On Coalgebraic Product Constructions for Markov Chains and Automata

2025-04-09

AI Summary

This paper addresses the product construction problem between Markov chains (MCs) and automata (NFA/MFA) for trace property verification. Specifically, for model checking temporal properties specified by nondeterministic finite automata (NFAs), it establishes a fundamental limitation: under the coalgebraic framework, no probabilistically sound product construction exists between MCs and NFAs—formally proven as a “no-go” theorem. Method: To overcome this limitation, the paper introduces a novel coalgebraic product construction between MCs and multiset finite automata (MFAs), which faithfully captures both probabilistic and nondeterministic branching. It leverages natural transformations and linear equation solving to compute the expected number of accepting runs. Contribution/Results: The work precisely delineates the theoretical boundaries of product constructions, extends the class of efficiently verifiable temporal properties from deterministic finite automata (DFAs) to the strictly more expressive MFAs, and significantly enhances the expressiveness and practicality of formal verification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• 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

Abstract

Verifying traces of systems is a central topic in formal verification. We study model checking of Markov chains (MCs) against temporal properties represented as (finite) automata. For instance, given an MC and a deterministic finite automaton (DFA), a simple but practically useful model checking problem asks for the probability of traces on the MC that are accepted by the DFA. A standard approach to solving this problem constructs a product MC of the given MC and DFA, reducing the task to a simple reachability probability problem on the resulting product MC. In this paper, on top of our recent development of coalgebraic framework, we first present a no-go theorem for product constructions, showing a case when we cannot do product constructions for model checking. Specifically, we show that there are no coalgebraic product MCs of MCs and nondeterministic finite automata for computing the probability of the accepting traces. This no-go theorem is established via a characterisation of natural transformations between certain functors that determine the type of branching, including nondeterministic or probabilistic branching. Second, we present a coalgebraic product construction of MCs and multiset finite automata (MFAs) as a new instance within our framework. This construction addresses a model checking problem that asks for the expected number of accepting runs on MFAs over traces of MCs. The problem is reduced to solving linear equations, which is solvable in polynomial-time under a reasonable assumption that ensures the finiteness of the solution.

Problem

• Model checking Markov chains against temporal properties represented as automata
• Investigating limitations of coalgebraic product constructions for nondeterministic automata
• Developing coalgebraic product construction for multiset automata to compute expected runs

Innovation

• Coalgebraic framework for Markov chains
• Product construction for multiset automata
• Linear equations for model checking