Deciding probabilistic simulation between probabilistic pushdown automata and finite-state systems

Abstract

Checking whether a pushdown automaton is simulated – in the sense of a simulation pre-order – by a finite-state automaton is EXPTIME-complete. This paper shows that the same computational complexity is obtained in a probabilistic setting. That is, the problem of deciding whether a probabilistic pushdown automaton (pPDA) is simulated by a finite Markov decision process (MDP) is EXPTIME-complete. The considered pPDA contain both probabilistic and non-deterministic branching. The EXPTIME-membership is shown by combining a partition-refinement algorithm with a tableaux system that is inspired by Stirling's technique for bisimilarity checking of ordinary pushdown automata. The hardness is obtained by exploiting the EXPTIME-hardness for the non-probabilistic case. Moreover, our decision problem is in PTIME when both the number of states of the pPDA and the number of states in the MDP are fixed.