Efficiently Checking Propositional Resolution Proofs in Isabelle / HOL

This paper describes the integration of zChaff and MiniSat, currently two leading SAT solvers, with Isabelle/HOL. Both SAT solvers generate resolution-style proofs for (instances of) propositional tautologies. These proofs are verified by the theorem prover. The presented approach significantly improves Isabelle’s performance on propositional problems, and exhibits counterexamples for unprovable conjectures. It is shown that an LCF-style theorem prover can serve as a viable proof checker even for large SAT problems. An efficient representation of the propositional problem in the theorem prover turns out to be crucial; several possible solutions are discussed.

[1]  G. S. Tseitin.  On the Complexity of Derivation in Propositional Calculus , 1983 .

[2]  Graham Wrightson,et al.  Automation of reasoning--classical papers on computational logic , 2012 .

[3]  Thomas Kropf,et al.  Integrating A First-order Automatic prover In The HOL Environment , 1991, 1991., International Workshop on the HOL Theorem Proving System and Its Applications.

[4]  Natarajan Shankar,et al.  PVS: A Prototype Verification System , 1992, CADE.

[5]  M. Gordon,et al.  Introduction to HOL: a theorem proving environment for higher order logic , 1993 .

[6]  Lawrence Charles Paulson,et al.  Isabelle: A Generic Theorem Prover , 1994 .

[7]  Christoph Weidenbach,et al.  On Generating Small Clause Normal Forms , 1998, CADE.

[8]  Yves Bertot,et al.  Theorem Proving in Higher Order Logics: 12th International Conference, TPHOLs'99, Nice, France, September 14-17, 1999, Proceedings , 1999 .

[9]  Joe Hurd.  Integrating Gandalf and HOL , 1999, TPHOLs.

[10]  Michael J. C. Gordon,et al.  From LCF to HOL: a short history , 2000, Proof, Language, and Interaction.

[11]  Thomas Stützle,et al.  SATLIB: An Online Resource for Research on SAT , 2000 .

[12]  Andreas Meier.  System description : TRAMP transformation of machine-found proofs into natural deduction proofs at the assertion level , 2000 .

[13]  M. Moskewicz,et al.  Chaff: engineering an efficient SAT solver , 2001, Proceedings of the 38th Design Automation Conference (IEEE Cat. No.01CH37232).

[14]  Natarajan Shankar,et al.  Using Decision Procedures with a Higher-Order Logic , 2001, TPHOLs.

[15]  Lawrence Charles Paulson,et al.  Isabelle/HOL: A Proof Assistant for Higher-Order Logic , 2002 .

[16]  Joe Hurd.  An LCF-Style Interface between HOL and First-Order Logic , 2002, CADE.

[17]  Piergiorgio Bertoli,et al.  A SAT Based Approach for Solving Formulas over Boolean and Linear Mathematical Propositions , 2002, CADE.

[18]  Ofer Strichman.  On Solving Presburger and Linear Arithmetic with SAT , 2002, FMCAD.

[19]  Armando Tacchella,et al.  Theory and Applications of Satisfiability Testing , 2003, Lecture Notes in Computer Science.

[20]  Niklas Sörensson,et al.  An Extensible SAT-solver , 2003, SAT.

[21]  Jia Meng.  Integration of Interactive and Automatic Provers , 2003 .

[22]  Sergey Berezin,et al.  A proof-producing boolean search engine , 2003 .

[23]  Sharad Malik,et al.  Validating SAT solvers using an independent resolution-based checker: practical implementations and other applications , 2003, 2003 Design, Automation and Test in Europe Conference and Exhibition.

[24]  Lawrence C. Paulson,et al.  Experiments on Supporting Interactive Proof Using Resolution , 2004, IJCAR.

[25]  Sergey Berezin,et al.  CVC Lite: A New Implementation of the Cooperating Validity Checker Category B , 2004, CAV.

[26]  Volker Sorge,et al.  Applying SAT Solving in Classification of Finite Algebras , 2005, Journal of Automated Reasoning.

[27]  Tjark Weber,et al.  Using a SAT Solver as a Fast Decision Procedure for Propositional Logic in an LCF-style Theorem Prover? , 2005 .

[28]  Warren A. Hunt,et al.  A SAT-Based Decision Procedure for the Subclass of Unrollable List Formulas in ACL2 (SULFA) , 2006, IJCAR.

[29]  Lawrence C. Paulson,et al.  Translating higher-order problems to first-order clauses , 2006 .

[30]  Hans de Nivelle,et al.  Geometric Resolution: A Proof Procedure Based on Finite Model Search , 2006, IJCAR.

[31]  Tjark Weber,et al.  Integrating a SAT Solver with an LCF-style Theorem Prover , 2005, PDPAR@CAV.

[32]  Stephan Merz,et al.  Practical Proof Reconstruction for First-order Logic and Set-Theoretical Constructions , 2006 .

[33]  Stephan Merz,et al.  Expressiveness + Automation + Soundness: Towards Combining SMT Solvers and Interactive Proof Assistants , 2006, TACAS.

[34]  Hasan Amjad.  Compressing Propositional Refutations , 2007, Electron. Notes Theor. Comput. Sci..