Abstract

The classical van Benthem theorem characterizes modal logic as the bisimulation-invariant fragment of first-order logic; put differently, modal logic is as expressive as full first-order logic on bisimulation-invariant properties. This result has recently been extended to two flavours of quantitative modal logic, viz. fuzzy modal logic and probabilistic modal logic. In both cases, the quantitative van Benthem theorem states that every formula in the respective quantitative variant of first-order logic that is bisimulation-invariant, in the sense of being nonexpansive w.r.t. behavioural distance, can be approximated by quantitative modal formulae of bounded rank. In the present paper, we unify and generalize these results in three directions: We lift them to full coalgebraic generality, thus covering a wide range of system types including, besides fuzzy and probabilistic transition systems as in the existing examples, e.g. also metric transition systems; and we generalize from real-valued to quantale-valued behavioural distances, e.g. nondeterministic behavioural distances on metric transition systems; and we remove the symmetry assumption on behavioural distances, thus covering also quantitative notions of simulation.

Highlights

  • Modal logic takes part of its popularity from the fact that it specifies transition systems at what for many purposes may be regarded as the right level of granularity; that is, it is invariant under the standard process-theoretic notion of bisimulation in the sense that bisimilar states satisfy the same modal formulae

  • There are two quite different well-known converses to this elementary property, which both witness the expressiveness of modal logic: By the Hennessy-Milner theorem [29], states in finitely branching systems that satisfy the same modal formulae are bisimilar, and by the van Benthem theorem, every first-order definable bisimulation-invariant property is expressible by a modal formula

  • We remove the assumption that distances need to be symmetric, so that we cover notions of quantitative simulation. At this level of generality, we prove both a Hennessy-Milner type theorem stating coincidence of logical and behavioural distance, effectively generalizing the existing coalgebraic quantitative Hennessy-Milner theorem [37] to quantale-valued distances; and, as our main result, a quantitative van Benthem theorem stating that all behaviourally non-expansive first-order properties can be modally approximated in bounded rank

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Summary

Introduction

Modal logic takes part of its popularity from the fact that it specifies transition systems at what for many purposes may be regarded as the right level of granularity; that is, it is invariant under the standard process-theoretic notion of bisimulation in the sense that bisimilar states satisfy the same modal formulae. These theorems take the form of approximability properties, and state that every behaviourally nonexpansive quantitative firstorder property is approximable by quantitative modal formulae of bounded rank The latter qualification is the key content of the respective theorems – without it, approximability is closer in flavour to Hennessy-Milner-type theorems, which apply to arbitrary rather than just first-order definable properties ( one should note that our van Benthem theorems do not assume compactness of the state space). We remove the assumption that distances need to be symmetric, so that we cover notions of quantitative simulation At this level of generality, we prove both a Hennessy-Milner type theorem stating coincidence of logical and behavioural distance, effectively generalizing the existing coalgebraic quantitative Hennessy-Milner theorem [37] to quantale-valued distances; and, as our main result, a quantitative van Benthem theorem stating that all behaviourally non-expansive first-order properties can be modally approximated in bounded rank. There is work on Hennessy-Milner theorems for certain Heyting-valued modal logics [21,18]; since Heyting algebras are quantales but often fail to meet a continuity assumption needed in our generic Hennessy-Milner theorem, we do not claim to subsume these results

Preliminaries
Quantale-Valued Distances and Lax Extensions
Convex-nondeterministic distances
Quantale-Valued Modal and Predicate Logics
Fuzzy modal logic
Probabilistic modal logic
Metric modal logic
Convex-nondeterministic metric modal logic
Behavioural Distance and Quantitative Bisimulation Invariance
Modal Approximation
Locality and Modal Characterization
Conclusions
Full Text
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