A Probabilistic Higher-order Fixpoint Logic

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