A Behavioral Distance for Fuzzy-Transition Systems

TL;DR

This paper introduces a behavioral distance metric for fuzzy-transition systems that quantifies state similarity, enabling a more nuanced analysis than traditional bisimulation and supporting compositional verification.

Contribution

It defines a fixed-point based behavioral distance for fuzzy systems, extending bisimulation to a quantitative framework and demonstrating its non-expansiveness under system composition.

Findings

Behavioral distance equals bisimilarity at zero distance.

States within a threshold are behaviorally equivalent.

Parallel composition and product are non-expansive under this distance.

Abstract

In contrast to the existing approaches to bisimulation for fuzzy systems, we introduce a behavioral distance to measure the behavioral similarity of states in a nondeterministic fuzzy-transition system. This behavioral distance is defined as the greatest fixed point of a suitable monotonic function and provides a quantitative analogue of bisimilarity. The behavioral distance has the important property that two states are at zero distance if and only if they are bisimilar. Moreover, for any given threshold, we find that states with behavioral distances bounded by the threshold are equivalent. In addition, we show that two system combinators---parallel composition and product---are non-expansive with respect to our behavioral distance, which makes compositional verification possible.