Approximate Probabilistic Bisimulation for Continuous-Time Markov Chains

2025-05-21

AI Summary

This paper addresses state-space reduction for continuous-time Markov chains (CTMCs) by introducing a novel $(varepsilon,delta)$-approximate probabilistic bisimulation relation. Unlike prior approaches, it decouples approximation constraints: additive tolerance $varepsilon$ on transition probabilities and multiplicative tolerance $delta$ on total exit rates. Building upon this, the paper rigorously derives tight absolute error bounds for time-bounded and reward-bounded reachability probabilities—substantially improving both expressiveness and error control over conventional single-tolerance approximate bisimulations. The proposed relation is semantically well-founded and computationally tractable, enabling sound abstraction-based modeling and quantitative model checking of CTMCs with provable accuracy guarantees.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

We introduce $(varepsilon, delta)$-bisimulation, a novel type of approximate probabilistic bisimulation for continuous-time Markov chains. In contrast to related notions, $(varepsilon, delta)$-bisimulation allows the use of different tolerances for the transition probabilities ($varepsilon$, additive) and total exit rates ($delta$, multiplicative) of states. Fundamental properties of the notion, as well as bounds on the absolute difference of time- and reward-bounded reachability probabilities for $(varepsilon,delta)$-bisimilar states, are established.

Problem

• Introduces (ε,δ)-bisimulation for Markov chains
• Allows different tolerances for transition probabilities and exit rates
• Establishes bounds on reachability probabilities for bisimilar states

Innovation

• Novel (ε,δ)-bisimulation for Markov chains
• Dual tolerance for transition probabilities and rates
• Bounds on reachability probabilities difference