A Tableau Calculus for Combining Non-disjoint Theories

The Nelson-Oppen combination method combines ground satisfiability checkers for first-order theories satisfying certain conditions into a single ground satisfiability checker for the union theory. The most significant restriction that the combined theories must satisfy, for the Nelson-Oppen combination method to be applicable, is that they must have disjoint signatures. Unfortunately, this is a very serious restriction since many combination problems concern theories over non-disjoint signatures.In this paper we present a tableau calculus for combining first-order theories over non-disjoint signatures. The calculus generalizes the Nelson-Oppen combination method to formulae with quantifiers and to the union of arbitrary theories over non necessarily disjoint signatures.

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

[2]  Melvin Fitting,et al.  First-Order Logic and Automated Theorem Proving , 1990, Graduate Texts in Computer Science.

[3]  Christophe Ringeissen,et al.  Unification in a Combination of Equational Theories with Shared Constants and its Application to Primal Algebras , 1992, LPAR.

[4]  Cesare Tinelli,et al.  Combining Decision Procedures for Positive Theories Sharing Constructors , 2002, RTA.

[5]  Greg Nelson,et al.  Simplification by Cooperating Decision Procedures , 1979, TOPL.

[6]  M. Fitting.  First-order logic and automated theorem proving (2nd ed.) , 1996 .

[7]  Enno Ohlebusch,et al.  Modular Properties of Composable Term Rewriting Systems , 1995, J. Symb. Comput..

[8]  Christophe Ringeissen,et al.  Combination Techniques for Non-Disjoint Equational Theories , 1994, CADE.

[10]  A. Tiwari,et al.  Decision procedures in automated deduction , 2000 .

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

[12]  R. Smullyan.  First-Order Logic , 1968 .

[13]  T. Käufl,et al.  Cooperation of decision procedures in a tableau-based theorem prover , 1990 .

[14]  Cesare Tinelli,et al.  Cooperation of Background Reasoners in Theory Reasoning by Residue Sharing , 2003, Journal of Automated Reasoning.

[15]  David A. Plaisted,et al.  Eliminating duplication with the hyper-linking strategy , 1992, Journal of Automated Reasoning.

[16]  Michaël Rusinowitch,et al.  Uniform Derivation of Decision Procedures by Superposition , 2001, CSL.

[17]  Bernhard Gramlich,et al.  On Termination and Confluence Properties of Disjoint and Constructor-Sharing Conditional Rewrite Systems , 1996, Theor. Comput. Sci..

[18]  Cesare Tinelli,et al.  Unions of non-disjoint theories and combinations of satisfiability procedures , 2003, Theor. Comput. Sci..

[19]  Christophe Ringeissen,et al.  Cooperation of Decision Procedures for the Satisfiability Problem , 1996, FroCoS.

[20]  M. Rusinowitch,et al.  Uniform Derivation of Decision Procedures by Superposition , 2001, CSL.

[21]  Cesare Tinelli,et al.  A New Correctness Proof of the {Nelson-Oppen} Combination Procedure , 1996, FroCoS.