Comparing Labeled Markov Chains: A Cantor-Kantorovich Approach

TL;DR

This paper investigates the Cantor-Kantorovich distance for comparing Labeled Markov Chains, analyzing its properties, computational complexity, and approximation methods to establish a rigorous theoretical foundation.

Contribution

It introduces a new perspective on the CK distance, analyzing its properties, complexity, and providing an approximation scheme, thus advancing understanding of probabilistic model comparison.

Findings

CK distance can be expressed as a discounted sum of finite-horizon Total Variation distances.

Exact computation of CK distance is #P-hard.

A computable approximation scheme for CK distance is also #P-hard.

Abstract

Labeled Markov Chains (or LMCs for short) are useful mathematical objects to model complex probabilistic languages. A central challenge is to compare two LMCs, for example to assess the accuracy of an abstraction or to quantify the effect of model perturbations. In this work, we study the recently introduced Cantor-Kantorovich (or CK) distance. In particular we show that the latter can be framed as a discounted sum of finite-horizon Total Variation distances, making it an instance of discounted linear distance, but arising from the natural Cantor topology. Building on the latter observation, we analyze the properties of the CK distance along three dimensions: computational complexity, continuity properties and approximation. More precisely, we show that the exact computation of the CK distance is #P-hard. We also provide an upper bound on the CK distance as a function of the approximation relation between the two LMCs, and show that a bounded CK distance implies a bounded error between probabilities of finite-horizon traces. Finally, we provide a computable approximation scheme, and show that the latter is also #P-hard. Altogether, our results provide a rigorous theoretical foundation for the CK distance and clarify its relationship with existing distances.