Some techniques for reasoning automatically on co-inductive data structures

Abstract

Some techniques are proposed for reasoning on co-inductive structures. First, we devise a sound axiomatization of (conservative extensions) of such structures, thus reducing the problem of checking whether a formula admits a co-inductive model to a first-order satisfiability test. We devise a class of structures, called regularly co-inductive, for which the axiomatization is complete (for other co-inductive structures, the proposed axiomatization is sound, but not complete). Then, we propose proof calculi for reasoning on such structures. We first show that some of the axioms mentioned above can be omitted if the inference rules are able to handle rational terms. Furthermore, under some conditions, some other axioms may be replaced by an additional inference rule that computes the solutions of fixpoint equations. Finally, we show that a stronger completeness result can be established under some additional conditions on the signature.