Topic

# Modal operator

About: Modal operator is a research topic. Over the lifetime, 1151 publications have been published within this topic receiving 22865 citations. The topic is also known as: modal connective.

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15 Nov 2007TL;DR: This book provides various logics to support formal specifications of multi-agent systems, including proof systems, and discusses various results on the expressive power of the logics presented.

Abstract: Dynamic Epistemic Logic is the logic of knowledge change. This is not about one logical system, but about a whole family of logics that allows us to specify static and dynamic aspects of multi-agent systems. This book provides various logics to support such formal specifications, including proof systems. Concrete examples and epistemic puzzles enliven the exposition. The book also contains exercises including answers and is eminently suitable for graduate courses in logic. A sweeping chapter-wise outline of the content of this book is the following. The chapter 'Introduction' informs the reader about the history of the subject, and its relation to other disciplines. 'Epistemic Logic' is an overview of multi-agent epistemic logic - the logic of knowledge - including modal operators for groups, such as general and common knowledge. 'Belief Revision' is an overview on how to model belief revision, both in the 'traditional' way and in a dynamic epistemic setting. 'Public Announcements' is a detailed and comprehensive introduction into the logic of knowledge to which dynamic operators for truthful public announcement are added. Many interesting applications are also presented in this chapter: a form of cryptography for ideal agents also known as 'the russian cards problem', the sum-and-product riddle, etc. 'Epistemic Actions' introduces a generalization of public announcement logic to more complex epistemic actions. A different perspective on that matter is independently presented in 'Action Models'. 'Completeness' gives details on the completeness proof for the logics introduced in 'Epistemic Logic', 'Public Announcements', and 'Action Models'. 'Expressivity' discusses various results on the expressive power of the logics presented.

1,008 citations

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TL;DR: It is shown that while the problem of deciding satisfiability of an S5 formula with one agent is NP-complete, the problem for many agents is PSPACE-complete and the problem becomes complete for exponential time once a common knowledge operator is added to the language.

886 citations

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TL;DR: A number of extended rough set models are proposed and examined based on the properties satisfied by a binary relation, such as serial, reflexive, symmetric, transitive, and Euclidean, which correspond to different modal logic systems.

Abstract: The theory of rough sets is an extension of set theory with two additional unary set-theoretic operators defined based on a binary relation on the universe. These two operators are related to the modal operators in modal logics. By exploring the relationship between rough sets and modal logics, this paper proposes and examines a number of extended rough set models. By the properties satisfied by a binary relation, such as serial, reflexive, symmetric, transitive, and Euclidean, various classes of algebraic rough set models can be derived. They correspond to different modal logic systems. With respect to graded and probabilistic modal logics, graded and probabilistic rough set models are also discussed.

458 citations

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01 Jan 2007TL;DR: In this article, the conjunctive interpretation of a family of disjunctive constructions, referred to as the free choice effect (FCE), is attested for all constructions in which a sentence appears under the scope of an existential quantifier and a universal quantifier.

Abstract: This chapter will be concerned with the conjunctive interpretation of a family of disjunctive constructions. The relevant conjunctive interpretation, sometimes referred to as a ‘free choice effect,’ (FC) is attested when a disjunctive sentence is embedded under an existential modal operator. I will provide evidence that the relevant generalization extends (with some caveats) to all constructions in which a disjunctive sentence appears under the scope of an existential quantifier, as well as to seemingly unrelated constructions in which conjunction appears under the scope of negation and a universal quantifier.

456 citations

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TL;DR: This work reconsiders the foundations of modal logic, following Martin-Löf's methodology of distinguishing judgments from propositions, and gives a new presentation of lax logic, finding that the lax modality is already expressible using possibility and necessity.

Abstract: We reconsider the foundations of modal logic, following Martin-Lof's methodology of distinguishing judgments from propositions. We give constructive meaning explanations for necessity and possibility, which yields a simple and uniform system of natural deduction for intuitionistic modal logic that does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic and find that the lax modality is already expressible using possibility and necessity. Through a computational interpretation of proofs in modal logic we further obtain a new formulation of Moggi's monadic metalanguage.

348 citations