Zero-Reachability in Probabilistic Multi-Counter Automata

TL;DR

This paper investigates zero-reachability problems in probabilistic multi-counter automata, identifying decidability boundaries and providing algorithms for qualitative and quantitative analysis, with implications for stochastic Petri nets.

Contribution

It establishes decidability results and approximation algorithms for zero-reachability in probabilistic multi-counter automata, including new techniques for complex cases.

Findings

Qualitative zero-reachability is decidable in polynomial time for one case.

Quantitative approximation is feasible with doubly exponential complexity in counters.

New proof techniques challenge classical results in ergodicity of stochastic Petri nets.

Abstract

We study the qualitative and quantitative zero-reachability problem in probabilistic multi-counter systems. We identify the undecidable variants of the problems, and then we concentrate on the remaining two cases. In the first case, when we are interested in the probability of all runs that visit zero in some counter, we show that the qualitative zero-reachability is decidable in time which is polynomial in the size of a given pMC and doubly exponential in the number of counters. Further, we show that the probability of all zero-reaching runs can be effectively approximated up to an arbitrarily small given error epsilon > 0 in time which is polynomial in log(epsilon), exponential in the size of a given pMC, and doubly exponential in the number of counters. In the second case, we are interested in the probability of all runs that visit zero in some counter different from the last counter. Here we show that the qualitative zero-reachability is decidable and SquareRootSum-hard, and the probability of all zero-reaching runs can be effectively approximated up to an arbitrarily small given error epsilon > 0 (these result applies to pMC satisfying a suitable technical condition that can be verified in polynomial time). The proof techniques invented in the second case allow to construct counterexamples for some classical results about ergodicity in stochastic Petri nets.