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f f f f f f f f f f f f f f f f f f f f SAT-based Finite Model Generation for Higher-Order Logic Tjark Weber Lehrstuhl fur¨ Software & Systems Engineering Institut fur¨ Informatik Technische Universit¨at Munchen¨ Lehrstuhl fur¨ Software & Systems Engineering Institut fur¨ Informatik Technische Universit¨at Munchen¨ SAT-based Finite Model Generation for Higher-Order Logic Tjark Weber Vollst¨andiger Abdruck der von der Fakult¨at fur¨ Informatik der Technischen Universit¨at Munchen¨ zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. F. J. Esparza Estaun Prufer¨ der Dissertation: 1. Univ.-Prof. T. Nipkow, Ph. D. 2. Univ.-Prof. Dr. H. Veith, Technische Universit¨at Darmstadt Die Dissertation wurde am 30. April 2008 bei der Technischen Universit¨at Munchen¨ eingereicht und durch die Fakult¨at fur¨ Informatik am 30. September 2008 angenommen. Kurzfassung Diese Arbeit pr¨asentiert zwei Erweiterungen des Theorembeweisers Isabelle/HOL, einem auf Logik h¨oherer Stufe basierenden System. Der Hauptbeitrag ist ein Modellgenerator fur¨ h¨oherstufige Logik, der seine Eingabeformel in Aussagenlogik ubersetzt,¨ so dass ein herk¨ommlicher SAT-Solver fur¨ die eigentliche Modellsuche verwendet werden kann. Die Korrektheit der Ubersetzung¨ wird gezeigt. Der Modellgenerator ist in das Isabelle-System integriert worden, unterstutzt¨ verschiedene definitorische Mechanismen, wie sie in Isabelle/HOL zur Verfugung¨ stehen, und ist auf mehrere Fallstudien angewandt worden. Daruber¨ hinaus sind SAT-Solver beweisgenerierend mit Isabelle integriert worden: Aussagen- logische Tautologien k¨onnen von einem SAT-Solver bewiesen werden, und der von dem Solver gefundene Resolutionsbeweis wird von Isabelle verifiziert. Eine gunstige¨ Repr¨asentation des Problems erlaubt die Verifikation von Beweisen mit mehreren Millionen Resolutionsschritten. i ii Abstract This thesis presents two extensions to the theorem prover Isabelle/HOL, a logical framework based on higher-order logic. The main contribution is a model generator for higher-order logic that proceeds by translating the input formula to propositional logic, so that a standard SAT solver can be employed for the actual model search. The translation is proved correct. The model generator has been integrated with the Isabelle system, extended to support some of the definitional mechanisms provided by Isabelle/HOL, and applied to various case studies. Moreover, SAT solvers have been integrated with Isabelle in a proof-producing fashion: propo- sitional tautologies can be proved by a SAT solver, and the resolution proof found by the solver is verified by Isabelle. An adequate representation of the problem allows to verify proofs with millions of resolution steps. iii iv Acknowledgements I would like to thank Tobias Nipkow, my supervisor, for his invaluable support, advice, and patience. The current and former members of his research group have contributed to a fruitful work environment: Clemens Ballarin, Gertrud Bauer, Stefan Berghofer, Amine Chaieb, Florian Haftmann, Gerwin Klein, Farhad Mehta, Steven Obua, Norbert Schirmer, Sebastian Skalberg, Martin Strecker, Markus Wenzel, and Martin Wildmoser. I am particularly indebted to Lars Ebert and Marco Helmers for proof-reading my thesis, to Michael Fortleff for providing shelter, and to Nora for distracting me. I thank Helmut Veith for acting as a referee. My thanks also go to Helmut Schwichtenberg, speaker of the \Graduiertenkolleg Logik in der Informatik", which provided both financial and intellectual support, and to the other members of the Graduiertenkolleg|in particular to Andreas Abel, Klaus Aehlig, Steffen Jost, and Ralph Matthes for inspiring discussions and more. Many other people have influenced this thesis in one way or another. Among them are Reinhold Letz, Gernot Stenz, Jan J¨urjens,and Manfred Broy from Technische Universit¨atM¨unchen, Martin Hofmann from Ludwig-Maximilians-Universit¨atM¨unchen, Hasan Amjad and Larry Paulson from the University of Cambridge, Sharad Malik and Zhaohui Fu from Princeton Uni- versity, Pascal Fontaine, Stephan Merz, and Alwen Tiu from INRIA Lorraine, John Harrison from Intel Corporation, John Matthews from Galois, Inc., David Aspinall from the Univer- sity of Edinburgh, Annabelle McIver from Macquarie University, and Moshe Vardi from Rice University. Finally I would like to thank everyone who has played a part in making the past years in Munich a pleasant and successful time for me. Friends and family have been a steady source of encouragement. v vi Contents 1 Introduction 1 1.1 Motivation . .1 1.2 Contributions . .2 1.3 Related Work . .3 1.4 Isabelle . .4 1.5 Overview . .6 2 Finite Model Generation 7 2.1 Introduction . .7 2.2 Higher-Order Logic . .8 2.2.1 Types . .8 2.2.2 Terms . 11 2.2.3 Satisfiability . 14 2.3 Translation to Propositional Logic . 15 2.3.1 Propositional Logic . 15 2.3.2 Interpretation of Types . 16 2.3.3 Interpretation of Terms . 22 2.3.4 Examples . 40 2.4 Model Generation . 41 2.4.1 Finding a Satisfying Assignment . 41 2.4.2 Type Environments and Type Models . 42 2.4.3 The Algorithm . 43 2.4.4 Building the HOL Model . 45 2.5 Conclusion . 45 3 Extensions and Optimizations 47 3.1 Introduction . 47 vii viii CONTENTS 3.2 Optimizations . 48 3.3 Isabelle's Meta-Logic . 52 3.4 Type and Constant Definitions, Overloading . 52 3.4.1 Type Definitions . 52 3.4.2 Constant Definitions and Overloading . 53 3.4.3 Definite Description and Hilbert's Choice . 54 3.5 Axiomatic Type Classes . 55 3.6 Datatypes and Recursive Functions . 56 3.6.1 Non-Recursive Datatypes . 57 3.6.2 Recursive Datatypes . 58 3.6.3 Recursive Functions . 64 3.7 Sets and Records . 67 3.8 HOLCF . 68 3.9 Conclusion . 69 4 Case Studies 71 4.1 Introduction . 71 4.2 The RSA-PSS Security Protocol . 72 4.2.1 Abstract Protocol Formalization . 72 4.2.2 Avoiding Confusion . 75 4.3 Probabilistic Programs . 77 4.3.1 The Probabilistic Model LS ......................... 78 4.3.2 The Abstract Model KS ........................... 86 4.3.3 Mechanization of Counterexample Search . 93 4.4 A SAT-based Sudoku Solver . 94 4.4.1 Implementation in Isabelle/HOL . 95 4.4.2 Translation to Propositional Logic . 96 4.5 Conclusion . 98 5 Integration of Proof-producing SAT Solvers 99 5.1 Introduction . 99 5.2 Related Work . 100 5.3 System Description . 101 5.3.1 Preprocessing . 101 5.3.2 Proof Reconstruction . 103 CONTENTS ix 5.3.3 Clause Representations . 110 5.4 Evaluation . 112 5.5 Persistent Theorems . 115 5.6 Conclusion . 117 6 Conclusion 119 6.1 Summary . 119 6.2 Future Work . 120 x CONTENTS All men by nature desire to know. Aristotle, 350 B.C. Chapter 1 Introduction 1.1 Motivation Interactive theorem proving is about identifying false conjectures almost as much as it is about finding proofs for theorems. False conjectures may occur for a number of reasons. They can be caused by a trivial typographical error in a formalization, but also by a possibly subtle flaw in the design of a complex system. During interactive proof, false conjectures may be produced by proof strategies that can potentially render a goal unprovable, e.g. by dropping an essential premise. False conjectures can be seen as intermediate stages of a validation. A true conjecture is often obtained from a sequence of false conjectures and their disproofs. Once identified, it is usually clear how a false conjecture needs to be fixed. Determining that a conjecture is false can however be a time-consuming process. Interactive theorem provers have been enhanced with numerous automatic proof procedures for various application domains over the past years, but failure of an automatic proof attempt is far from sufficient to conclude that a statement is false. It may just be that an additional lemma needs to be proved, or that an induction hypothesis needs to be generalized. The user typically gets little useful information about these reasons, and it may take hours (sometimes days even) of failed proof attempts before he realizes that the conjecture he is trying to prove was false to begin with. In such cases an automatic tool that can refute non-theorems could be highly useful. One particularly illuminative way to identify a false conjecture is by providing a counterexam- ple. More specifically, when the conjecture is a logical formula, a (counter-)example is given by a (counter-)model. A finite model generator is an automatic tool that attempts to find a finite model which satisfies (or, equivalently, refutes) a given formula. Finite model genera- tion is an instance of the constraint satisfaction problem, with|apart from the generation of counterexamples|numerous applications in artificial intelligence [92], operations research [34], and finite mathematics [102, 175]. 1 2 CHAPTER 1. INTRODUCTION This thesis presents (in Chapters 2 and 3) a finite model generator for higher-order logic. A translation to propositional logic is employed and proved correct, so that a standard SAT solver can be used to search for a satisfying model. Such a model constitutes a counterexample to a false conjecture, which can be displayed to the user together with a suitable diagnostic message. This thesis is among the first to consider finite model generation in the context of interactive theorem proving; see Section 1.3 for a discussion of related work. Interactive theorem provers can benefit from efficient SAT solvers in a second way. Many problems that occur in hardware or software verification can be encoded in propositional logic. The performance of an interactive theorem prover on propositional formulae can therefore be of great practical significance, and the use of SAT solvers for model generation naturally raises the question if they can be used for finding proofs as well. In Chapter 5 we give a positive answer by considering proof-generating SAT solvers, which output either a satisfying assignment for a propositional formula, or a resolution-style proof of unsatisfiability.
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