Simultaneous Rigid Sorted Unification for Tableaux

In this paper we integrate a sorted unification calculus into free variable tableau methods for logics with term declarations. The calculus we define is used to close a tableau at once, unifying a set of equations derived from pairs of potentially complementary literals occurring in its branches. Apart from making the deduction system sound and complete, the calculus is terminating and so, it can be used as a decision procedure. In this sense we have separated the complexity of sorts from the undecidability of first order logic.     

Ground and Free-Variable Tableaux for Variants of Quantified Modal Logics

In this paper we study proof procedures for some variants of first-order modal logics, where domains may be either cumulative or freely varying and terms may be either rigid or non-rigid, local or non-local. We define both ground and free variable tableau methods, parametric with respect to the variants of the considered logics. The treatment of each variant is equally simple and is based on the annotation of functional symbols by natural numbers, conveying some semantical information on the worlds where they are meant to be interpreted.This paper is an extended version of a previous work where full proofs were not included. Proofs are in some points rather tricky and may help in understanding the reasons for some details in basic definitions.     

Connection-Driven Inductive Theorem Proving

We present a method for integrating rippling-based rewriting into matrix-based theorem proving as a means for automating inductive specification proofs. The selection of connections in an inductive matrix proof is guided by symmetries between induction hypothesis and induction conclusion. Unification is extended by decision procedures and a rippling/reverse-rippling heuristic. Conditional substitutions are generated whenever a uniform substitution is impossible. We illustrate the integrated method by discussing several inductive proofs for the integer square root problem as well as the algorithms extracted from these proofs.