Game Characterization of Probabilistic Bisimilarity, and Applications to Pushdown Automata

TL;DR

This paper characterizes probabilistic bisimilarity for pushdown automata and subclasses, providing complexity bounds and showing that probabilistic versions retain the complexity of their non-probabilistic counterparts.

Contribution

It introduces a game-based characterization of probabilistic bisimilarity and applies it to derive complexity bounds for various subclasses of probabilistic pushdown automata.

Findings

Bisimilarity reduces to standard PDA bisimilarity via a two-player game.

Decidability of probabilistic bisimilarity is established despite non-elementary complexity.

Complexity bounds for subclasses match those of non-probabilistic versions, e.g., PSPACE, EXPTIME.

Abstract

We study the bisimilarity problem for probabilistic pushdown automata (pPDA) and subclasses thereof. Our definition of pPDA allows both probabilistic and non-deterministic branching, generalising the classical notion of pushdown automata (without epsilon-transitions). We first show a general characterization of probabilistic bisimilarity in terms of two-player games, which naturally reduces checking bisimilarity of probabilistic labelled transition systems to checking bisimilarity of standard (non-deterministic) labelled transition systems. This reduction can be easily implemented in the framework of pPDA, allowing to use known results for standard (non-probabilistic) PDA and their subclasses. A direct use of the reduction incurs an exponential increase of complexity, which does not matter in deriving decidability of bisimilarity for pPDA due to the non-elementary complexity of the problem. In the cases of probabilistic one-counter automata (pOCA), of probabilistic visibly pushdown automata (pvPDA), and of probabilistic basic process algebras (i.e., single-state pPDA) we show that an implicit use of the reduction can avoid the complexity increase; we thus get PSPACE, EXPTIME, and 2-EXPTIME upper bounds, respectively, like for the respective non-probabilistic versions. The bisimilarity problems for OCA and vPDA are known to have matching lower bounds (thus being PSPACE-complete and EXPTIME-complete, respectively); we show that these lower bounds also hold for fully probabilistic versions that do not use non-determinism.