Uniform Inductive Reasoning in Transitive Closure Logic via Infinite Descent

Authors Liron Cohen, Reuben N. S. Rowe

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Liron Cohen
  • Dept. of Computer Science, Cornell University, NY, USA
Reuben N. S. Rowe
  • School of Computing, University of Kent, Canterbury, UK

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Liron Cohen and Reuben N. S. Rowe. Uniform Inductive Reasoning in Transitive Closure Logic via Infinite Descent. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof theory
  • Theory of computation → Automated reasoning
  • Induction
  • Transitive Closure
  • Infinitary Proof Systems
  • Cyclic Proof Systems
  • Soundness
  • Completeness
  • Standard Semantics
  • Henkin Semantics


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