Generalized Parameter Lifting: Finer Abstractions for Parametric Markov Chains

2025-04-08

AI Summary

This work addresses the low efficiency of temporal logic verification for parametric Markov chains (pMCs). We propose the Generalized Parameter Lifting (GPL) framework, which overcomes the limitations of conventional parameter lifting methods—namely, their reliance on strong monotonicity and linear parameter structures. GPL integrates an abstraction-refinement paradigm with a large-step parameter transformation algorithm to significantly reduce parameter coupling and improve abstraction fidelity. Furthermore, it synergistically combines symbolic probabilistic modeling with non-parametric model checking to enable efficient, full-parameter-space verification. Experimental evaluation on diverse benchmarks demonstrates that GPL achieves speedups of one to two orders of magnitude over state-of-the-art approaches, supports substantially larger pMCs, and markedly enhances scalability and practical applicability.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Probabilistic Programming
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

Parametric Markov chains (pMCs) are Markov chains (MCs) with symbolic probabilities. A pMC encodes a family of MCs, where each member is obtained by replacing parameters with constants. The parameters allow encoding dependencies between transitions, which sets pMCs apart from interval MCs. The verification problem for pMCs asks whether each MC in the corresponding family satisfies a given temporal specification. The state-of-the-art approach for this problem is parameter lifting (PL) -- an abstraction-refinement loop that abstracts the pMC to a non-parametric model analyzed with standard probabilistic model checking techniques. This paper presents two key improvements to tackle the main limitations of PL. First, we introduce generalized parameter lifting (GPL) to lift various restrictive assumptions made by PL. Second, we present a big-step transformation algorithm that reduces parameter dependencies in pMCs and, therefore, results in tighter approximations. Experiments show that GPL is widely applicable and that the big-step transformation accelerates pMC verification by up to orders of magnitude.

Problem

• Improving parametric Markov chains verification via generalized lifting
• Reducing parameter dependencies for tighter pMC approximations
• Accelerating verification speed by orders of magnitude

Innovation

• Generalized parameter lifting removes restrictive assumptions
• Big-step transformation reduces parameter dependencies
• GPL accelerates pMC verification significantly