Bisimilarity of Probabilistic Pushdown Automata

TL;DR

This paper investigates the bisimilarity problem for probabilistic pushdown automata and subclasses, providing decidability results, complexity bounds, and reductions to simpler models, advancing understanding of probabilistic automata equivalence.

Contribution

It introduces a reduction from probabilistic to non-deterministic bisimilarity, establishing decidability and tight complexity bounds for various subclasses of probabilistic pushdown automata.

Findings

Decidability of bisimilarity for pPDA

EXPTIME-completeness for probabilistic visibly pushdown automata

PSPACE-completeness for probabilistic one-counter automata

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). Our first contribution is a general construction that reduces checking bisimilarity of probabilistic transition systems to checking bisimilarity of non-deterministic transition systems. This construction directly yields decidability of bisimilarity for pPDA, as well as an elementary upper bound for the bisimilarity problem on the subclass of probabilistic basic process algebras, i.e., single-state pPDA. We further show that, with careful analysis, the general reduction can be used to prove an EXPTIME upper bound for bisimilarity of probabilistic visibly pushdown automata. Here we also provide a matching lower bound, establishing EXPTIME-completeness. Finally we prove that deciding bisimilarity of probabilistic one-counter automata, another subclass of pPDA, is PSPACE-complete. Here we use a more specialised argument to obtain optimal complexity bounds.