Flow Faster: Efficient Decision Algorithms for Probabilistic Simulations

TL;DR

This paper introduces significantly more efficient algorithms for deciding simulation relations in Markov chains and probabilistic automata, leveraging parametric maximum flow and LP techniques to improve computational complexity.

Contribution

It presents novel algorithms with improved efficiency for strong and weak simulation decisions in Markov chains and probabilistic automata, including polynomial-time solutions.

Findings

Algorithms for Markov chain simulation are drastically improved.

New polynomial-time algorithm for probabilistic automata simulation.

Complexity remains consistent when extending to continuous-time models.

Abstract

Strong and weak simulation relations have been proposed for Markov chains, while strong simulation and strong probabilistic simulation relations have been proposed for probabilistic automata. However, decision algorithms for strong and weak simulation over Markov chains, and for strong simulation over probabilistic automata are not efficient, which makes it as yet unclear whether they can be used as effectively as their non-probabilistic counterparts. This paper presents drastically improved algorithms to decide whether some (discrete- or continuous-time) Markov chain strongly or weakly simulates another, or whether a probabilistic automaton strongly simulates another. The key innovation is the use of parametric maximum flow techniques to amortize computations. We also present a novel algorithm for deciding strong probabilistic simulation preorders on probabilistic automata, which has polynomial complexity via a reduction to an LP problem. When extending the algorithms for probabilistic automata to their continuous-time counterpart, we retain the same complexity for both strong and strong probabilistic simulations.