Modal Fixpoint Logic: Some model theoretic questions [PhD Thesis]

263 Pages·2010·1.296 MB·English

by  Gaelle Fontaine

Modal fixpoint logic:
some model theoretic questions
Ga¨elle Fontaine

Modal fixpoint logic:
some model theoretic questions
ILLC Dissertation Series DS-2010-09
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Modal fixpoint logic:
some model theoretic questions
Academisch Proefschrift
ter verkrijging van de graad van doctor aan de
Universiteit van Amsterdam
op gezag van de Rector Magnificus
prof.dr. D.C. van den Boom
ten overstaan van een door het college voor
promoties ingestelde commissie, in het openbaar
te verdedigen in de Agnietenkapel
op donderdag 9 december 2010, te 10.00 uur
door
Ga¨elle Marie Marguerite Fontaine
geboren te Messancy, Belgi¨e.
Promotiecommissie:
Promotores:
Prof.dr. J. F. A. K. van Benthem
Prof.dr. Y. Venema
Overige Leden:
Prof.dr. E. Gra¨del
Prof.dr. L. Santocanale
Prof.dr. J. Va¨a¨n¨anen
Prof.dr. I. Walukiewicz
Dr. B. D. ten Cate
Dr. M. Marx
Dr. D. Niwin´ski
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
The investigations were supported by VICI grant 639.073.501 of the Netherlands
Organization for Scientific Research (NWO).
Copyright (cid:13)c 2010 by Ga¨elle Fontaine
Printed and bound by Printondemand-worldwide.
ISBN: 978-90-5776-215-4
Contents
Acknowledgments ix
1 Introduction 1
2 Preliminaries 11
2.1 Syntax of the µ-calculus . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Semantics for the µ-calculus . . . . . . . . . . . . . . . . . . . . . 15
2.3 Game terminology and the evaluation game . . . . . . . . . . . . 21
2.4 µ-Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1 ω-Automata . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.2 µ-automata . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3 Disjunctive formulas . . . . . . . . . . . . . . . . . . . . . 31
2.5 Axiomatization of the µ-calculus . . . . . . . . . . . . . . . . . . . 32
2.6 Expressivity of the µ-calculus . . . . . . . . . . . . . . . . . . . . 33
2.6.1 Bisimulation . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6.2 Expressivity results . . . . . . . . . . . . . . . . . . . . . . 35
2.6.3 Expressivity results for µ-programs . . . . . . . . . . . . . 39
2.7 Graded µ-calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Completeness for the µ-calculus on finite trees 45
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Rank of a formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Completeness for generalized models . . . . . . . . . . . . . . . . 50
3.4 Completeness for finite trees . . . . . . . . . . . . . . . . . . . . . 55
3.5 Adding shallow axioms to Kµ +µx.(cid:50)x . . . . . . . . . . . . . . . 58
3.6 Graded µ-calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
v
4 The µ-calculus and frame definability on trees 67
4.1 µMLF-definability on trees . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Graded µ-calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Preservation under p-morphic images . . . . . . . . . . . . . . . . 84
4.4 Local definability . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5 Negative and projective definability . . . . . . . . . . . . . . . . . 108
4.5.1 Negative definability . . . . . . . . . . . . . . . . . . . . . 108
4.5.2 Projective definability . . . . . . . . . . . . . . . . . . . . 110
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5 Characterizations of fragments of the µ-calculus 115
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.1.1 Structures and games . . . . . . . . . . . . . . . . . . . . . 117
5.1.2 Guarded and disjunctive formulas . . . . . . . . . . . . . . 118
5.1.3 Expansion of a formula . . . . . . . . . . . . . . . . . . . . 119
5.1.4 Monotonicity and positivity . . . . . . . . . . . . . . . . . 120
5.2 Finite path property . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3 Finite width property . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4.1 Link with Scott continuity . . . . . . . . . . . . . . . . . . 139
5.4.2 Constructivity . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.4.3 Semantic characterization of continuity . . . . . . . . . . . 141
5.5 Complete additivity . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6 CoreXPath restricted to the descendant relation 165
6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.1.1 XML trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.1.2 CoreXPath, the navigational core of XPath 1.0 . . . . . . 167
6.1.3 Connections with modal logic . . . . . . . . . . . . . . . . 169
6.2 CoreXPath(↓+) node expressions . . . . . . . . . . . . . . . . . . 171
6.3 CoreXPath(↓+) path expressions . . . . . . . . . . . . . . . . . . 179
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7 Automata for coalgebras: an approach using predicate liftings 189
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.1.1 Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.1.2 Graph games . . . . . . . . . . . . . . . . . . . . . . . . . 194
7.2 Automata for the coalgebraic µ-calculus . . . . . . . . . . . . . . 197
7.3 Finite model property . . . . . . . . . . . . . . . . . . . . . . . . 203
7.4 One-step tableau completeness . . . . . . . . . . . . . . . . . . . . 214
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
vi
8 Conclusion 223
Bibliography 227
Index 237
List of symbols 241
Samenvatting 243
Abstract 245
vii