Consequence Finding in Modal Logic [PhD Thesis]

214 Pages·2009·2.088 MB·English

by  Meghyn Bienvenu

TTHHEESSEE

En vue de l'obtention du

DDOOCCTTOORRAATT DDEE LL’’UUNNIIVVEERRSSIITTÉÉ DDEE TTOOUULLOOUUSSEE

Délivré par l'Université Toulouse III - Paul Sabatier

Discipline ou spécialité : Informatique

Présentée et soutenue par Meghyn BIENVENU
Le 7 mai 2009

Titre : La génération de conséquences en logique modale

JURY
BLACKBURN Patrick, DR INRIA Nancy (membre)
GASQUET Olivier PR Université Paul Sabatier (membre)
HERZIG Andreas, DR Université Paul Sabatier (directeur de thèse)
LANG Jérôme, DR Université Paris Dauphine (directeur de thèse)
MARQUIS Pierre, PR Université d'Artois (rapporteur)
MENGIN Jérôme, MCF Université Paul Sabatier (directeur de thèse)
ROUSSET Marie-Christine, PR Université de Grenoble (membre)
WOLTER Frank, PR University of Liverpool (rapporteur)

Ecole doctorale : Mathématiques, Informatique, et Télécommunications

Unité de recherche : Institut de Recherche en Informatique de Toulouse

Directeur(s) de Thèse : Andreas Herzig, Jérôme Lang, et Jérôme Mengin
Rapporteurs : Pierre Marquis et Frank Wolter
Consequence Finding in Modal Logic
Meghyn Bienvenu
To Morfar

Acknowledgements
First of all, I would like to thank my thesis advisors Andreas Herzig, J´erome Lang,
and J´erome Mengin for all of the advice, support, and encouragement they have
provided me over these past few years. I feel truly lucky to have had such excellent
thesis advisors, and I sincerely hope that we will find find opportunities to work
together again in the future.
I would also like to thank Pierre Marquis and Frank Wolter for kindly accepting
to review this thesis, and Patrick Blackburn, Olivier Gasquet, and Marie-Christine
Rousset for agreeing to participate in my jury.
A special thanks to Sheila McIlraith, my undergraduatesummerproject supervisor
andfirstco-author, forhelpingmetake my firststeps asaresearcher andfor always
looking out for me as if I were one of her students.
To my friends and colleagues from the LILaC and RPDMP teams at IRIT, thank
you for all of the lunches, coffee breaks, and evenings we shared together. I only
regret that I was not able to spend more time in Toulouse during my thesis.
To my family, thank you for your continued support over the years, and for flying
all the way across the ocean to attend my defense. It meant so much to me to have
you all there.
Finally, to Laurent, thank you not only for helping me through the stressful mo-
ments, but most of all, for being there to share the happy ones.
v

Contents
1 Introduction 1
2 The Modal Logic K 11
n
2.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Logical Consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Basic Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Basic Reasoning Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Uniform Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7 Relation to First-Order Logic . . . . . . . . . . . . . . . . . . . . . . 41
2.8 Relation to Description Logics. . . . . . . . . . . . . . . . . . . . . . 42
2.8.1 A short introduction to description logics . . . . . . . . . . . 43
2.8.2 The description logic ALC . . . . . . . . . . . . . . . . . . . . 44
2.8.3 The description logic ALE . . . . . . . . . . . . . . . . . . . . 46
3 Prime Implicates and Prime Implicants in K 51
n
3.1 Defining Clauses and Terms in K . . . . . . . . . . . . . . . . . . . 51
n
3.1.1 Impossibility result . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.2 Analysis of candidate definitions . . . . . . . . . . . . . . . . 54
3.1.3 Summary and discussion . . . . . . . . . . . . . . . . . . . . . 66
3.2 Defining Prime Implicates and Prime Implicants in K . . . . . . . . 67
n
3.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.2 Desirable properties . . . . . . . . . . . . . . . . . . . . . . . 68
3.2.3 Analysis of candidate definitions . . . . . . . . . . . . . . . . 69
4 Generating and Recognizing Prime Implicates 79
4.1 Prime Implicate Generation . . . . . . . . . . . . . . . . . . . . . . . 79
4.1.1 Prime implicate generation in propositional logic . . . . . . . 79
vii
viii
4.1.2 The algorithm GenPI . . . . . . . . . . . . . . . . . . . . . . 80
4.1.3 Correctness of GenPI . . . . . . . . . . . . . . . . . . . . . . 82
4.1.4 Bounds on prime implicate size . . . . . . . . . . . . . . . . . 85
4.1.5 Bounds on the number of prime implicates . . . . . . . . . . 92
4.1.6 Improving the efficiency of GenPI . . . . . . . . . . . . . . . 95
4.2 Prime Implicate Recognition . . . . . . . . . . . . . . . . . . . . . . 100
4.2.1 Lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.2 Na¨ıve approach . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.3 Decomposition theorem . . . . . . . . . . . . . . . . . . . . . 101
4.2.4 Prime implicate recognition for propositional clauses . . . . . 106
4.2.5 Prime implicate recognition for 2-formulae . . . . . . . . . . 107
4.2.6 Prime implicate recognition for 3-formulae . . . . . . . . . . 108
4.2.7 The algorithm TestPI . . . . . . . . . . . . . . . . . . . . . . 113
5 Restricted Consequence Finding 119
5.1 New prime implicates . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.1.1 Properties of new prime implicates . . . . . . . . . . . . . . . 120
5.1.2 Generating and recognizing new prime implicates . . . . . . . 122
5.2 Signature-bounded prime implicates . . . . . . . . . . . . . . . . . . 123
5.2.1 Properties of signature-bounded prime implicates . . . . . . . 124
5.2.2 Generating signature-bounded prime implicates . . . . . . . . 128
5.2.3 Recognizing signature-bounded prime implicates . . . . . . . 128
6 Prime Implicate Normal Form 131
6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Definition of Prime Implicate Normal Form . . . . . . . . . . . . . . 132
6.3 Properties of Prime Implicate Normal Form . . . . . . . . . . . . . . 135
6.3.1 Tractable entailment . . . . . . . . . . . . . . . . . . . . . . . 135
6.3.2 Tractable uniform interpolation . . . . . . . . . . . . . . . . . 153
6.3.3 Canonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.4 Computing Prime Implicate Normal Form . . . . . . . . . . . . . . . 170
6.5 Spatial Complexity of Prime Implicate Normal Form . . . . . . . . . 173
6.6 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.6.1 Disjunctive form . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.6.2 Linkless normal form . . . . . . . . . . . . . . . . . . . . . . . 179
7 Conclusion 183
A Complexity Theory 187
0. Contents ix
Bibliography 189
Index 197