Non-well-founded Proof Theory of Transitive Closure Logic

Liron Cohen, Reuben N.S. Rowe

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


Supporting inductive reasoning is an essential component is any framework of use in computer science. To do so, the logical framework must extend that of first-order logic. Transitive closure logic is a known extension of first-order logic that is particularly straightforward to automate. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of a single transitive closure operator has the advantage of uniformly capturing all finitary inductive definitions. To further improve the reasoning techniques for transitive closure logic, we here present an infinitary proof system for it, which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that the 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.

Original languageEnglish
Article number31
JournalACM Transactions on Computational Logic
Issue number4
StatePublished - 1 Oct 2020


  • Henkin semantics
  • Induction
  • completeness
  • cyclic proof systems
  • infinitary proof systems
  • soundness
  • standard semantics
  • transitive closure

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science
  • Logic
  • Computational Mathematics


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