Probabilistic Verification Beyond Context-Freeness

Abstract

Probabilistic pushdown automata (recursive state machines) are a widely known model of probabilistic computation associated with many decidable problems concerning termination (time) and lineartime model checking. Higher-order recursion schemes (HORS) are a prominent formalism for the analysis of higher-order computation.Recent studies showed that, for the probabilistic variant of HORS, even the basic problem of determining whether a scheme terminates almost surely is undecidable. Moreover, the undecidability already holds for order-2 schemes (order-1 schemes are known to correspond to pushdown automata).Motivated by these results, we study restricted probabilistic treestack automata (rPTSA), which in the nondeterministic setting are known to characterise a proper extension of context-free languages, namely, the multiple context-free languages. We show that several verification problems, such as almost-sure termination, positive almost-sure termination and 𝜔-regular model checking are decidable for this class.At the level of higher-order recursion schemes, this corresponds to being able to verify a probabilistic version of MAHORS (which are a multiplicative-additive version of higher-order recursion schemes). MAHORS extend order-1 recursion schemes and are incomparable with order-2 schemes. CCS CONCEPTS• Theory of computation → Grammars and context-free languages; Probabilistic computation; Program verification. KEYWORDS probabilistic (affine additive) higher-order recursion schemes; restricted (probabilistic) tree stack automata; computing termination probability; deciding almost sure termination; first-order theory of the reals