Encoding First Order Proofs in SMT

We present a method for encoding first order proofs in SMT. Our implementation, called ChewTPTP-SMT, transforms a set of first order clauses into a propositional encoding (modulo theories) of the existence of a rigid first order connection tableau and the satisfiability of unification constraints, which is then fed to Yices. For the unification constraints, terms are represented as recursive datatypes, and unification constraints are equations on terms. The finiteness of the tableau is encoded by linear real arithmetic inequalities. We compare our implementation with our previous implementation ChewTPTP-SAT, encoding rigid connection tableau in SAT, and show that for Horn clauses many fewer propositional clauses are generated by ChewTPTP-SMT, and ChewTPTP-SMT is much faster than ChewTPTP-SAT. This is not the case for our non-Horn clause encoding. We explain this, and we conjecture a rule of thumb on when to use theories in encoding a problem.

[1]  John N. Hooker,et al.  Partial Instantiation Methods for Inference in First-Order Logic , 2002, Journal of Automated Reasoning.

[2]  Peter B. Andrews.  Theorem Proving via General Matings , 1981, JACM.

[3]  Cesare Tinelli,et al.  Solving SAT and SAT Modulo Theories: From an abstract Davis--Putnam--Logemann--Loveland procedure to DPLL(T) , 2006, JACM.

[4]  Reinhold Letz,et al.  Automated Theorem Proving Proof and Model Generation with Disconnection Tableaux , 2001, LPAR.

[5]  J. Bell,et al.  Models and ultraproducts , 1971 .

[6]  Hai Lin,et al.  Encoding First Order Proofs in SAT , 2007, CADE.

[7]  Donald W. Loveland,et al.  A machine program for theorem-proving , 2011, CACM.

[8]  Cesare Tinelli,et al.  DPLL( T): Fast Decision Procedures , 2004, CAV.

[9]  John L. Bell,et al.  Models and Ultraproducts: An Introduction. , 1969 .

[10]  Stéphanie Delaune,et al.  Protocol Verification Via Rigid/Flexible Resolution , 2007, LPAR.

[11]  Richard C. T. Lee,et al.  Symbolic logic and mechanical theorem proving , 1973, Computer science classics.

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

[13]  Reinhold Letz,et al.  Model Elimination and Connection Tableau Procedures , 2001, Handbook of Automated Reasoning.

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

[15]  Inês Lynce,et al.  Local Search for Unsatisfiability , 2006, SAT.

[16]  Alan Robinson,et al.  Handbook of automated reasoning , 2001 .

[17]  Geoff Sutcliffe,et al.  The TPTP Problem Library , 1994, Journal of Automated Reasoning.

[18]  Jean-Paul Billon,et al.  The Disconnection Method - A Confluent Integration of Unification in the Analytic Framework , 1996, TABLEAUX.

[19]  Reiner Hähnle,et al.  Tableaux and Related Methods , 2001, Handbook of Automated Reasoning.

[20]  Jean Goubault,et al.  The Complexity of Resource-Bounded First-Order Classical Logic , 1994 .

[21]  Peter Baumgartner,et al.  FDPLL - A First Order Davis-Putnam-Longeman-Loveland Procedure , 2000, CADE.