Computing Distinguishing Formulae for Threshold-Based Behavioural Distances

2026-02-12

AI Summary

This work addresses the efficient generation of distinguishing formulas that characterize behavioral distances between quantitative systems with respect to a given threshold. To this end, the authors propose a unified framework that lifts classical Boolean predicates to quantitative ones and integrates quantitative modal logic with ε-satisfaction semantics. By leveraging metric tools such as the Lévy–Prokhorov and Hausdorff distances, the framework formally captures behavioral distances and their logical characterizations. Notably, it enables, for the first time, the extraction of distinguishing formulas in polynomial time across a range of system models—including Markov chains and metric transition systems—encompassing canonical instances such as ε-bisimulation distance. This advancement significantly enhances the efficiency of verification and analysis for quantitative systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

Behavioural distances generally offer more fine-grained means of comparing quantitative systems than two-valued behavioural equivalences. They often relate to quantitative modalities, which generate quantitative modal logics that characterize a given behavioural distance in terms of the induced logical distance. We develop a unified framework for behavioural distances and logics induced by a special type of modalities that lift two-valued predicates to quantitative predicates. A typical example is the probability operator, which maps a two-valued predicate $A$ to a quantitative predicate on probability distributions assigning to each distribution the respective probability of $A$. Correspondingly, the prototypical example of our framework is $\epsilon$-bisimulation distance of Markov chains, which has recently been shown to coincide with the behavioural distance induced by the popular L\'evy-Prokhorov distance on distributions. Other examples include behavioural distance on metric transition systems and Hausdorff behavioural distance on fuzzy transition systems. Our main generic results concern the polynomial-time extraction of distinguishing formulae in two characteristic modal logics: A two-valued logic with a notion of satisfaction up to $\epsilon$, and a quantitative modal logic. These results instantiate to new results in many of the mentioned examples. Notably, we obtain polynomial-time extraction of distinguishing formulae for $\epsilon$-bisimulation distance of Markov chains in a quantitative logic featuring a `generally'modality used in probabilistic knowledge representation.

Problem

• behavioural distance
• distinguishing formulae
• modal logic
• epsilon-bisimulation
• quantitative systems

Innovation

• behavioural distance
• quantitative modal logic
• distinguishing formulae
• epsilon-bisimulation
• polynomial-time extraction