[Paper Review] Topologies of Stochastic Markov Models: Computational Aspects

Giorgio Bacci, Giovanni Bacci+2 more|arXiv (Cornell University)|Mar 24, 2014

Formal Methods in Verification22 references3 citations

TL;DR

This paper introduces two behavioral distances for Stochastic Markov Models (SMMs) to measure dissimilarity with respect to linear real-time specifications expressed in Metric Temporal Logic (MTL) or Deterministic Timed Automata (DTA). It proves that exact computation of the natural variation distance is NP-hard, but proposes a Kantorovich-based over-approximation that can be computed in polynomial time under mild distributional assumptions, enabling efficient approximate model checking.

ABSTRACT

In this paper we propose two behavioral distances that support approximate reasoning on Stochastic Markov Models (SMMs), that are continuous-time stochastic transition systems where the residence time on each state is described by a generic probability measure on the positive real line. In particular, we study the problem of measuring the behavioral dissimilarity of two SMMs against linear real-time specifications expressed as Metric Temporal Logic (MTL) formulas or Deterministic Timed-Automata (DTA). The most natural choice for such a distance is the one that measures the maximal difference that can be observed comparing two SMMs with respect to their probability of satisfying an arbitrary specification. We show that computing this metric is NP-hard. In addition, we show that any algorithm that approximates the distance within a certain absolute error, depending on the size of the SMMs, is NP-hard. Nevertheless, we introduce an alternative distance, based on the Kantorovich metric, that is an over-approximation of the former and we show that, under mild assumptions on the residence time distributions, it can be computed in polynomial time.

Motivation & Objective

• To define behavioral metrics for continuous-time stochastic systems that support approximate reasoning under real-time specifications.
• To analyze the computational complexity of measuring behavioral dissimilarity in SMMs with respect to MTL and DTA specifications.
• To develop a tractable over-approximation of the exact variation distance that is computable in polynomial time.
• To establish theoretical foundations for approximating model checking results using 1-RDTA specifications.
• To extend fixed-point and linear programming techniques from discrete-time Markov chains to continuous-time SMMs.

Proposed method

• Introduces Stochastic Markov Models (SMMs) with generic residence time distributions on the positive real line, generalizing CTMCs and discrete-time MCs.
• Defines two pseudometrics—MTL and DTA variation distances—measuring maximal difference in satisfaction probability across all linear real-time specifications.
• Proves that the exact variation distance coincides for MTL and DTA specifications and is fully characterized by 1-RDTA specifications.
• Proposes a Kantorovich-based fixed-point distance as an over-approximation of the exact variation distance.
• Reduces the problem of computing the Kantorovich distance to a linear program with a polynomial-time separation oracle via the ellipsoid method.
• Leverages transportation polytopes and vertex enumeration to efficiently compute optimal couplings in the Kantorovich formulation.

Experimental results

Research questions

• RQ1 Is the exact computation of the behavioral variation distance between SMMs NP-hard?
• RQ2 Can the variation distance be approximated within a fixed absolute error in polynomial time?
• RQ3 Does the Kantorovich-based distance provide a tractable over-approximation of the exact variation distance?
• RQ4 Can 1-RDTA specifications approximate any measurable set of timed paths in the σ-algebra generated by MTL or DTA specifications?
• RQ5 Under what distributional assumptions can the Kantorovich distance be computed in polynomial time?

Key findings

• The exact MTL and DTA variation distances are NP-hard to compute, even within a fixed absolute error, via reduction from Max Clique.
• The variation distance coincides for MTL and DTA specifications and is fully determined by 1-RDTA specifications, which are dense in the generated σ-algebra.
• The Kantorovich-based fixed-point distance is an over-approximation of the exact variation distance and can be computed in polynomial time using the ellipsoid method.
• The linear program formulation of the Kantorovich distance admits a polynomial-time separation oracle based on solving transportation problems.
• The method extends fixed-point bisimilarity techniques from discrete-time MCs to continuous-time SMMs under mild assumptions on residence time distributions.
• The proposed approach enables efficient approximate model checking for CTMCs by leveraging efficient algorithms for single-clock DTAs.