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Comparing Labeled Markov Chains: A Cantor-Kantorovich Approach

Adrien Banse, Alessandro Abate, Raphaël M. Jungers

TL;DR

The paper addresses how to compare Labeled Markov Chains (LMCs) by introducing the Cantor-Kantorovich (CK) distance, which arises from the Cantor topology on $A^\omega$ and can be written as a discounted sum of finite-horizon Total Variation distances: $d(\Sigma_1,\Sigma_2) = \sum_{i=1}^\infty \frac{m-1}{m^i} \mathrm{TV}(p_1^i, p_2^i)$. It establishes that exact CK computation is $\#\mathrm{P}$-hard, proves strong continuity properties and finite-horizon guarantees, and proposes a computable approximation scheme that is itself $\#\mathrm{P}$-hard to compute. The CK distance provides a rigorous, trace-based notion that captures transient behavior and aligns with Cantor-topology intuition, offering practical relevance for abstraction and perturbation analysis in probabilistic systems. Open questions remain on whether CK distance is computable in general and whether there exist efficient polynomial-time approximation methods.

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.

Comparing Labeled Markov Chains: A Cantor-Kantorovich Approach

TL;DR

The paper addresses how to compare Labeled Markov Chains (LMCs) by introducing the Cantor-Kantorovich (CK) distance, which arises from the Cantor topology on and can be written as a discounted sum of finite-horizon Total Variation distances: . It establishes that exact CK computation is -hard, proves strong continuity properties and finite-horizon guarantees, and proposes a computable approximation scheme that is itself -hard to compute. The CK distance provides a rigorous, trace-based notion that captures transient behavior and aligns with Cantor-topology intuition, offering practical relevance for abstraction and perturbation analysis in probabilistic systems. Open questions remain on whether CK distance is computable in general and whether there exist efficient polynomial-time approximation methods.

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.

Paper Structure

This paper contains 22 sections, 15 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline
  4. Notations
  5. Preliminaries
  6. Distances between Probability Distributions
  7. Language theory
  8. Labeled Markov Chains
  9. Computational complexity classes
  10. The Cantor-Kantorovich distance
  11. Continuity Properties
  12. Approximation of CK Distance
  13. Conclusions and Open Questions
  14. Proof of Theorem \ref{['thm:ck-is-tv']}
  15. Proof of Corollary \ref{['cor:ck-bounds']}
  16. ...and 7 more sections

Key Result

Lemma 1

Any B-LTL definable property $\phi \subseteq A^\omega$ is a finite union of cylinder sets.

Figures (4)

  • Figure 1: Biased Markov's Eugene Onegin text model markov1913example.
  • Figure 2: Upper bound \ref{['eq:ck-bound']} on the CK distance between two $\delta$-approximately bisimilar LMCs. The upper bound is given as a function $\delta \in (0, 1)$, and for different number of labels.
  • Figure 3: Approximation of the CK distance $d(\Sigma, \tilde{\Sigma}_{\varepsilon})$. The plain line is $S_k(\Sigma_1, \Sigma_2)$, and the dashed line is the upper bound \ref{['eq:ck-bound']}. As expected from \ref{['eq:approx']}, $S_k(\Sigma_1, \Sigma_2)$ converges.
  • Figure 4: The LMC $\Sigma_p$ that encodes the product distribution $p$ described by parameters $p_1, \dots, p_k$.

Theorems & Definitions (46)

  • Example 1: Markov's Eugene Onegin text model markov1913example
  • Definition 1: Total Variation distance
  • Definition 2: Kantorovich distance
  • Definition 3: Cantor distance
  • Remark 1
  • Definition 4: LTL syntax
  • Definition 5: B-LTL syntax Jaeger2014
  • Lemma 1: Jaeger2014
  • Remark 2
  • Definition 6: Labeled Markov Chain
  • ...and 36 more