Non-cyclic Sorts for First-Order Satisfiability

In this paper we investigate the finite satisfiability problem for first-order logic. We show that the finite satisfiability problem can be represented as a sequence of satisfiability problems in a fragment of many-sorted logic, which we call the non-cyclic fragment. The non-cyclic fragment can be seen as a generalisation of the effectively propositional fragment (EPR) in the many-sorted setting. We show that the non-cyclic fragment is decidable by instantiation-based methods and present a linear time algorithm for checking whether a given clause set is in this fragment. One of the distinctive features of our finite satisfiability translation is that it avoids unnecessary flattening of terms, which can be crucial for efficiency. We implemented our finite model finding translation in iProver and evaluated it over the TPTP library. Using our translation it was possible solve a large class of problems which could not be solved by other systems.

[1]  Konstantin Korovin,et al.  iProver-Eq: An Instantiation-Based Theorem Prover with Equality , 2010, IJCAR.

[2]  Harald Ganzinger,et al.  Integrating Equational Reasoning into Instantiation-Based Theorem Proving , 2004, CSL.

[3]  Andrei Voronkov,et al.  The design and implementation of VAMPIRE , 2002, AI Commun..

[4]  Larry Wos,et al.  What Is Automated Reasoning? , 1987, J. Autom. Reason..

[5]  Koen Claessen,et al.  Sort It Out with Monotonicity - Translating between Many-Sorted and Unsorted First-Order Logic , 2011, CADE.

[6]  Cesare Tinelli,et al.  Quantifier Instantiation Techniques for Finite Model Finding in SMT , 2013, CADE.

[7]  Peter Baumgartner,et al.  Implementing the Model Evolution Calculus , 2006, Int. J. Artif. Intell. Tools.

[8]  Frank Wolter,et al.  Monodic fragments of first-order temporal logics: 2000-2001 A.D , 2001, LPAR.

[9]  Leonardo Mendonça de Moura,et al.  Computation in Real Closed Infinitesimal and Transcendental Extensions of the Rationals , 2013, CADE.

[10]  Nikolaj Bjørner,et al.  Deciding Effectively Propositional Logic Using DPLL and Substitution Sets , 2008, IJCAR.

[11]  Geoff Sutcliffe.  The TPTP Problem Library and Associated Infrastructure , 2009, Journal of Automated Reasoning.

[12]  Koen Claessen.  The Anatomy of Equinox - An Extensible Automated Reasoning Tool for First-Order Logic and Beyond - (Talk Abstract) , 2011, CADE.

[13]  Andrei Voronkov,et al.  Preprocessing techniques for first-order clausification , 2012, 2012 Formal Methods in Computer-Aided Design (FMCAD).

[14]  Rajeev Alur,et al.  A Temporal Logic of Nested Calls and Returns , 2004, TACAS.

[15]  Jacques D. Fleuriot,et al.  IsaPlanner: A Prototype Proof Planner in Isabelle , 2003, CADE.

[16]  K. Claessen,et al.  New Techniques that Improve MACE-style Finite Model Finding , 2007 .

[17]  Stephan Schulz,et al.  System Description: E 0.81 , 2004, IJCAR.

[18]  Pascal Fontaine.  Combinations of Theories and the Bernays-Schönfinkel-Ramsey Class , 2007, VERIFY.

[19]  Alexander Moshe Rabinovich,et al.  Decidable fragments of many-sorted logic , 2010, J. Symb. Comput..

[20]  Alex K. Simpson,et al.  Computational Adequacy in an Elementary Topos , 1998, CSL.

[21]  Harald Ganzinger,et al.  Elimination of Equality via Transformation with Ordering Constraints , 1998, CADE.

[22]  Ian Green,et al.  System Description: Proof Planning in Higher-Order Logic with Lambda-Clam , 1998, CADE.

[23]  Daniel Brand,et al.  Proving Theorems with the Modification Method , 1975, SIAM J. Comput..

[24]  Konstantin Korovin,et al.  iProver - An Instantiation-Based Theorem Prover for First-Order Logic (System Description) , 2008, IJCAR.

[25]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[26]  Cesare Tinelli,et al.  Finite Model Finding in SMT , 2013, CAV.

[27]  Peter Baumgartner,et al.  The Model Evolution Calculus , 2003, CADE.

[28]  Nikolaj Bjørner,et al.  Automated Deduction - CADE-23 - 23rd International Conference on Automated Deduction, Wroclaw, Poland, July 31 - August 5, 2011. Proceedings , 2011, CADE.

[29]  Pascal Fontaine,et al.  Decidability of Invariant Validation for Paramaterized Systems , 2003, TACAS.

[30]  Konstantin Korovin,et al.  Inst-Gen - A Modular Approach to Instantiation-Based Automated Reasoning , 2013, Programming Logics.

[31]  Harald Ganzinger,et al.  New directions in instantiation-based theorem proving , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

[32]  Peter Baumgartner,et al.  Computing finite models by reduction to function-free clause logic , 2009, J. Appl. Log..

[33]  Konstantin Korovin,et al.  Labelled Unit Superposition Calculi for Instantiation-Based Reasoning , 2010, LPAR.

[34]  Pascal Fontaine.  Techniques for verification of concurrent systems with invariants , 2004 .

[35]  Leonardo Mendonça de Moura,et al.  Complete Instantiation for Quantified Formulas in Satisfiabiliby Modulo Theories , 2009, CAV.