On-the-Fly Computation of Bisimilarity Distances

TL;DR

This paper introduces a polynomial-time linear program for computing bisimilarity distances in continuous-time Markov chains and proposes an efficient on-the-fly algorithm that outperforms existing methods in practice.

Contribution

It presents a novel polynomial-sized linear program for bisimilarity distance computation and an on-the-fly algorithm that efficiently computes distances without exhaustive state space exploration.

Findings

Linear program can compute distances in polynomial time

On-the-fly algorithm significantly outperforms iterative and LP methods

Algorithm reduces state space exploration by refining approximations

Abstract

We propose a distance between continuous-time Markov chains (CTMCs) and study the problem of computing it by comparing three different algorithmic methodologies: iterative, linear program, and on-the-fly. In a work presented at FoSSaCS'12, Chen et al. characterized the bisimilarity distance of Desharnais et al. between discrete-time Markov chains as an optimal solution of a linear program that can be solved by using the ellipsoid method. Inspired by their result, we propose a novel linear program characterization to compute the distance in the continuous-time setting. Differently from previous proposals, ours has a number of constraints that is bounded by a polynomial in the size of the CTMC. This, in particular, proves that the distance we propose can be computed in polynomial time. Despite its theoretical importance, the proposed linear program characterization turns out to be inefficient in practice. Nevertheless, driven by the encouraging results of our previous work presented at TACAS'13, we propose an efficient on-the-fly algorithm, which, unlike the other mentioned solutions, computes the distances between two given states avoiding an exhaustive exploration of the state space. This technique works by successively refining over-approximations of the target distances using a greedy strategy, which ensures that the state space is further explored only when the current approximations are improved. Tests performed on a consistent set of (pseudo)randomly generated CTMCs show that our algorithm improves, on average, the efficiency of the corresponding iterative and linear program methods with orders of magnitude.