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.

@article{paperbot1259,
  title = {A Probabilistic Higher-order Fixpoint Logic},
  author = {Yo Mitani and Naoki Kobayashi and Takeshi Tsukada},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 17, Issue 4},
  year = {2021},
  doi = {10.46298/lmcs-17(4:15)2021}
}