On Variable-inactivity and Polynomial $\mathcal T$-Satisfiability Procedures

Verification problems require to reason in theories of data structures and fragments of arithmetic. Thus, decision procedures for such theories are needed, to be embedded in, or interfaced with, proof assistants or software model checkers. Such decision procedures ought to be sound and complete, to avoid false negatives and false positives, efficient, to handle large problems, and easy to combine, because most problems involve multiple theories. The rewrite-based approach to decision procedures aims at addressing these sometimes conflicting issues in a uniform way, by harnessing the power of general first-order theorem proving. In this article, we generalize the rewrite-based approach from deciding the satisfiability of sets of ground literals to deciding that of arbitrary ground formulæ in the theory. Next, we present polynomial rewrite-based satisfiability procedures for the theories of records with extensionality and integer offsets. The generalization of the rewrite-based approach to arbitrary ground formulæ and the polynomial satisfiability procedure for the theory of records with extensionality use the same key property—termed variable-inactivity—that allows one to combine theories in a simple way in the rewrite-based approach.