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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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TL;DR: In this paper, it was shown that the unification problem is undecidable for basic modal logics K and K4 extended with the universal modality, and it follows that the admissibility problem for inference rules for these logics as well as the problem of proving whether there is a substitution instance of a given formula that is provable in a given logic is also undecidable.
Abstract: We show that the unification problem `is there a substitution instance of a given formula that is provable in a given logic?' is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems are undecidable for basic hybrid logics. Recently, unification has been introduced as an important reasoning service for description logics. The undecidability proof for K with nominals can be used to show the undecidability of unification for boolean description logics with nominals (such as ALCO and SHIQO). The undecidability proof for K with the universal modality can be used to show that the unification problem relative to role boxes is undecidable for Boolean description logic with transitive roles, inverse roles, and role hierarchies (such as SHI and SHIQ).
39 citations
01 Oct 2018
39 citations
TL;DR: The machinery of coalgebraic modal logic becomes applicable to arbitrary rank-1 modal logics, without regard to their semantic status, and the methods lead to the new result that these logics are strongly complete.
Abstract: Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatized in rank 1. Here we establish the converse, i.e. every rank-1 modal logic has a sound and strongly complete coalgebraic semantics. This is achieved by constructing for a given modal logic a canonical coalgebraic semantics, consisting of a signature functor and interpretations of modal operators, which turns out to be final among all such structures. The canonical semantics may be seen as a coalgebraic reconstruction of neighbourhood semantics, broadly construed. A finitary restriction of the canonical semantics yields a canonical weakly complete semantics which moreover enjoys the Hennessy–Milner property. As a consequence, the machinery of coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, becomes applicable to arbitrary rank-1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of such results. As an extended example, we apply our framework to recently defined deontic logics. In particular, our methods lead to the new result that these logics are strongly complete.
39 citations
07 Dec 2011
TL;DR: Two modal operators are introduced into the type theory to provide detailed control about how much information is communicated: one based on traditional proof irrelevance and one integrating digital signatures.
Abstract: Dependent session types allow us to describe not only properties of the I/O behavior of processes but also of the exchanged data. In this paper we show how to exploit dependent session types to express proof-carrying communication. We further introduce two modal operators into the type theory to provide detailed control about how much information is communicated: one based on traditional proof irrelevance and one integrating digital signatures.
38 citations
01 Jan 2002
TL;DR: In this article, the authors give a physicist's view on the framework of branching space-time and give a formal semantics for the modal operators ''possibly'' and ''necessarily'' and for the counterfactual conditional.
Abstract: The paper gives a physicist's view on the framework of branching space-time (Belnap, Synthese 92 (1992), 385--434). Branching models are constructed from physical state assignments. The models are then employed to give a formal semantics for the modal operators ``possibly'' and ``necessarily'' and for the counterfactual conditional. The resulting formal language can be used to analyze quantum correlation experiments. As an application sketch, Stapp's premises LOC1 and LOC2 from his purported proof of non-locality (Am. J. Phys. 65 (1997), 300--304) are analyzed.
38 citations