Behavioural Conformances based on Lax Couplings

TL;DR

This paper introduces a flexible approach to defining behavioural distances in coalgebraic systems using lax couplings, enabling broader applicability across various system types and addressing limitations of traditional coupling methods.

Contribution

It proposes a novel lax coupling framework for behavioural distances, extending the applicability of Wasserstein-like metrics to more system types and scenarios.

Findings

Broadened the class of behavioural conformances expressible via couplings.

Applied lax couplings to modal transition systems and metric labelled Markov chains.

Enhanced the robustness of behavioural distance definitions in coalgebraic models.

Abstract

Behavioural conformances -- e.g. behavioural equivalences, distances, preorders -- on a wide range of system types (non-deterministic, probabilistic, weighted etc.) can be dealt with uniformly in the paradigm of universal coalgebra. One of the most commonly used constructions for defining behavioural distances on coalgebras arises as a generalization of the well-known Wasserstein metric. In this construction, couplings of probability distributions are replaced with couplings of more general objects, depending on the functor describing the system type. In many cases, however, the set of couplings of two functor elements is empty, which causes such elements to have infinite distance even in situations where this is not desirable. We propose an approach to defining behavioural distances and preorders based on a more liberal notion of coupling where the coupled elements are matched laxly rather than on-the-nose. We thereby substantially broaden the range of behavioural conformances expressible in terms of couplings, covering, e.g., refinement of modal transition systems and behavioural distance on metric labelled Markov chains.