Computing Distances between Probabilistic Automata

TL;DR

This paper introduces epsilon-relaxed simulation and bisimulation for Probabilistic Automata, providing logical characterisations, efficient computation methods, and new distance metrics that account for approximation errors.

Contribution

It defines epsilon-approximate notions of simulation and bisimulation, offers logical characterisations, and develops efficient algorithms for computing distances between states in probabilistic automata.

Findings

Efficient PTIME algorithms for computing epsilon-bisimulation relations.

Introduction of non-discounted and discounted distance metrics for PAs.

Logical characterisations using modified modal logics.

Abstract

We present relaxed notions of simulation and bisimulation on Probabilistic Automata (PA), that allow some error epsilon. When epsilon is zero we retrieve the usual notions of bisimulation and simulation on PAs. We give logical characterisations of these notions by choosing suitable logics which differ from the elementary ones, L with negation and L without negation, by the modal operator. Using flow networks, we show how to compute the relations in PTIME. This allows the definition of an efficiently computable non-discounted distance between the states of a PA. A natural modification of this distance is introduced, to obtain a discounted distance, which weakens the influence of long term transitions. We compare our notions of distance to others previously defined and illustrate our approach on various examples. We also show that our distance is not expansive with respect to process algebra operators. Although L without negation is a suitable logic to characterise epsilon-(bi)simulation on deterministic PAs, it is not for general PAs; interestingly, we prove that it does characterise weaker notions, called a priori epsilon-(bi)simulation, which we prove to be NP-difficult to decide.