Fragments of Fixpoint Logics. Automata and Expressiveness [PhD Thesis]

270 Pages·2015·2.061 MB·English

by  Facundo Matias Carreiro

Fragments of Fixpoint Logics
Automata and Expressiveness
Facundo Matías Carreiro

Fragments of Fixpoint Logics
Automata and Expressiveness
ILLC Dissertation Series DS-2015-05
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Copyright c 2015 by Facundo Matías Carreiro
(cid:13)
Cover design by Facundo Matías Carreiro.
Printed and bound by GVO drukkers & vormgevers B.V.
ISBN: 978-90-6464-919-6
Fragments of Fixpoint Logics
Automata and Expressiveness
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 Aula der Universiteit
op vrijdag 11 december 2015, te 11.00 uur
door
Facundo Matías Carreiro
geboren te Buenos Aires, Argentinië.
Promotor: Prof.dr. Y. Venema Universiteit van Amsterdam
Overige leden: Prof.dr. J.F.A.K. van Benthem Universiteit van Amsterdam
Prof.dr. D.J.N. van Eijck Universiteit van Amsterdam
Prof.dr. M. Otto TU Darmstadt
Prof.dr. I. Walukiewicz Université de Bordeaux
Dr. A. Facchini IDSIA
Dr. M.J. Marx Universiteit van Amsterdam
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
Para los abuelos Tinos,
ejemplo de valores, coraje y perseverancia.
v

Contents
Acknowledgments xi
1 Introduction 1
1.1 Featuring logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Fragments of fixpoint logics . . . . . . . . . . . . . . . . . . . . . 9
1.3 Logic and automata . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Expressiveness modulo bisimilarity . . . . . . . . . . . . . . . . . 19
1.5 Source of the material . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Preliminaries 23
2.1 Terminology, transition systems and trees . . . . . . . . . . . . . . 23
2.2 Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Parity automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 The modal µ-calculus . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Logics of programs and games . . . . . . . . . . . . . . . . . . . . 31
2.5.1 Propositional Dynamic Logic . . . . . . . . . . . . . . . . 31
2.5.2 Concurrent PDL . . . . . . . . . . . . . . . . . . . . . . . 33
2.5.3 Game Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Bisimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.7 First-order logic and extensions . . . . . . . . . . . . . . . . . . . 36
2.7.1 First-order logic with generalized quantifiers . . . . . . . . 37
2.7.2 Fixpoint extension of first-order logic . . . . . . . . . . . . 38
2.7.3 First-order logic with transitive closure . . . . . . . . . . . 39
2.8 Second-order logics . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.9 Notational convention . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Fragments of fixpoint logics 43
3.1 Completely additive fragments . . . . . . . . . . . . . . . . . . . . 44
3.1.1 Fixpoint theory of completely additive maps . . . . . . . . 47
vii
3.1.2 Characterization of PDL inside µML . . . . . . . . . . . . 51
3.1.3 Characterization of FO(TC1) inside FO(LFP1) . . . . . . . 61
3.2 Continuous fragments . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.1 Fixpoint theory of continuous maps . . . . . . . . . . . . . 65
3.2.2 Characterization of CPDL inside µML . . . . . . . . . . . 67
3.2.3 Finiteness, µ FOE and WMSO . . . . . . . . . . . . . . 72
c ∞
3.3 The question of Game Logic . . . . . . . . . . . . . . . . . . . . . 76
3.4 Conclusions and open problems . . . . . . . . . . . . . . . . . . . 82
4 Subclasses of parity automata 85
4.1 Weak parity automata . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2 Continuous-weak parity automata . . . . . . . . . . . . . . . . . . 87
4.3 Additive-weak parity automata . . . . . . . . . . . . . . . . . . . 88
4.4 Partial unraveling of parity automata . . . . . . . . . . . . . . . . 89
4.5 Variants of parity automata . . . . . . . . . . . . . . . . . . . . . 92
4.6 Conclusions and open problems . . . . . . . . . . . . . . . . . . . 94
5 One-step model theory 97
5.1 Single-sorted first-order languages . . . . . . . . . . . . . . . . . . 99
5.1.1 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.1.2 One-step monotonicity . . . . . . . . . . . . . . . . . . . . 108
5.1.3 One-step continuity . . . . . . . . . . . . . . . . . . . . . . 114
5.1.4 One-step additivity . . . . . . . . . . . . . . . . . . . . . . 118
5.1.5 Dual fragments . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 Selected multi-sorted first-order languages . . . . . . . . . . . . . 125
5.2.1 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.2 One-step monotonicity . . . . . . . . . . . . . . . . . . . . 132
5.2.3 One-step additivity . . . . . . . . . . . . . . . . . . . . . . 134
5.2.4 One-step multiplicativity . . . . . . . . . . . . . . . . . . . 139
5.3 Selected modal languages . . . . . . . . . . . . . . . . . . . . . . . 139
5.3.1 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3.2 One-step monotonicity . . . . . . . . . . . . . . . . . . . . 141
5.3.3 One-step continuity . . . . . . . . . . . . . . . . . . . . . . 142
5.3.4 One-step additivity . . . . . . . . . . . . . . . . . . . . . . 143
5.3.5 Dual fragments . . . . . . . . . . . . . . . . . . . . . . . . 144
5.4 Effectiveness of the normal forms . . . . . . . . . . . . . . . . . . 146
5.5 Conclusions and open problems . . . . . . . . . . . . . . . . . . . 147
6 Concrete modal automata 149
6.1 Automata for test-free PDL . . . . . . . . . . . . . . . . . . . . . 150
6.1.1 From formulas to automata . . . . . . . . . . . . . . . . . 151
6.1.2 From automata to formulas . . . . . . . . . . . . . . . . . 153
6.2 Automata for PDL . . . . . . . . . . . . . . . . . . . . . . . . . . 158
viii