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Ryo Kashima - One of the best experts on this subject based on the ideXlab platform.
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Advances in Modal Logic - Completeness Proof by Semantic Diagrams for Transitive Closure of Accessibility Relation.
2020Co-Authors: Ryo KashimaAbstract: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.
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completeness proof by semantic diagrams for transitive closure of Accessibility Relation
Advances in Modal Logic, 2010Co-Authors: Ryo KashimaAbstract: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 - One of the best experts on this subject based on the ideXlab platform.
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Reducing model checking from multi-valued CTL* to CTL
Lecture Notes in Computer Science, 2020Co-Authors: Beata Konikowska, Wojciech PenczekAbstract: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.
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Model checking for multi-valued computation tree logics
Beyond Two: Theory and Applications of Multiple-Valued Logic, 2020Co-Authors: Beata Konikowska, Wojciech PenczekAbstract: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.
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On Designated Values in Multi-valued CTL^* Model Checking
Fundamenta Informaticae, 2003Co-Authors: Beata Konikowka, Wojciech PenczekAbstract: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.
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CONCUR - Reducing Model Checking from Multi-valued {\rm CTL}^{\ast} to {\rm CTL}^{\ast}
2002Co-Authors: Beata Konikowska, Wojciech PenczekAbstract: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.
Dov M. Gabbay - One of the best experts on this subject based on the ideXlab platform.
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Pillars of Computer Science - Introducing reactive Kripke semantics and arc Accessibility
2020Co-Authors: Dov M. GabbayAbstract: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.
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Reactive Kripke Semantics and Arc Accessibility
2020Co-Authors: Dov M. GabbayAbstract: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
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Global view on reactivity: switch graphs and their logics
Annals of Mathematics and Artificial Intelligence, 2012Co-Authors: Dov M. Gabbay, Sérgio MarcelinoAbstract: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.
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A Theory of Hierarchical Consequence and Conditionals
Journal of Logic Language and Information, 2009Co-Authors: Dov M. Gabbay, Karl SchlechtaAbstract:We introduce $${\mathcal{A}}$$ -ranked preferential structures and combine them with an Accessibility Relation. $${\mathcal{A}}$$ -ranked preferential structures are intermediate between simple preferential structures and ranked structures. The additional Accessibility Relation allows us to consider only parts of the overall $${\mathcal{A}}$$ -ranked structure. This framework allows us to formalize contrary to duty obligations, and other pictures where we have a hierarchy of situations, and maybe not all are accessible to all possible worlds. Representation results are proved.
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A theory of hierarchical consequence and conditionals
arXiv: Logic, 2008Co-Authors: Dov M. Gabbay, Karl SchlechtaAbstract:We introduce A-ranked preferential structures and combine them with an Accessibility Relation. This framework allows us to formalize contrary to duty obligations. Representation results are proved.
Corrado Priami - One of the best experts on this subject based on the ideXlab platform.
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A Logical Approach to Security in the Context of Ambient Calculus
Electronic Notes in Theoretical Computer Science, 2004Co-Authors: Radu Mardare, Corrado PriamiAbstract:In this paper 2 we advocate the use of a CTL* logic, built upon Ambient Calculus to analyze security properties. Our logic is a more expressive alternative to Ambient Logic, based on a single modality, but still powerful enough to handle mobility and dynamic hierarchies of locations. Moreover, having a temporal logic to express properties of computation, we can reuse the algorithms for model checking temporal logics in analyzing models for security problems. We resort to syntax trees of Ambient Calculus and enrich them with some labeling functions in order to obtain what we called labeled syntax trees. The labeled syntax trees will be used as possible worlds in a Kripke structure developed for a propositional branching temporal logic. The Accessibility Relation is generated by the reduction of Ambient Calculus considered as reduction between syntax trees. Providing the algorithms for calculating the Accessibility Relation between states, we open the perspective of model checking Ambient Calculus by using our algorithms together with the algorithms for model checking temporal logic.
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CMSB - Model checking biological systems described using ambient calculus
Computational Methods in Systems Biology, 2004Co-Authors: Radu Mardare, Corrado Priami, Paola Quaglia, Oleksandr VaginAbstract:We propose a way of performing model checking analysis for biological systems. The technics were developed for a CTL* logic built upon Ambient Calculus. We introduce labeled syntax trees for ambient processes and use them as possible worlds in a Kripke structure developed for a propositional branching temporal logic. The Accessibility Relation over labeled syntax trees is generated by the reduction over corresponding Ambient Calculus processes. Providing the algorithms for calculating the Accessibility Relation between states, we open the perspective of using model checking algorithms developed for temporal logics in analyzing any phenomena described in Ambient Calculus.
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A logical approach to security in the context of ambient calculus
Electronic Notes in Theoretical Computer Science, 2004Co-Authors: Radu Mardare, Corrado PriamiAbstract:In this paper2 we advocate the use of a CTL* logic, built upon Ambient Calculus to analyze security properties. Our logic is a more expressive alternative to Ambient Logic, based on a single modality, but still powerful enough to handle mobility and dynamic hierarchies of locations. Moreover, having a temporal logic to express properties of computation, we can reuse the algorithms for model checking temporal logics in analyzing models for security problems. We resort to syntax trees of Ambient Calculus and enrich them with some labeling functions in order to obtain what we called labeled syntax trees. The labeled syntax trees will be used as possible worlds in a Kripke structure developed for a prepositional branching temporal logic. The Accessibility Relation is generated by the reduction of Ambient Calculus considered as reduction between syntax trees. Providing the algorithms for calculating the Accessibility Relation between states, we open the perspective of model checking Ambient Calculus by using our algorithms together with the algorithms for model checking temporal logic. © 2004 Elsevier B.V. All rights reserved.
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computing the Accessibility Relation for the ambient calculus
2003Co-Authors: Radu Mardare, Corrado PriamiAbstract:We present some algorithms to compute the reductions of Ambient Calculus and to describe mobility and dynamic hierarchies of locations. The main idea is to treat each ambient program as a complex of a set-theoretical structure for describing the hierarchy of locations, a function for naming the nodes of the structure, and a function for registering the capabilities of each node. These complexes, named state-processes, can be seen as states for a propositional branching tree logic built upon the Ambient Calculus. We develop here, the algorithms for implementing the Accessibility Relation between the states of this logic. Our algorithms compute the evolution of the truth values for the atomical propositions during the firing of capabilities. The model presented in this paper permits to re-use algorithms for model checking temporal logic to approach the Ambient Calculus.
Ylva Søvik - One of the best experts on this subject based on the ideXlab platform.
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The semantics of preference-based belief operators
Memorandum (institute of Pacific Relations American Council), 2020Co-Authors: Geir B. Asheim, Ylva SøvikAbstract:We show how different kinds of belief operators derived from preferences can be defined in terms an Accessibility Relation of epistemic priority, and characterized by means of a vector of nested Accessibility Relations. The semantic structure is used to reconcile and compare certain non-standard notions of belief that have recently been used in epistemic analyses of games.
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Preference-based belief operators
Mathematical Social Sciences, 2005Co-Authors: Geir B. Asheim, Ylva SøvikAbstract:Abstract We show how different kinds of belief operators derived from preferences can be defined in terms an Accessibility Relation of epistemic priority, and characterized by means of a vector of nested Accessibility Relations. The semantic structure enables us to compare and reconcile certain non-standard notions of belief that have recently been used in epistemic analyses of games.
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TARK - The semantics of preference-based belief operators
Proceedings of the 9th conference on Theoretical aspects of rationality and knowledge - TARK '03, 2003Co-Authors: Geir B. Asheim, Ylva SøvikAbstract:We show how different kinds of belief operators derived from preferences can be defined in terms an Accessibility Relation of epistemic priority, and characterized by means of a vector of nested Accessibility Relations. The semantic structure is used to reconcile and compare certain non-standard notions of belief that have recently been used in epistemic analyses of games.
