A Cyclic Proof System for HFLN
This work addresses the need for semi-automated verification of higher-order programs, though it appears incremental as it extends cyclic proof systems to a specific higher-order logic.
The paper tackles the problem of inductive reasoning in higher-order predicate logic with natural numbers and alternating fixed-points (HFLN) by proposing the first cyclic proof system for such a logic, proving its decidability, soundness, and a restricted form of standard completeness.
A cyclic proof system allows us to perform inductive reasoning without explicit inductions. We propose a cyclic proof system for HFLN, 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 HFLN validity checking.