Quantitative Graded Semantics and Spectra of Behavioural Metrics

2023-06-02

arXiv.org

AI Summary

This paper addresses the challenge of characterizing behavioral distances in quantitative systems—such as probabilistic and fuzzy transition systems—using modal logic, where existing logical frameworks lack precise expressive power. Methodologically, we first prove that probabilistic trace distance is not definable by unary modal logic; then introduce a graded monadic algebraic construction to establish general criteria for real-valued modal logics to characterize behavioral distances; finally, leveraging category theory and coalgebraic techniques, we develop a novel characteristic logic for trace distance in fuzzy metric transition systems. Our main contribution is a unified, scalable, and multi-granular framework for behavioral distances that overcomes the expressive limitations of classical modal logic. This framework constitutes the first metric semantic unification for quantitative transition systems—spanning both probabilistic and fuzzy settings—with rigorous theoretical foundations and broad modeling applicability.

Abstract

Behavioural metrics provide a quantitative refinement of classical two-valued behavioural equivalences on systems with quantitative data, such as metric or probabilistic transition systems. In analogy to the linear-time/branching-time spectrum of two-valued behavioural equivalences on transition systems, behavioural metrics vary in granularity, and are often characterized by fragments of suitable modal logics. In the latter respect, the quantitative case is, however, more involved than the two-valued one; in fact, we show that probabilistic metric trace distance cannot be characterized by any compositionally defined modal logic with unary modalities. We go on to provide a unifying treatment of spectra of behavioural metrics in the emerging framework of graded monads, working in coalgebraic generality, that is, parametrically in the system type. In the ensuing development of quantitative graded semantics, we introduce algebraic presentations of graded monads on the category of metric spaces. Moreover, we provide a general criterion for a given real-valued modal logic to characterize a given behavioural distance. As a case study, we apply this criterion to obtain a new characteristic modal logic for trace distance in fuzzy metric transition systems.

Problem

• Semantic Difference
• Behavioral Similarity
• Modal Logic

Innovation

• Hierarchical Monads
• Behavioral Distance
• Modal Logic