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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26 Jun 1995
TL;DR: This paper considers several classes of models for ITL which make different assumptions about time and constructs a complete and sound proof system for each class.
Abstract: Different interval modal logics have been proposed for reasoning about the temporal behaviour of digital systems. Some of them are purely propositional and only enable the specification of qualitative time requirements. Others, such as ITL and the duration calculus, are first order logics which support the expression of quantitative, real-time requirements. These two logics have in common the presence of a binary modal operator 'chop' interpreted as the action of splitting an interval into two parts. Proof systems for ITL or the duration calculus have been proposed but little is known about their power. This paper present completeness results for a variant of ITL where 'chop' is the only modal operator. We consider several classes of models for ITL which make different assumptions about time and we construct a complete and sound proof system for each class.
77 citations
16 Jul 1979
TL;DR: The Temporal interpretation of Modal Logic is adopted and the variety of program properties expressible within the modal formalism is demonstrated.
Abstract: We explore the general framework of Modal Logic and its applicability to program reasoning. We relate the basic concepts of Modal Logic to the programming environment: the concept of "world" corresponds to a program state, and the concept of "accessibility relation" corresponds to the relation of derivability between states during execution. Thus we adopt the Temporal interpretation of Modal Logic. The variety of program properties expressible within the modal formalism is demonstrated.
77 citations
21 Feb 2006
TL;DR: Modal Kleene algebras provide a unifying semantics for various program calculi and enhance efficient cross-theory reasoning in this class, often in a very concise pointfree style.
Abstract: Modal Kleene algebras are Kleene algebras enriched by forward and backward box and diamond operators. We formalise the symmetries of these operators as Galois connections, complementarities and dualities. We study their properties in the associated operator algebras and show that the axioms of relation algebra are theorems at the operator level. Modal Kleene algebras provide a unifying semantics for various program calculi and enhance efficient cross-theory reasoning in this class, often in a very concise pointfree style. This claim is supported by novel algebraic soundness and completeness proofs for Hoare logic and by connecting this formalism with an algebraic decision procedure.
77 citations
11 Aug 2009
TL;DR: A logic LSLR is defined, which is inspired by Plotkin and Abadi's logic for parametricity, but also supports recursively defined relations by means of the modal"later" operator from Appel et al.'s "very modal model" paper.
Abstract: We show how to reason about "step-indexed" logical relations in an abstract way, avoiding the tedious, error-prone, and proof-obscuring step-index arithmetic that seems superficially to be an essential element of the method. Specifically, we define a logic LSLR, which is inspired by Plotkin and Abadi's logic for parametricity, but also supports recursively defined relations by means of the modal"later" operator from Appel et al.'s "very modal model" paper. We encode in LSLR a logical relation for reasoning(in-)equationally about programs in call-by-value System F extended with recursive types. Using this logical relation, we derive a useful set of rules with which we can prove contextual (in-)equivalences without mentioning step indices.
75 citations
01 Jan 1987
TL;DR: In this article, the authors describe a formal system that uses Kripke-style possible worlds to express the three wise men and the puzzle of Mr. S and Mr. P. Their axioms are written in first-order logic and use Krike-like possible worlds directly rather than modal operators or imitations thereof.
Abstract: This paper describes a formal system and uses it to express the puzzle of the three wise men and the puzzle of Mr. S and Mr. P. Four innovations in the axiomatization of knowledge were required: the ability to express joint knowledge of several people, the ability to express the initial non-knowledge, the ability to describe knowing what rather than merely knowing that, and the ability to express the change which occurs when someone learns something. Our axioms are written in first order logic and use Kripke-style possible worlds directly rather than modal operators or imitations thereof. We intend to use functions imitating modal operators and taking ``propositions'' and ``individual concepts'' as operands, but we haven't yet solved the problem of how to treat learning in such a formalism.
74 citations