DTMC Model Checking by Path Abstraction Revisited (extended version)

2025-09-02

AI Summary

This paper addresses the problem of efficiently computing the probability mass of all finite paths reaching a target state set (G) in a discrete-time Markov chain (DTMC), particularly for analyzing subsets of paths—e.g., during counterexample refinement. The proposed method employs path abstraction-based decomposition. Its key contributions are: (i) a proof that path abstraction can be applied along any sequence of non-target states, eliminating reliance on strongly connected component structure; and (ii) an algebraic framework over the free monoid, enabling concise, rigorous formal modeling and correctness proofs. A compact, reusable open-source reference implementation is provided in PARI/GP, formally verified to yield results identical to direct path enumeration. The approach significantly enhances flexibility and interpretability in path-level probabilistic analysis.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Computing the probability of reaching a set of goal states G in a discrete-time Markov chain (DTMC) is a core task of probabilistic model checking. We can do so by directly computing the probability mass of the set of all finite paths from the initial state to G; however, when refining counterexamples, it is also interesting to compute the probability mass of subsets of paths. This can be achieved by splitting the computation into path abstractions that calculate "local" reachability probabilities as shown by Ábrahám et al. in 2010. In this paper, we complete and extend their work: We prove that splitting the computation into path abstractions indeed yields the same result as the direct approach, and that the splitting does not need to follow the SCC structure. In particular, we prove that path abstraction can be performed along any finite sequence of sets of non-goal states. Our proofs proceed in a novel way by interpreting the DTMC as a structure on the free monoid on its state space, which makes them clean and concise. Additionally, we provide a compact reference implementation of path abstraction in PARI/GP.

Problem

• Computing reachability probability in DTMC model checking
• Verifying path abstraction equivalence with direct computation
• Extending path abstraction beyond SCC structure constraints

Innovation

• Path abstraction splits computation into local probabilities
• Splitting does not require following SCC structure
• Novel monoid interpretation enables clean concise proofs