Proof Search in Intuitionistic Logic with Equality, or Back to Simultaneous Rigid E-Unification

We characterize provability in intuitionistic logic with equality in terms of a constraint calculus. This characterization uncovers close connections between provability in intuitionistic logic with equality and solutions to simultaneous rigid E-unification. We show that the problem of existence of a sequent proof with a given skeleton is polynomial-time equivalent to simultaneous rigid E-unifiability. This gives us a proof procedure for intuitionistic logic with equality modulo simultaneous rigid E-unification. We also show that simultaneous rigid E-unifiability is polynomial-time reducible to intuitionistic logic with equality. Thus, any proof procedure for intuitionistic logic with equality can be considered as a procedure for simultaneous rigid E-unifiability. In turn, any procedure for simultaneous rigid E-unifiability gives a procedure for establishing provability in intuitionistic logic with equality.

[1]  David A. Plaisted,et al.  Special Cases and Substitutes for Rigid E-Unification , 2000, Applicable Algebra in Engineering, Communication and Computing.

[2]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

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

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

[5]  Andrei Voronkov.  Proof-Search in Intuitionistic Logic Based on Constraint Satisfaction , 1996, TABLEAUX.

[6]  Andrei Voronkov,et al.  Decidability problems for the prenex fragment of intuitionistic logic , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

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

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

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

[10]  Carla Limongelli,et al.  Design and Implementation of Symbolic Computation Systems , 1996, Lecture Notes in Computer Science.

[11]  Andrei Voronkov,et al.  Monadic Simultaneous Rigid E-Unification and Related Problems , 1997, ICALP.

[12]  A. P. Bel'tyukov.  Decidability of the universal theory of natural numbers with addition and divisibility , 1980 .

[13]  Vladimir Yu. Sazonov,et al.  A Logical Approach to the Problem "P=NP?" , 1980, MFCS.

[14]  Margus Veanes,et al.  On Simultaneous Rigid E-Unification , 1997 .

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

[16]  J. Girard.  Proof Theory and Logical Complexity , 1989 .

[17]  V. P. Orevkov.  Solvable classes of pseudoprenex formulas , 1980 .

[18]  Andrei Voronkov,et al.  Simultaneous rigid E-unification and related algorithmic problems , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[19]  Donald W. Loveland,et al.  Automated theorem proving: a logical basis , 1978, Fundamental studies in computer science.

[20]  G. E. Mints.  Choice of Terms in Quantifier Rules of Constructive Predicate Calculus , 1969 .

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

[22]  Tanel Tammet,et al.  A Resolution Theorem Prover for Intuitonistic Logic , 1996, CADE.

[23]  V. A. Lifshits.  PROBLEM OF DECIDABILITY FOR SOME CONSTRUCTIVE THEORIES OF EQUALITIES , 1969 .

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

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

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

[27]  Margus Veanes,et al.  Some undecidable problems related to the Herbrand theorem , 1997 .

[28]  Margus Veanes.  Uniform representation of recursively enumerable sets with simultaneous rigid E-unification , 1996 .

[29]  James W. Garson,et al.  Quantification in Modal Logic , 1984 .

[30]  Andrei Voronkov,et al.  General Connections via Equality Elimination , 1995, WOCFAI.

[31]  S. Yu Maslov.  An Inverse Method for Establishing Deducibility of Nonprenex Formulas of the Predicate Calculus , 1967 .

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

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

[34]  Natarajan Shankar,et al.  Proof search in first-order linear logic and other cut-free sequent calculi , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[35]  Seif Haridi,et al.  An Intuitionistic Predicate Logic Theorem Prover , 1989, J. Log. Comput..

[36]  R. L. Goodstein,et al.  Provability in logic , 1959 .

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

[38]  Andrei Voronkov.  Proof-Search in Intuitionistic Logic with Equality, or Back to Simultaneous Rigid E-Unification , 2004, Journal of Automated Reasoning.

[39]  L. Wos,et al.  Paramodulation and Theorem-Proving in First-Order Theories with Equality , 1983 .

[40]  Andrei Voronkov,et al.  Equality Elimination for the Tableau Method , 1996, DISCO.