A Probabilistic Higher-Order Fixpoint Logic

Authors Yo Mitani, Naoki Kobayashi , Takeshi Tsukada

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Yo Mitani
  • The University of Tokyo, Japan
Naoki Kobayashi
  • The University of Tokyo, Japan
Takeshi Tsukada
  • The University of Tokyo, Japan


We would like to thank anonymous referees for useful comments.

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Yo Mitani, Naoki Kobayashi, and Takeshi Tsukada. A Probabilistic Higher-Order Fixpoint Logic. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 19:1-19:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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 μ^p-calculus. We show that PHFL is strictly more expressive than the μ^p-calculus, and that the PHFL model-checking problem for finite Markov chains is undecidable even for the μ-only, order-1 fragment of PHFL. Furthermore the full PHFL is far more expressive: we give a translation from Lubarsky’s μ-arithmetic to PHFL, which implies that PHFL model checking is Π^1₁-hard and Σ^1₁-hard. As a positive result, we characterize a decidable fragment of the PHFL model-checking problems using a novel type system.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Probabilistic logics
  • higher-order fixpoint logic
  • model checking


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