Infinitary and Cyclic Proof Systems for Transitive Closure Logic

TL;DR

This paper introduces an infinitary proof system for transitive closure logic, enabling effective automated inductive reasoning by leveraging cyclic proofs and establishing completeness and semantic soundness.

Contribution

It presents a novel infinitary proof system for transitive closure logic that is complete, subsumes existing systems, and facilitates automation through cyclic proofs.

Findings

The infinitary system is complete for standard semantics.

Cyclic proofs enable effective automation of inductive reasoning.

The system subsumes existing explicit induction proof systems.

Abstract

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.