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.

On the relative succinctness of modal logics with union, intersection and quantification

Abstract: In the study of knowledge representation formalisms, there is a current interest in the question of how different formal languages compare in their ability to compactly express semantic properties. Recently, French et al. have shown that modal logics with a modality for public announcement, for everybody knows, and for somebody knows are all exponentially more succinct than basic modal logic. In this paper we compare the above mentioned logics not with basic modal logic but with each other and also with modal logics that have a modality for distributed knowledge. Interestingly, modal logic with such a modality is more expressive than the other modal logics mentioned, but still we can show that some of those weaker logics are exponentially more succinct than the former. Additionally, we prove that the opposite is also possible: indeed, we show that modal logic with a modality for distributed knowledge is more succinct than modal logic with a modality for everybody knows.

Circumscription in a Modal Logic

Abstract: In this paper, we extend circumscription [McCarthy, 80] to a propositional modal logic of knowledge of one agent. Instead of circumscribing a predicate, we circumscribe the knowledge operator "K" in a formula. In order to have a nontrivial circumscription schema, we extend S5 modal logic of knowledge by adding another modality "V al" and a universal quantifier over base sentences (sentences which do not contain modality). Intuitively, "V al(P)" means that P is a valid formula. It turns out that by circumscribing the knowledge operator in a formula, we completely characterize the maximally ignorant models of the formula (models of the formula where agents have minimal knowledge).

Algebra, proof theory and applications for an intuitionistic logic of propositions, actions and adjoint modal operators

Abstract: We develop a cut-free nested sequent calculus as basis for a proof search procedure for an intuitionistic modal logic of actions and propositions. The actions act on propositions via a dynamic modality (the weakest precondition of program logics), whose left adjoint we refer to as “update” (the strongest postcondition). The logic has agent-indexed adjoint pairs of epistemic modalities: the left adjoints encode agents' uncertainties and the right adjoints encode their beliefs. The rules for the “update” modality encode learning as a result of discarding uncertainty. We prove admissibility of Cut, and hence the soundness and completeness of the logic with respect to an algebraic semantics. We interpret the logic on epistemic scenarios that consist of honest and dishonest communication actions, add assumption rules to encode them, and prove that the calculus with the assumption rules still has the admissibility results. We apply the calculus to encode (and allow reasoning about) the classic epistemic puzzles of dirty children (a.k.a. “muddy children”) and drinking logicians and some versions with dishonesty or noise; we also give an application where the actions are movements of a robot rather than announcements.

On Modal Probability and Belief

Abstract: We investigate a simple modal logic of probability with a unary modal operator expressing that a proposition is more probable than its negation. Such an operator is not closed under conjunction, and its modal logic is therefore non-normal. Within this framework we study the relation of probability with other modal concepts: belief and action.