A Cyclic Proof System for HFL_ℕ

Authors Mayuko Kori , Takeshi Tsukada , Naoki Kobayashi

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Author Details

Mayuko Kori
  • Department of Informatics, The Graduate University for Advanced Studies (SOKENDAI), Hayama, Japan
  • National Institute of Informatics, Tokyo, Japan
Takeshi Tsukada
  • Graduate School of Science, Chiba University, Japan
Naoki Kobayashi
  • The University of Tokyo, Japan


We would like to thank anonymous referees for useful comments.

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Mayuko Kori, Takeshi Tsukada, and Naoki Kobayashi. A Cyclic Proof System for HFL_ℕ. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 29:1-29:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


A cyclic proof system allows us to perform inductive reasoning without explicit inductions. We propose a cyclic proof system for HFL_ℕ, which is a higher-order predicate logic with natural numbers and alternating fixed-points. Ours is the first cyclic proof system for a higher-order logic, to our knowledge. Due to the presence of higher-order predicates and alternating fixed-points, our cyclic proof system requires a more delicate global condition on cyclic proofs than the original system of Brotherston and Simpson. We prove the decidability of checking the global condition and soundness of this system, and also prove a restricted form of standard completeness for an infinitary variant of our cyclic proof system. A potential application of our cyclic proof system is semi-automated verification of higher-order programs, based on Kobayashi et al.’s recent work on reductions from program verification to HFL_ℕ validity checking.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof theory
  • Cyclic proof
  • higher-order logic
  • fixed-point logic
  • sequent calculus


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