# Polynomial-time Algorithms for Computing Distances of Fuzzy Transition Systems

## TL;DR

This paper introduces polynomial-time algorithms for computing behavioral distances in fuzzy transition systems, extending existing definitions with discounting and providing efficient methods for both discounted and non-discounted cases.

## Contribution

It presents the first polynomial-time algorithms for calculating behavioral distances in fuzzy transition systems, including a strongly polynomial algorithm for the non-discounted case.

## Key findings

- Polynomial-time algorithms for non-discounted distances
- Polynomial-time algorithms for discounted distances
- Efficient computation of bisimulation as zero-distance cases

## Abstract

Behaviour distances to measure the resemblance of two states in a (nondeterministic) fuzzy transition system have been proposed recently in the literature. Such a distance, defined as a pseudo-ultrametric over the state space of the model, provides a quantitative analogue of bisimilarity. In this paper, we focus on the problem of computing these distances. We first extend the definition of the pseudo-ultrametric by introducing discount such that the discounting factor being equal to 1 captures the original definition. We then provide polynomial-time algorithms to calculate the behavioural distances, in both the non-discounted and the discounted setting. The algorithm is strongly polynomial in the former case. Furthermore, we give a polynomial-time algorithm to compute bisimulation over fuzzy transition systems which captures the distance being equal to 0.