Infinitary and Cyclic Proof Systems for Transitive Closure Logic

We formulate an infinitary proof system for transitive closure logic, which is the logic obtained by augmenting first-order logics with a transitive closure operator. Our system is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first order logic with inductive definitions, our infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides the basis for an effective system for automating inductive reasoning.