Improved Second-Order Quantifier Elimination in Modal Logic

This paper introduces improvements for second-order quantifier elimination methods based on Ackermann’s Lemma and investigates their application in modal correspondence theory. In particular, we define refined calculi and procedures for solving the problem of eliminating quantified propositional symbols from modal formulae. We prove correctness results and use the approach to compute first-order frame correspondence properties for modal axioms and modal rules. Our approach can solve two new classes of formulae which have wider scope than existing classes known to be solvable by second-order quantifier elimination methods.

[1]  Harald Ganzinger,et al.  Refutational theorem proving for hierarchic first-order theories , 1994, Applicable Algebra in Engineering, Communication and Computing.

[2]  Valentin Goranko,et al.  Elementary canonical formulae: extending Sahlqvist's theorem , 2006, Ann. Pure Appl. Log..

[3]  Leslie M. Smith,et al.  The renormalization group, the ɛ-expansion and derivation of turbulence models , 1992 .

[4]  Patrick Doherty,et al.  Computing Circumscription Revisited: A Reduction Algorithm , 1997, Journal of Automated Reasoning.

[5]  Valentin Goranko,et al.  Algorithmic Correspondence and Completeness in Modal Logic. II. Polyadic and Hybrid Extensions of the Algorithm SQEMA , 2006, J. Log. Comput..

[6]  Dov M. Gabbay,et al.  Quantifier Elimination in Second-Order Predicate Logic , 1992, KR.

[7]  Valentin Goranko,et al.  Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA , 2006, Log. Methods Comput. Sci..

[8]  Valentin Goranko,et al.  Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA , 2006, Log. Methods Comput. Sci..

[9]  W. Ackermann.  Untersuchungen über das Eliminationsproblem der mathematischen Logik , 1935 .

[10]  Dov M. Gabbay,et al.  Second-Order Quantifier Elimination - Foundations, Computational Aspects and Applications , 2008, Studies in logic : Mathematical logic and foundations.

[11]  Henrik Sahlqvist.  Completeness and Correspondence in the First and Second Order Semantics for Modal Logic , 1975 .