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.

Optimal Regression for Reasoning about Knowledge and Actions.

Abstract: We show how in the propositional case both Reiter's and Scherl & Levesque's solutions to the frame problem can be modelled in dynamic epistemic logic (DEL), and provide an optimal regression algorithm for the latter. Our method is as follows: we extend Reiter's framework by integrating observation actions and modal operators of knowledge, and encode the resulting formalism in DEL with announcement and assignment operators. By extending Lutz' recent satisfiability-preserving reduction to our logic, we establish optimal decision procedures for both Reiter's and Scherl & Levesque's approaches: satisfiability is NP-complete for one agent, PSPACE-complete for multiple agents and EXPTIME-complete when common knowledge is involved.

A modal logic of relations

Abstract: Treating existential quantifiers as modal diamonds, we study the n-variable fragment Ln of first-order logic, as if it were a modal formalism. In order to deal with atomic formulas adequately, to the modal version of the language we add operators corresponding to variable substitution. Since every modal language comes with an abstract Kripke-style semantics, this modal viewpoint on Ln provides an alternative, far more general semantics for the latter. One may impose conditions on the Kripke models, for instance approximating the standard Tarskian semantics. In this way one finds that some theorems of first-order logic are ‘more valid’ than others. As an example, we consider a class of generalized assignment frames called local cubes; here the basic idea is that only certain assignments are admissible. We show that the theory of this class is finitely axiomatizable and decidable. ∗Department of Mathematics and Computer Science, Free University, De Boelelaan 1081, 1081 HV Amsterdam. The research of the first author has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. †Department of Computing, Imperial College, 180 Queen’s Gate, London, UK.

Omega algebra, demonic refinement algebra and commands

Abstract: Weak omega algebra and demonic refinement algebra are two ways of describing systems with finite and infinite iteration. We show that these independently introduced kinds of algebras can actually be defined in terms of each other. By defining modal operators on the underlying weak semiring, that result directly gives a demonic refinement algebra of commands. This yields models in which extensionality does not hold. Since in predicate-transformer models extensionality always holds, this means that the axioms of demonic refinement algebra do not characterise predicate-transformer models uniquely. The omega and the demonic refinement algebra of commands both utilise the convergence operator that is analogous to the halting predicate of modal μ-calculus. We show that the convergence operator can be defined explicitly in terms of infinite iteration and domain if and only if domain coinduction for infinite iteration holds.

Quantified Modal Logic and the Plural De Re

Abstract: within a language whose only modal operators are the box and the diamond; other modal idioms cannot be expressed within such a language at all. Nonetheless, quantified modal logic has enjoyed considerable success in uncovering and explaining ambiguities in modal sentences and fallacies in modal reasoning. A prime example of this success is the now standard analysis of the distinction between modality de dido and modality de re. The analysis has been applied first and foremost to modal sentences containing definite descriptions. Such sentences are often ambiguous between an interpretation de dicto, according to which a modal property is attributed to a proposition (or, on some views, a sentence), and an interpretation de re, according to which a modal property is attributed to an individual. When these sentences are translated into the language of quantified modal logic, the de dicfo/de re ambiguity turns out to involve an ambiguity of scope. If the definite description is within the scope of the modal operator, then the operator attaches to a complete sentence, and the resulting sentence is de dicto. If the definite description is outside the scope of the modal operator, then the operator attaches to a predicate to form a modal predicate, and the resulting sentence is de re. Quantified modal logic has the resources to clarify and disambiguate English modal sentences containing definite descriptions. In this paper, I explore to what extent the analysis in terms of scope can be applied to modal sentences containing denoting phrases other than definite descriptions, phrases such as ‘some F’ and ‘every F.1 I will focus upon categorical modal sentences of the following two forms: mo s a1 idioms must be artificially restructured if they are to be expressed

Extended Full Computation-Tree Logic with Sequence Modal Operator: Representing Hierarchical Tree Structures

Abstract: An extended full computation-tree logic, CTLS*, is introduced as a Kripke semantics with a sequence modal operator. This logic can appropriately represent hierarchical tree structures where sequence modal operators in CTLS* are applied to tree structures. An embedding theorem of CTLS* into CTL* is proved. The validity, satisfiability and model-checking problems of CTLS* are shown to be decidable. An illustrative example of biological taxonomy is presented using CTLS* formulas.