Van Benthem

14,000,000 Leading Edge Experts on the ideXlab platform

Scan Science and Technology

Contact Leading Edge Experts & Companies

Scan Science and Technology

Contact Leading Edge Experts & Companies

The Experts below are selected from a list of 1737 Experts worldwide ranked by ideXlab platform

Schilthuizen Menno - One of the best experts on this subject based on the ideXlab platform.

Liew Thor-seng - One of the best experts on this subject based on the ideXlab platform.

Vermeulen Jaap - One of the best experts on this subject based on the ideXlab platform.

Lutz Schroder - One of the best experts on this subject based on the ideXlab platform.

  • a quantified coalgebraic Van Benthem theorem
    Foundations of Software Science and Computation Structure, 2021
    Co-Authors: Paul Wild, Lutz Schroder
    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.

  • a Van Benthem theorem for quantitative probabilistic modal logic
    arXiv: Logic in Computer Science, 2018
    Co-Authors: Paul Wild, Lutz Schroder, Dirk Pattinson, Barbara Konig
    Abstract:

    In probabilistic transition systems, behavioural metrics provide a more fine-grained and stable measure of system equivalence than crisp notions of bisimilarity. They correlate strongly to quantitative probabilistic logics, and in fact the distance induced by a probabilistic modal logic taking values in the real unit interval has been shown to coincide with behavioural distance. For probabilistic systems, probabilistic modal logic thus plays an analogous role to that of Hennessy-Milner logic on classical labelled transition systems. In the quantitative setting, invariance of modal logic under bisimilarity becomes non-expansivity of formula evaluation w.r.t. behavioural distance. In the present paper, we provide a characterization of the expressive power of probabilistic modal logic based on this observation: We prove a probabilistic analogue of the classical Van Benthem theorem, which states that modal logic is precisely the bisimulation-invariant fragment of first-order logic. Specifically, we show that quantitative probabilistic modal logic lies dense in the bisimulation-invariant fragment, in the indicated sense of non-expansive formula evaluation, of quantitative probabilistic first-order logic; more precisely, bisimulation-invariant first-order formulas are approximable by modal formulas of bounded rank. For a description logic perspective on the same result, see arXiv:1906.00784.

  • a Van Benthem theorem for fuzzy modal logic
    arXiv: Logic in Computer Science, 2018
    Co-Authors: Paul Wild, Lutz Schroder, Dirk Pattinson, Barbara Konig
    Abstract:

    We present a fuzzy (or quantitative) version of the Van Benthem theorem, which characterizes propositional modal logic as the bisimulation-invariant fragment of first-order logic. Specifically, we consider a first-order fuzzy predicate logic along with its modal fragment, and show that the fuzzy first-order formulas that are non-expansive w.r.t. the natural notion of bisimulation distance are exactly those that can be approximated by fuzzy modal formulas.

  • A Characterization Theorem for a Modal Description Logic
    International Joint Conferences on Artificial Intelligence Organization, 2017
    Co-Authors: Paul Wild, Lutz Schroder
    Abstract:

    Modal description logics feature modalities that capture dependence of knowledge on parameters such as time, place, or the information state of agents. E.g., the logic S5-ALC combines the standard description logic ALC with an S5-modality that can be understood as an epistemic operator or as representing (undirected) change. This logic embeds into a corresponding modal first-order logic S5-FOL. We prove a modal characterization theorem for this embedding, in analogy to results by Van Benthem and Rosen relating ALC to standard first-order logic: We show that S5-ALC with only local roles is, both over finite and over unrestricted models, precisely the bisimulation-invariant fragment of S5-FOL, thus giving an exact description of the expressive power of S5-ALC with only local roles.

  • a Van Benthem rosen theorem for coalgebraic predicate logic
    Journal of Logic and Computation, 2015
    Co-Authors: Lutz Schroder, Dirk Pattinson, Tadeusz Litak
    Abstract:

    Coalgebraic modal logic serves as a unifying framework to study a wide range of modal logics beyond the relational realm, including probabilistic and graded logics as well as conditional logics and logics based on neighbourhoods and games. Coalgebraic predicate logic (CPL), a generalization of a neighbourhood-based first-order logic introduced by Chang, has been identified as a natural first-order extension of coalgebraic modal logic, which in particular coincides with the standard first-order correspondence language when instantiated to Kripke-style relational modal operators. Here, we generalize to the CPL setting the classical Van Benthem/Rosen theorem stating that both over arbitrary and over finite models, modal logic is precisely the bisimulation-invariant fragment of first-order logic. As instances of this generic result, we obtain corresponding characterizations for, e.g. conditional logic, neighbourhood logic (i.e. classical modal logic) and monotone modal logic.

Marzuki, Mohammad Effendi - One of the best experts on this subject based on the ideXlab platform.