Equality Elimination for the Tableau Method

We apply the equality elimination method to semantic tableaux with equality. The resulting logical system is a combination of a goal-directed tableau calculus with a basic superposition calculus. Unlike most other known methods of adding equality to semantic tableaux, equality elimination does not use simultaneous rigid E-unification or its modifications. For controlling redundancy, we can use powerful strategies of subsumption and simplification. We also make an extensive comparison with related works in the area.

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

[2]  Natarajan Shankar,et al.  Proof Search in the Intuitionistic Sequent Calculus , 1992, CADE.

[3]  Wayne Snyder,et al.  Theorem Proving Using Rigid E-Unification Equational Matings , 1987, LICS.

[4]  Andrei Voronkov,et al.  Equality Elimination for the Inverse Method and Extension Procedures , 1995, IJCAI.

[5]  Andrei Voronkov,et al.  Equality control methods in machine theorem proving , 1986, Cybernetics.

[6]  Bernhard Beckert,et al.  A Completion-Based Method for Mixed Universal and Rigid E-Unification , 1994, CADE.

[7]  Andrei Voronkov,et al.  Theorem Proving in Non-Standard Logics Based on the Inverse Method , 1992, CADE.

[8]  Wayne Snyder,et al.  Basic Paramodulation , 1995, Inf. Comput..

[9]  Wolfgang Bibel,et al.  On Matrices with Connections , 1981, JACM.

[10]  E. Beth,et al.  The Foundations of Mathematics , 1961 .

[11]  Robert Demolombe.  An Efficient Strategy for Non-Horn Deductive Databases , 1991, Theor. Comput. Sci..

[12]  S. Maslov,et al.  Mechanical Proof-Search and the Theory of Logical Deduction in the Ussr , 1971 .

[13]  Andrei Voronkov.  A proof-search method for the first-order logic , 1988, Conference on Computer Logic.

[14]  Uwe Petermann.  A Complete Connection Calculus with Rigid E-Unification , 1994, JELIA.

[15]  Albert Rubio,et al.  Theorem Proving with Ordering and Equality Constrained Clauses , 1995, J. Symb. Comput..

[16]  Andrei Voronkov,et al.  What You Always Wanted to Know About Rigid E-Unification , 1996, JELIA.

[17]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

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

[19]  Andrei Voronkov,et al.  The Undecidability of Simultaneous Rigid E-Unification , 1996, Theor. Comput. Sci..

[20]  J. Goubault.  Rigid E/spl I.oarr/-unifiability is DEXPTIME-complete , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[21]  Andrei Voronkov,et al.  Handling Equality in Logic Programming via Basic Folding , 1996, ELP.

[22]  Jorge Lobo,et al.  Theory of Disjunctive Logic Programs , 1991, Computational Logic - Essays in Honor of Alan Robinson.

[23]  Paliath Narendran,et al.  Theorem proving using equational matings and rigid E-unification , 1992, JACM.

[24]  Hao Wang,et al.  Toward Mechanical Mathematics , 1960, IBM J. Res. Dev..

[25]  Andrei Voronkov,et al.  Simultaneous Regid E-Unification Is Undecidable , 1995, CSL.

[26]  Donald W. Loveland,et al.  Mechanical Theorem-Proving by Model Elimination , 1968, JACM.

[27]  A. Voronkov,et al.  Automatic theorem proving. II , 1987 .

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

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

[30]  Bernhard Beckert,et al.  An Improved Method for Adding Equality to Free Variable Semantic Tableaux , 1992, CADE.

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

[32]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[33]  S. Yu. Maslov,et al.  Invertible Sequential Variant of Constructive Predicate Calculus , 1969 .

[34]  Stig Kanger,et al.  A Simplified Proof Method for Elementary Logic , 1959 .

[35]  Ewing L. Lusk,et al.  Controlling Redundancy in Large Search Spaces: Argonne-Style Theorem Proving Through the Years , 1992, LPAR.

[36]  Andrei Voronkov.  LISS - The Logic Inference Search System , 1990, CADE.

[37]  Andrei Voronkov,et al.  A New Procedural Interpretation of Horn Clauses with Equality , 1995, ICLP.

[38]  Frank van Harmelen,et al.  Extensions to the Rippling-Out Tactic for Guiding Inductive Proofs , 1990, CADE.