Game Characterization of Probabilistic Bisimilarity, and Applications to Pushdown Automata

Abstract

We study the bisimilarity problem for probabilistic pushdown automata (pPDA)and subclasses thereof. Our definition of pPDA allows both probabilistic andnon-deterministic branching, generalising the classical notion of pushdownautomata (without epsilon-transitions). We first show a generalcharacterization of probabilistic bisimilarity in terms of two-player games,which naturally reduces checking bisimilarity of probabilistic labelledtransition systems to checking bisimilarity of standard (non-deterministic)labelled transition systems. This reduction can be easily implemented in theframework of pPDA, allowing to use known results for standard(non-probabilistic) PDA and their subclasses. A direct use of the reductionincurs an exponential increase of complexity, which does not matter in derivingdecidability of bisimilarity for pPDA due to the non-elementary complexity ofthe problem. In the cases of probabilistic one-counter automata (pOCA), ofprobabilistic visibly pushdown automata (pvPDA), and of probabilistic basicprocess algebras (i.e., single-state pPDA) we show that an implicit use of thereduction can avoid the complexity increase; we thus get PSPACE, EXPTIME, and2-EXPTIME upper bounds, respectively, like for the respective non-probabilisticversions. The bisimilarity problems for OCA and vPDA are known to have matchinglower bounds (thus being PSPACE-complete and EXPTIME-complete, respectively);we show that these lower bounds also hold for fully probabilistic versions thatdo not use non-determinism