Uniform Inductive Reasoning in Transitive Closure Logic via Infinite Descent

Transitive closure logic is a known extension of first-order logic obtained by introducing a

transitive closure operator. While other extensions of first-order logic with inductive definitions

are a priori parametrized by a set of inductive definitions, the addition of the transitive closure

operator uniformly captures all finitary inductive definitions. In this paper we present an

infinitary proof system for transitive closure logic which 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.