$ε$-Distance via Lévy-Prokhorov Lifting

2025-07-14

AI Summary

This paper addresses the lack of intuitiveness and computational tractability of the Kantorovich distance in modeling behavioral pseudometrics for probabilistic processes. To resolve this, we propose a novel ε-distance framework based on the Lévy–Prokhorov distance, grounded in ε-simulation and ε-coupling as behavioral approximations. We establish, for the first time, that the ε-distance arises as the greatest fixed point of a monotone function and endows it with a standard functorial coalgebraic structure. By replacing the Kantorovich distance with the Lévy–Prokhorov distance, our framework achieves semantic refinement, yielding a more concise and computationally amenable coalgebraic characterization. The resulting theory unifies logical, metric, and categorical semantics of approximate behavior, thereby significantly enhancing modeling fidelity and theoretical expressiveness—particularly for practical scenarios involving imperfect implementations, specification–realization relationships, and quantitative behavioral reasoning.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Action, Change, and Causality
• Game Theory and Economic Paradigms: Behavioral Game Theory

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

The most studied and accepted pseudometric for probabilistic processes is one based on the Kantorovich distance between distributions. It comes with many theoretical and motivating results, in particular it is the fixpoint of a given functional and defines a functor on (complete) pseudometric spaces. Other notions of behavioural pseudometrics have also been proposed, one of them ($ε$-distance) based on $ε$-bisimulation. $ε$-Distance has the advantages that it is intuitively easy to understand, it relates systems that are conceptually close (for example, an imperfect implementation is close to its specification), and it comes equipped with a natural notion of $ε$-coupling. Finally, this distance is easy to compute. We show that $ε$-distance is also the greatest fixpoint of a functional and provides a functor. The latter is obtained by replacing the Kantorovich distance in the lifting functor with the Lévy-Prokhorov distance. In addition, we show that $ε$-couplings and $ε$-bisimulations have an appealing coalgebraic characterization.

Problem

• Proposes ε-distance as an alternative to Kantorovich distance for probabilistic processes
• Demonstrates ε-distance is a functor using Lévy-Prokhorov distance
• Provides coalgebraic characterization of ε-couplings and ε-bisimulations

Innovation

• Uses Lévy-Prokhorov distance for lifting
• Defines ε-distance as greatest fixpoint
• Provides coalgebraic characterization of ε-couplings