A Cyclic Proof System for HFL_?

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.Comment: 27 page

A Probabilistic Higher-Order Fixpoint Logic

We introduce PHFL, a probabilistic extension of higher-order fixpoint logic, which can also be regarded as a higher-order extension of probabilistic temporal logics such as PCTL and the $\mu^p$-calculus. We show that PHFL is strictly more expressive than the $\mu^p$-calculus, and that the PHFL model-checking problem for finite Markov chains is undecidable even for the $\mu$-only, order-1 fragment of PHFL. Furthermore the full PHFL is far more expressive: we give a translation from Lubarsky's $\mu$-arithmetic to PHFL, which implies that PHFL model checking is $\Pi^1_1$-hard and $\Sigma^1_1$-hard. As a positive result, we characterize a decidable fragment of the PHFL model-checking problems using a novel type system

A Probabilistic Higher-order Fixpoint Logic

We introduce PHFL, a probabilistic extension of higher-order fixpoint logic, which can also be regarded as a higher-order extension of probabilistic temporal logics such as PCTL and the $\mu^p$-calculus. We show that PHFL is strictly more expressive than the $\mu^p$-calculus, and that the PHFL model-checking problem for finite Markov chains is undecidable even for the $\mu$-only, order-1 fragment of PHFL. Furthermore the full PHFL is far more expressive: we give a translation from Lubarsky's $\mu$-arithmetic to PHFL, which implies that PHFL model checking is $\Pi^1_1$-hard and $\Sigma^1_1$-hard. As a positive result, we characterize a decidable fragment of the PHFL model-checking problems using a novel type system