# Uniform Inductive Reasoning in Transitive Closure Logic via Infinite Descent

Cohen, Liron, Rowe, Reuben (2018) Uniform Inductive Reasoning in Transitive Closure Logic via Infinite Descent. In: Leibniz International Proceedings in Informatics. Proceedings of the 27th EACSL Annual Conference on Computer Science Logic, CSL 2018. . LIPICS (doi:10.4230/LIPIcs.CSL.2018.16)

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http://dx.doi.org/10.4230/LIPIcs.CSL.2018.16

## 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.

Item Type: Conference or workshop item (Proceeding) 10.4230/LIPIcs.CSL.2018.16 Induction, Transitive Closure, Infinitary Proof Systems, Cyclic Proof Systems, Soundness, Completeness, Standard Semantics, Henkin Semantics Q Science > QA Mathematics (inc Computing science) > QA 76 Software, computer programming, Faculties > Sciences > School of Computing > Programming Languages and Systems Group Reuben Rowe 29 Jun 2018 13:45 UTC 09 Jul 2019 11:24 UTC https://kar.kent.ac.uk/id/eprint/67460 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-4271-9078