for numerous technical discussions and for making my stay in graduate school an enjoyable experience.

ABSTRACT A DESCRIPTIVE APPROACH TO DATABASE LANGUAGES AND DYNAMIC COMPLEXITY SEPTEMBER 1994 SUSHANT PATNAIK, B.S., INDIAN INSTITUTE OF TECHNOLOGY DELHI M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph .D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Neil Immerman In the first part of the thesis, we characterize exactly the expressiveness of a family of typed database transaction languages based on a traversal (or recursion) scheme devised by Fegaras, Sheard and Stemple (in [FSS92]), and we capture the complexity of reflection in a first order calculus framework. Reflection in a programming language refers to the ability to generate a program and execute it in the same environment. Using a first-order interpretation from PSPACE to the quantified Boolean formula value problem, we are able to derive an easy proof of the fact that reflection when added to first-order algebra captures exactly the problems in PSPACE. Further, we give a partial answer toward resolving the complexity of higher order reflection. In the second part, eschewing traditional (static) complexity classes for measur­ ing the complexity of database query languages, we propose a complexity theoretic framework for studying dynamic complexity classes. In this thesis we define the requisite dynamic complexity classes. In particular, we introduce and investigate a natural logic for a parallel dynamic complexity class, Dynamic First-Order Logic (Dyn-FO). We show that many interesting graph problems are in Dyn-FO, including, among others, graph connectivity and the computation of minimum spanning trees. We prove that certain standard complete problems for static complexity classes, such

vi

as AGAP for P remain, complete via these new reductions. On the other hand, we prove that other such problems including GAP for NL and 1GAP for L are no longer complete via bounded expansion reductions. Our results shed light on some of the interesting differences between static and dynamic complexity. We examine the dynamic complexity of maintaining approximate solutions to NP optimization problems in the descriptive framework. We introduce a new approxima­ tion class called BMAXSNP that is a subset of MAX SNP and includes interesting problems such as MAX CUT, MAX SAT and MAX 3SAT. We define reductions that honor dynamic complexity while preserving approximations. We then show the completeness, via such reductions, of a number of NP optimization problems for BMAXSNP, and we further show that this class has constant approximable solutions which can be maintained in Dyn-FO.

vii

TABLE OF CONTENTS Page

ACKNOWLEDGEMENTS

v

ABSTRACT

vi

LIST OP FIGURES

x

CHAPTER 1. INTRODUCTION 1.1 1.2 1.3

1

Background Motivation Organization

1 3 9

2. DESCRIPTIVE COMPLEXITY 2.1 2.2

13

Background First-Order Reductions

13 14

3. COMPLEXITY OF UTC 3.1 3.2 3.3 3.4 3.5

19

Introduction Definitions Results Proofs Conclusion

19 22 28 30 45

4. COMPLEXITY OF REFLECTION 4.1 4.2 4.3 4.4 4.5 4.6

46

Introduction Coding Quantified Boolean Formulas PSPACE-completeness of QBF Definitions and Reflection Constructs Proofs Conclusion

5. DYN-FO: A DYNAMIC COMPLEXITY CLASS 5.1

5.2

46 49 51 59 64 67

68

Introduction

68

5.1.1

Related Work

70

Dynamic Complexity Classes

71

viii

5.2.1 5.3 5.4

Definition of Dyn-C

72

Problems in Dyn-FO Dyn-FO versus NC1, L and NL

74 84

6. DYN-TIME COMPLEXITY CLASSES

91

6.1 6.2

Introduction Dynamic Language Recognition

7. BFO REDUCTIONS AND COMPLETENESS 7.1 7.2 7.3 7.4

91 95

102

Introduction Dynamic Reductions NP Complete Problems L, NL and P Completeness

102 102 106 113

8. LOWER BOUNDS IN DYN-TIME

120

8.1 8.2

Previous Work Lower Bounds

120 121

9. DYNAMIC APPROXIMATION CLASSES 9.1 9.2

Introduction Dynamic Approximation Classes

129 130

9.2.1

139

The Class BMAXSNP0

10.BEL REDUCTIONS AND COMPLETENESS 10.1 Introduction 10.2 Dynamic Approximation Preserving Reductions 10.3 Completeness under BEL reductions 10.3.1

Definition of BMAXSNP

10.4 Conclusions

143 143 143 160 160 162

11.CONCLUSIO N 11.1 11.2 11.3 11.4 11.5

129

163

Expressiveness Issues Dynamic Complexity Open Questions Dynamic Approximation Classes Future Work

163 164 166 168 170

APPENDICES A. NPO PROBLEMS B. ABBREVIATIONS

173 176

BIBLIOGRAPHY

177

IX

I

LIST OF FIGURES Figure

Page

5.1

GAP

77

7.1

Edge to Vertex Connectivity

105

7.2

3SAT to Max Ind Set

107

7.3

3SAT to Min Dominator

108

7.4

Vertex Cover to Min Dominator

110

7.5

3SAT to Min Coloring

Ill

8.1

DFA for Prefix Sum

122

8.2

Prefix Sum to UGAP

124

8.3

Prefix Sum Gadget

125

8.4

Planarity to Prefix Sum

126

9.1

Triangular TSP - Insertion

137

9.2

Triangular TSP - Deletion

137

10.1 NAE 3SAT to Max Cut

151

10.2 Bounded-degree Graphs

153

10.3 Dynamic Expander Graph

157

x

CHAPTER

1

INTRODUCTION 1.1

Background Complexity classes are usually defined by referring to computation models and

by putting suitable restrictions on them. The most commonly used models are deter­ ministic, non deterministic or alternating Turing machines, and the usual restrictions involve limiting time and/or space used in checking that a given input satisfies some property. However an alternate notion of complexity, called Descriptive Complexity, ensues from considering the power of a language needed to express the property. The complexity measures are not tied to a machine model of computation but to a language such as first order logic. The resources that are measured are typically number of variables, quantifier depth, size of the formulas, etc. A decision problem is viewed as a set of structures instead of the usual determination of membership in a set of strings over some alphabet. The descriptive approach was initiated by Fagin in [Fa74] where he gave a char­ acterization of non-deterministic polynomial time (NP) as the set of properties ex­ pressible in second-order existential logic. Later, Immerman in a series of papers [181, 182, 186, 187, 15, I89a, I89b, 191] showed that the computational complexity of checking a property is closely tied to the complexity of expressing the property in first-order

(FO) logic. He showed that the classical complexity classes - L, NL,

P and even parallel complexity classes such as NC have precise characterizations as properties expressible in first-order logic with suitable operators. (Vardi in [Var] independently came up with the same logical characterization for P.) The resources of space and time relate to the quantifier depth and number of variables respectively.

2

The existence of such natural languages for each important complexity class gave a new perspective on long standing problems in complexity theory. In particular, it led to the proof that NL is closed under complementation [I89a]. Immerman, inspired by the notion of interpretations between theories, also defined FO projections that induce maps between structures. These reductions preserve the relevant resources and are very weak. He showed that the usual problems complete for L, NL, P such as 1GAP, GAP and AGAP, remain complete under these very low-level reductions, thereby exhibiting a far closer similarity of the complete problems for each class than was known before. All the results that relate logics to complexity classes below NP rely on the presence of an ordering on the domain. This is an inherent feature of polynomial-time computation.

In the absence of an ordering, a number of lower bounds on the

quantifier depth have been proved using Ehrenfreucht-Fraisse games by Immerman, Fagin, Gurevich and others. But the main question posed by Immerman has so far remained unanswered, and that is, "Does there exist a "natural" logic for capturing order-independent L, NL or P?" Order independent P is the class of feasible properties that are generic. A possible approach is to consider candidate logics for capturing it, say for instance, extending first-order logic (w.o.