On the Complexity of the Equivalence Problem for Probabilistic Automata

TL;DR

This paper explores the complexity of the equivalence problem for probabilistic automata, introducing efficient randomized and deterministic algorithms for various cases, and establishing complexity equivalences with arithmetic circuit problems.

Contribution

It presents new algorithms for probabilistic automata equivalence, including randomized NC procedures and polynomial-time algorithms for fixed reward counters, and relates automata equivalence to circuit identity testing.

Findings

Randomized NC procedure for automata equivalence with counterexample output

Deterministic polynomial-time algorithm for automata with fixed reward counters

Equivalence problem for probabilistic visibly pushdown automata is logspace equivalent to circuit identity testing

Abstract

Checking two probabilistic automata for equivalence has been shown to be a key problem for efficiently establishing various behavioural and anonymity properties of probabilistic systems. In recent experiments a randomised equivalence test based on polynomial identity testing outperformed deterministic algorithms. In this paper we show that polynomial identity testing yields efficient algorithms for various generalisations of the equivalence problem. First, we provide a randomized NC procedure that also outputs a counterexample trace in case of inequivalence. Second, we show how to check for equivalence two probabilistic automata with (cumulative) rewards. Our algorithm runs in deterministic polynomial time, if the number of reward counters is fixed. Finally we show that the equivalence problem for probabilistic visibly pushdown automata is logspace equivalent to the Arithmetic Circuit Identity Testing problem, which is to decide whether a polynomial represented by an arithmetic circuit is identically zero.