A Quantified Coalgebraic van Benthem Theorem

Wild, P.; Schröder, Lutz

doi:10.1007/978-3-030-71995-1_28

Cited by 4 publications

(5 citation statements)

References 48 publications

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“…The set V = {⊥, N, B, } is an example of a frame quantale (Example 2). It is finite, and hence our general Corollary 33 applies to it, but not the previously existing expressivity theorem for quantale-valued distances [32], for this quantale is not a value-quantale. The induced logic is a paraconsistent four-valued logic with L(Λ) instantiated as follows:…”

Section: ( -Valued Powerset)mentioning

confidence: 85%

“…For simplicity, we have worked exclusively with symmetric V-categories throughout; nevertheless, we stress that our results carry over straightforwardly to the non-symmetric case, which covers quantitative analogues of simulation preorders (indeed, some of the existing quantitative coalgebraic Hennessy-Milner theorems already do apply to non-symmetric distances [32,20,33]). In fact, we expect our main expressivity theorem to be easily transported to topological categories that admit an initial dense object (which takes the role of V s ).…”

Section: Conclusion and Further Workmentioning

confidence: 98%

“…For every quantale V and every partially ordered set X, the set of monotone maps Pos(X, V) ordered pointwise becomes a quantale with tensor defined pointwise. For instance, Pos(2, [0, 1] ⊕ ) with discrete 2 yields the quantale [0, 1] 2 ⊕ , and by replacing 2 with the two-element chain 0 ≥ 1 we obtain the quantale I([0, 1] ⊕ ) of non-empty closed subintervals of [0, 1] [32].…”

Section: Every Left Continuous T-normmentioning

confidence: 99%

“…Kantorovich liftings are crucial prerequisites for existing expressiveness results of quantitative coalgebraic logics for Set-functors (e.g. [31,32,20]). However, it turns out that the Kantorovich property can be usefully detached from the notion of functor lifting: Definition 9 (Kantorovich Functor).…”

Section: Quantitative Coalgebraic Modal Logicsmentioning

confidence: 99%

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Quantitative Hennessy-Milner Theorems via Notions of Density

Jonas¹,

Goncharov²,

Hofmann³

et al. 2022

Preprint

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The classical Hennessy-Milner theorem is an important tool in the analysis of concurrent processes; it guarantees that any two non-bisimilar states in finitely branching labelled transition systems can be distinguished by a modal formula. Numerous variants of this theorem have since been established for a wide range of logics and system types, including quantitative versions where lower bounds on behavioural distance (e.g. in weighted, metric, or probabilistic transition systems) are witnessed by quantitative modal formulas. Both the qualitative and the quantitative versions have been accommodated within the framework of coalgebraic logic, with distances taking values in quantales, subject to certain restrictions, such as being so-called value quantales. While previous quantitative coalgebraic Hennessy-Milner theorems apply only to liftings of set functors to (pseudo-)metric spaces, in the present work we provide a quantitative coalgebraic Hennessy-Milner theorem that applies more widely to functors native to metric spaces; notably, we thus cover, for the first time, the well-known Hennessy-Milner theorem for continuous probabilistic transition systems, where transitions are given by Borel measures on metric spaces, as an instance. In the process, we also relax the restrictions imposed on the quantale, and additionally parametrize the technical account over notions of closure and, hence, density, providing associated variants of the Stone-Weierstraß theorem; this allows us to cover, for instance, behavioural ultrametrics. ACM Subject Classification Theory of computation → Modal and temporal logicsKeywords and phrases behavioural distances; characteristic modal logics; closure operators; density; Hennessy-Milner theorems; quantale-enriched categories; Stone-Weierstraß theorems.

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Section: ( -Valued Powerset)mentioning

confidence: 85%

Section: Conclusion and Further Workmentioning

confidence: 98%

Section: Every Left Continuous T-normmentioning

confidence: 99%

Section: Quantitative Coalgebraic Modal Logicsmentioning

confidence: 99%

See 2 more Smart Citations

Quantitative Hennessy-Milner Theorems via Notions of Density

Jonas¹,

Goncharov²,

Hofmann³

et al. 2022

Preprint

Self Cite

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show abstract

“…This principle has been extended to a large class of lax extensions [Marti and Venema, 2015], and further to the quantative setting [Wild and Schröder, 2020]. Conversely, lax extensions can be constructed from predicate liftings using the so-called Kantorovich extension [Wild and Schröder, 2020], even in quantalic generality [Wild and Schröder, 2021]. Finally, Moss liftings and the Kantorovich extension lead to a representation theorem (see Theorem 1.2 below) for specific "fuzzy" lax extensions [Wild and Schröder, 2020], which is instrumental in deriving a quantitative Henessy-Milner-type theorem.…”

Section: Introductionmentioning

confidence: 99%

A Point-free Perspective on Lax extensions and Predicate liftings

Goncharov¹,

Hofmann²,

Nora³

et al. 2021

Preprint

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In this paper we have a fresh look at the connection between lax extensions and predicate liftings of a functor from the point of view of quantale-enriched relations. Using this perspective, in particular we show that various fundamental concepts and results arise naturally and their proofs become very elementary. Ultimately, we prove that every lax extension is represented by a class of predicate liftings and discuss several implications of this result.

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Kantorovich Functors and Characteristic Logics for Behavioural Distances

Goncharov

Hofmann

Nora

et al. 2023

Lecture Notes in Computer Science

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Behavioural distances measure the deviation between states in quantitative systems, such as probabilistic or weighted systems. There is growing interest in generic approaches to behavioural distances. In particular, coalgebraic methods capture variations in the system type (nondeterministic, probabilistic, game-based etc.), and the notion of quantale abstracts over the actual values distances take, thus covering, e.g., two-valued equivalences, (pseudo)metrics, and probabilistic (pseudo)metrics. Coalgebraic behavioural distances have been based either on liftings of $$\textsf{Set}$$ Set -functors to categories of metric spaces, or on lax extensions of $$\textsf{Set}$$ Set -functors to categories of quantitative relations. Every lax extension induces a functor lifting but not every lifting comes from a lax extension. It was shown recently that every lax extension is Kantorovich, i.e. induced by a suitable choice of monotone predicate liftings, implying via a quantitative coalgebraic Hennessy-Milner theorem that behavioural distances induced by lax extensions can be characterized by quantitative modal logics. Here, we essentially show the same in the more general setting of behavioural distances induced by functor liftings. In particular, we show that every functor lifting, and indeed every functor on (quantale-valued) metric spaces, that preserves isometries is Kantorovich, so that the induced behavioural distance (on systems of suitably restricted branching degree) can be characterized by a quantitative modal logic.

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A Quantified Coalgebraic van Benthem Theorem

Cited by 4 publications

References 48 publications

Quantitative Hennessy-Milner Theorems via Notions of Density

Quantitative Hennessy-Milner Theorems via Notions of Density

A Point-free Perspective on Lax extensions and Predicate liftings

Kantorovich Functors and Characteristic Logics for Behavioural Distances

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