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Accessibility Relation

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Ryo Kashima – 1st expert on this subject based on the ideXlab platform

  • Advances in Modal Logic – Completeness Proof by Semantic Diagrams for Transitive Closure of Accessibility Relation.
    , 2020
    Co-Authors: Ryo Kashima

    Abstract:

    We treat the smallest normal modal propositional logic with two modal operators 2 and 2+. While 2 is interpreted in Kripke models by the Accessibility Relation R, 2+ is interpreted by the transitive closure of R. Intuitively the formula 2+φ means the infinite conjunction 2φ ∧ 22φ ∧ 222φ ∧ · · · . There is a Hilbert style axiomatization of this logic (a characteristic axiom is 2φ ∧ 2+(φ → 2φ) → 2+φ, called “induction axiom”), and its completeness with respect to finite models was shown by the canonical model method. This paper gives an alternative proof of this completeness. We use the method of “semantic diagram”, which is a variant of semantic tableaux, as follows. Given an unprovable formula φ, we first make a small model (consisting of one world that forces φ to be false); then we add worlds step by step using the Hilbert system as an oracle, and finally we get a finite countermodel for φ. The point is how to handle 2+ in this construction.

  • completeness proof by semantic diagrams for transitive closure of Accessibility Relation
    Advances in Modal Logic, 2010
    Co-Authors: Ryo Kashima

    Abstract:

    We treat the smallest normal modal propositional logic with two modal operators 2 and 2+. While 2 is interpreted in Kripke models by the Accessibility Relation R, 2+ is interpreted by the transitive closure of R. Intuitively the formula 2+φ means the infinite conjunction 2φ ∧ 22φ ∧ 222φ ∧ · · · . There is a Hilbert style axiomatization of this logic (a characteristic axiom is 2φ ∧ 2+(φ → 2φ) → 2+φ, called “induction axiom”), and its completeness with respect to finite models was shown by the canonical model method. This paper gives an alternative proof of this completeness. We use the method of “semantic diagram”, which is a variant of semantic tableaux, as follows. Given an unprovable formula φ, we first make a small model (consisting of one world that forces φ to be false); then we add worlds step by step using the Hilbert system as an oracle, and finally we get a finite countermodel for φ. The point is how to handle 2+ in this construction.

Wojciech Penczek – 2nd expert on this subject based on the ideXlab platform

  • Reducing model checking from multi-valued CTL* to CTL
    Lecture Notes in Computer Science, 2020
    Co-Authors: Beata Konikowska, Wojciech Penczek

    Abstract:

    A multi-valued version of CTL* (mv-CTL*), where both the propositions and the Accessibility Relation are multi-valued taking values in a finite quasi-boolean algebra, is considered. A general translation from mv-CTL* to CTL* model checking is defined. An application of the translation is shown for the most commonly used quasi-boolean algebras.

  • Model checking for multi-valued computation tree logics
    Beyond Two: Theory and Applications of Multiple-Valued Logic, 2020
    Co-Authors: Beata Konikowska, Wojciech Penczek

    Abstract:

    A multi-valued version of CTL* (mv-CTL*), where both the propositions and the Accessibility Relation are multi-valued taking values in a finite quasi-Boolean algebra, is defined. A translation from mv-CTL* model checking to CTL* model checking is investigated. First, the case where the elements of quasi-Boolean algebras are totally ordered is considered. Secondly, it is shown how to design a translation algorithm for the two most commonly applied quasi-Boolean algebras. This construction suggests the way one can deal with more complex quasi-Boolean algebras if necessary.

  • On Designated Values in Multi-valued CTL^* Model Checking
    Fundamenta Informaticae, 2003
    Co-Authors: Beata Konikowka, Wojciech Penczek

    Abstract:

    A multi-valued version of CTL^a (mv-CTL^a), where both the propositions and the Accessibility Relation are multi-valued, taking values in a complete lattice with a complement, is considered. Contrary to all the existing model checking results for multi-valued modal logics, our lattices are not required to be finite. A set of restrictions is provided under which there is a direct translation from mv-CTL^a to CTL^a model checking problem for designated values. Bisimulation induced by mv-CTL^a is characterized.

Dov M. Gabbay – 3rd expert on this subject based on the ideXlab platform

  • Pillars of Computer Science – Introducing reactive Kripke semantics and arc Accessibility
    , 2020
    Co-Authors: Dov M. Gabbay

    Abstract:

    Ordinary Kripke models are not reactive. When we evaluate (test/measure) a formula A at a model m, the model does not react, respond or change while we evaluate. The model is static and unchanged. This paper studies Kripke models which react to the evaluation process and change themselves during the process. The additional device we add to Kripke semantics to make it reactive is to allow the Accessibility Relation to access itself. Thus the Accessibility Relation R of a reactive Kripke model contains not only pairs (a, b) ∈ R of possible worlds (b is accessible to a, i.e. there is an Accessibility arc from a to b) but also pairs of the form (t, (a, b)) ∈ R, meaning that the arc (a, b) is accessible to t, or even connections of the form ((a, b), (c, d)) ∈ R.

    This new kind of Kripke semantics allows us to characterise more axiomatic modal logics (with one modality □) by a class of reactive frames. There are logics which cannot be characterised by ordinary frames but which can be characterised by reactive frames.

    We also discuss the manifestation of the ‘reactive’ idea in the context of automata theory, where we allow the automaton to react and change it’s own definition as it responds to input, and in graph theory, where the graph can change under us as we manipulate it.

  • Reactive Kripke Semantics and Arc Accessibility
    , 2020
    Co-Authors: Dov M. Gabbay

    Abstract:

    Ordinary Kripke models are not reactive. When we evaluate (test/measure) a formula A at a model m, the model does not react, respond or change while we evaluate. The model is static and unchanged. This paper studies Kripke models which react to the evaluation process and change themselves during the process. This is reminiscent of game theoretic semantics where the two sides react to each other. However, reactive Kripke models do not go as far as that. The only additional device we add to Kripke semantics to make it reactive is to allow the Accessibility Relation to access itself. Thus the Accessibility Relation R of a reactive Kripke model contains not only pairs (a, b) ∈ R of possible worlds (b is accessible to a, i.e. there is an Accessibility arc from a to b) but also pairs of the form (t, (a, b)) ∈ R, the arc (a, b) is accessible to t. This new kind of Kripke semantics allows us to characterise more axiomatic modal logics (with one modality ) by a class of reactive frames. There are logics which cannot be characterised by ordinary frames but which can be characterised by reactive frames. We use such models to fibre logics which disagree on their common language. 1 Motivation and Background Traditional modal logic uses possible world semantics with Accessibility Relation R. When we evaluate a formula such as B = 2 p∧ 3q in a Kripke model m = (S ,R, a, h) (S is the set of possible worlds, a ∈ S ,R ⊆ S 2 and h is the assignment) the model m does not change in the course of evaluation of B. We say the model m is not reactive. It stays the same during the process of evaluation. To make this point absolutely clear, consider the situation in Figure 1 below To evaluate a 3q, we have to check b 2q. We can also check another formula at b, say, b 2 p. In either case the world accessible to b are c and d. We do not say that since b 2q started its evaluation at world a as a 3q and continued to b 2q, then the accessible worlds to b are now different. In other words the model does not react to our starting the evaluation of a 3q by changing

  • Global view on reactivity: switch graphs and their logics
    Annals of Mathematics and Artificial Intelligence, 2012
    Co-Authors: Dov M. Gabbay, Sérgio Marcelino

    Abstract:

    The notion of reactive graph generalises the one of graph by allowing the base Accessibility Relation to change when its edges are traversed. Can we represent these more general structures using points and arrows? We prove this can be done by introducing higher order arrows: the switches. The possibility of expressing the dependency of the future states of the Accessibility Relation on individual transitions by the use of higher-order Relations, that is, coding meta-Relational concepts by means of Relations, strongly suggests the use of modal languages to reason directly about these structures. We introduce a hybrid modal logic for this purpose and prove its completeness.