Pure and Applied Fixed-Point Logics Von der Fakult¨at f¨ur Mathematik, Informatik und Naturwissenschaften der Rheinisch-Westf¨alischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Informatiker Stephan Kreutzer aus Aachen, Deutschland Berichter: Universit¨atsprofessor Dr. Erich Gr¨adel Universit¨atsprofessor Dr. Wolfgang Thomas Tag der m¨undlichen Pr¨ufung: 17.12.2002 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verf¨ugbar. 2 3 Abstract Fixed-point logics are logics with an explicit operator for forming fixed points of definable mappings. They are particularly well suited for modelling recur- sion in logical languages and consequently they have found applications in various areas of theoretical computer science such as database theory, finite model theory, and computer-aided verification. The topic of this thesis is the study of fixed-point logics with respect to their expressive power. Of particular interest are logics based on inflationary fixed points and their comparison to least fixed-point logics. The first part focuses on fixed-point extensions of first-order logic. In the main result we show that inflationary and least fixed-point logic – the extensions of first-order logic by least and inflationary fixed points – have the same expressive power on all structures, i.e. LFP = IFP. In the second part of this thesis, we study fixed-point extensions of modal logic. Such logics are widely used in the field of computer-aided verification. Again, the least fixed-point extension of modal logic, the modal µ-calculus, is of particular interest and is among the best studied logics in this area. The main contribution of the second part is the introduction and study of the corresponding inflationary fixed-point logic. Contrary to the case of first-order logic mentioned above, where least and inflationary fixed points lead to equivalent logics, it is shown that in the context of modal logic, inflationary fixed points are far more expressive than least fixed points. On the other hand, they are algorithmically far more complex. Besides the two main results, we study a variety of different fixed-point logics and develop methods to compare their expressive power. Finally, in the third part, we study fixed-point logics as query languages for constraint databases. It is shown that already relatively simple logics such as the transitive closure logic lead to undecidable query languages on constraint databases. Therefore we consider suitable restrictions of fixed- point logics to obtain tractable query languages, i.e. languages with polyno- mial time evaluation. A detailed overview of the results presented in this thesis can be found in the second part of the introduction. Zusammenfassung Die vorliegende Dissertation besch¨aftigt sich mit der Untersuchung von Fix- punktlogiken hinsichtlich ihrer Ausdrucksst¨arke. Der Schwerpunkt liegt da- bei auf inflation¨aren Fixpunktlogiken und ihrer Abgrenzung von Logiken, die auf kleinsten Fixpunkten basieren. Im ersten Teil der Arbeit werden da- zu die seit langem bekannten Fixpunkterweiterungen der Pr¨adikatenlogik untersucht. Das Hauptergebnis ist der Beweis, daß die Logiken LFP und 4 IFP, also die Erweiterung der Pr¨adikatenlogik um kleinste und inflation¨are Fixpunkte, die gleiche Ausdrucksst¨arke haben. Es gilt also LFP = IFP. Im zweiten Teil der Arbeit stehen dann Fixpunkterweiterungen der Mo- dallogik im Vordergrund, wie sie intensiv im Bereich der automatischen Ve- rifikation studiert werden. W¨ahrend der modale µ-Kalk¨ul (Lµ), die Erwei- terung der Modallogik um kleinste Fixpunkte, schon seit Anfang der 80er Jahre eingehend untersucht wird, wird hier zum ersten Mal die entsprechen- de inflation¨are Logik, der modale Iterationskalk¨ul (MIC), betrachtet. Es zeigt sich, daß, im Gegensatz zum Fall der Pr¨adikatenlogik, inflation¨are Fixpunk- te im modallogischen Kontext eine sehr viel gr¨oßere Ausdrucksst¨arke bieten als kleinste. MIC ist also sehr viel ausdrucksst¨arker als Lµ, allerdings im Hinblick auf algorithmische Probleme auch erheblich komplexer. Neben diesen beiden Hauptergebnissen werden in den ersten beiden Tei- len der Arbeit noch weitere Arten von Fixpunktlogiken studiert und Metho- den zum Vergleich ihrer Ausdrucksst¨arke entwickelt. Im dritten und letzten Teil der Dissertation stehen sogenannte constraint Datenbanken im Zentrum der Betrachtungen. Hierbei handelt es sich um ein relativ neues Datenbankmodell, das sich besonders zur Speicherung geome- trischer Daten eignet. Ahnlich¨ wie bei relationalen Datenbanken k¨onnen auch hier Fixpunktlogiken als Grundlage von Abfragesprachen dienen. In Teil III wird gezeigt, daß in diesem Bereich allerdings schon relativ einfache Fixpunktlogiken, wie die transitive Hullenlogik,¨ unentscheidbare Sprachen liefern. Anhand zweier auf kleinsten Fixpunkten basierenden Logiken wird jedoch demonstriert, daß durch geeignete Definition der Logiken auch im constraint Datenbankbereich algorithmisch handhabbare Abfragesprachen mit Hilfe von Fixpunktlogiken definiert werden k¨onnen. Eine ausfuhrlichere¨ Darstellung der in dieser Dissertation pr¨asentierten Ergebnisse findet sich im zweiten Teil der Einleitung. 5 Acknowledgements I have many people to thank and acknowledge. First of all, I want to thank my advisor, Erich Gr¨adel, for all his support and encouragement during the last four years. I am grateful to Anuj Dawar for the very enjoyable collabo- ration on several parts of this thesis and in particular for his patience during the discussions about the equivalence of least and inflationary fixed-point logic. Many thanks also to my colleagues, in particular Achim Blumensath and Dietmar Berwanger. Often enough they had the misfortune of being in their office – Achim mostly before anyone else arrived and Dietmar primarily after everybody else left – when I needed someone to bother with questions. Further, I want to thank the database group in Limburg, in particular Jan Van den Bussche and Floris Geerts for the pleasant discussions we had on constraint databases. Very special thanks go to Jan Van den Bussche for attracting my attention to the expressive power of stratified fixed-point logic and transitive-closure logic on constraint databases. Unknowingly, his ques- tions raised at an AFM-seminar in Aachen gave me the impulse to study fixed-point logics on infinite structures and in this sense made the results reported in the first part of this thesis possible. Finally, I want to thank all the other people who contributed to this thesis in some way, in particular Colin Hirsch and David Richerby for proofreading parts of the manuscript. 6 Contents 1 Introduction 11 2 Preliminaries 27 I Fixed-Point Extensions of First-Order Logic 29 3 Least and Inflationary Fixed-Point Logic 31 3.1 Definitions by Monotone Inductions . 31 3.2 Elementary Inductive Definitions . 33 3.3 Least and Monotone Fixed-Point Logic . 37 3.3.1 Simultaneous Inductions . 39 3.3.2 Alternation and Nesting in Least Fixed-Point Logic . 41 3.3.3 ComparingtheStages . 43 3.4 Inflationary Fixed-Point Logic . 44 3.5 Inductive Fixed Points and Second-Order Logic . 50 4 Other Fixed-Point Logics 53 4.1 Fragments of Least Fixed-Point Logic . 53 4.1.1 Transitive Closure Logics . 53 4.1.2 Existential and Stratified Fixed-Point Logic . 54 4.2 Partial Fixed-Point Logic . 56 5 Descriptive Complexity 59 5.1 Evaluation Complexity of Fixed-Point Formulae . 61 5.2 Logics Capturing Complexity Classes . 62 6 Fixed-Point Logics with Choice 65 6.1 Choice Fixed-Point Logic . 66 6.1.1 Simplifying the First-Order Quantifier Structure . 69 6.1.2 Choice Fixed-Point and Second-Order Logic . 71 6.1.3 Arity-Restricted CFP and Transitive-Closure Logic . 73 6.2 Fixed-Point Logics with Alternating Choice . 75 7 8 CONTENTS 7 A General Semantics for Partial Fixed-Point Logic 77 7.1 An Alternative Semantics for Partial Fixed-Point Logic . 79 7.2 Separating Partial and Inflationary Fixed-Point Logic . 85 7.2.1 Acceptable Structures, Coding, and Diagonalisation . 85 7.2.2 Separating Partial and Inflationary Fixed-Point Logic 87 8 Equivalence of Least and Inflationary Fixed-Point Logic 91 8.1 Equivalence on Finite Structures . 91 8.2 Equivalence in the General Case . 94 8.3 OpenProblems .......................... 100 II Modal Fixed-Point Logics 101 9 Modal Logics and Bisimulation 103 9.1 TransitionSystems . 104 9.2 Bisimulation............................ 106 9.3 ModalLogic............................108 10 The Modal µ-Calculus 111 11 The Modal Iteration Calculus 115 11.1 The Satisfiability Problem for MIC . 118 11.2 TheModelCheckingProblemforMIC . 122 11.3 LanguagesandMIC . 124 11.4 SimpleInductions . 126 11.4.1 Simple vs. Simultaneous Inductions . 126 11.4.2 Infinity Axioms and the Satisfiability Problem . 130 11.5 Comparison of Least and Inflationary Fixed-Point Inductions 138 11.6 Perspectives and Open Problems . 141 12 The Modal Partial Iteration Calculus 143 12.1 Semantics for Modal Partial Fixed-Point Inductions . 143 12.2 Expressive Power and Complexity . 148 13 Labelling Indices on Acyclic Transition Systems 151 13.1 TheRankofTrees . 153 13.2 LabellingSystems . 154 13.3 Labelling Indices of Modal Logics . 157 13.3.1 Modal Logic and the Modal µ-Calculus . 157 13.3.2 The Modal Iteration Calculus . 158 13.3.3 Higher Dimensional µ-Calculus . 161 13.4 Monadic Inflationary Fixed-Point Logic . 162 CONTENTS 9 14 Labelling Indices on Arbitrary Transition Systems 165 14.1 TheRankofArbitraryStructures . 165 14.2 The Labelling Index of Modal Logics . 168 14.3 The Modal µ-Calculus ......................169 14.4 Labelling Index and Complexity . 174 14.5 The Trace-Equivalence Problem . 177 III Constraint Databases 181 15 Constraint Databases 183 15.1 The Constraint Database Model . 183 15.2 ConstraintQueries . 186 16 The Linear Constraint Database Model 191 16.1 Linear Constraint Databases . 192 16.2 Semi-LinearSets . 193 16.2.1 Structural Properties . 193 16.2.2 Arrangements . 194 16.3 First-Order Logic as Linear Constraint Query Language . 197 16.3.1 Extending First-Order Logic by Convex Hulls . 199 16.3.2 Extending First-Order Logic by Multiplication .