7,685 research outputs found
Post Completeness in Congruential Modal Logics
Well-known results due to David Makinson show that there are exactly two Post complete normal modal logics, that in both of them, the modal operator is truth-functional, and that every consistent normal modal logic can be extended to at least one of them. Lloyd Humberstone has recently shown that a natural analog of this result in congruential modal logics fails, by showing that not every congruential modal logic can be extended to one in which the modal operator is truth-functional. As Humberstone notes, the issue of Post completeness in congruential modal logics is not well understood. The present article shows that in contrast to normal modal logics, the extent of the property of Post completeness among congruential modal logics depends on the background set of logics. Some basic results on the corresponding properties of Post completeness are established, in particular that although a congruential modal logic is Post complete among all modal logics if and only if its modality is truth-functional, there are continuum many modal logics Post complete among congruential modal logics
Modal logics are coalgebraic
Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. The coalgebraic approach is generic and compositional: tools and techniques simultaneously apply to a large class of application areas and can moreover be combined in a modular way. In particular, this facilitates a pick-and-choose approach to domain specific formalisms, applicable across the entire scope of application areas, leading to generic software tools that are easier to design, to implement, and to maintain. This paper substantiates the authors' firm belief that the systematic exploitation of the coalgebraic nature of modal logic will not only have impact on the field of modal logic itself but also lead to significant progress in a number of areas within computer science, such as knowledge representation and concurrency/mobility
On Nested Sequents for Constructive Modal Logics
We present deductive systems for various modal logics that can be obtained
from the constructive variant of the normal modal logic CK by adding
combinations of the axioms d, t, b, 4, and 5. This includes the constructive
variants of the standard modal logics K4, S4, and S5. We use for our
presentation the formalism of nested sequents and give a syntactic proof of cut
elimination.Comment: 33 page
Modal Logics of Topological Relations
Logical formalisms for reasoning about relations between spatial regions play
a fundamental role in geographical information systems, spatial and constraint
databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's
modal logic of time intervals based on the Allen relations, we introduce a
family of modal logics equipped with eight modal operators that are interpreted
by the Egenhofer-Franzosa (or RCC8) relations between regions in topological
spaces such as the real plane. We investigate the expressive power and
computational complexity of logics obtained in this way. It turns out that our
modal logics have the same expressive power as the two-variable fragment of
first-order logic, but are exponentially less succinct. The complexity ranges
from (undecidable and) recursively enumerable to highly undecidable, where the
recursively enumerable logics are obtained by considering substructures of
structures induced by topological spaces. As our undecidability results also
capture logics based on the real line, they improve upon undecidability results
for interval temporal logics by Halpern and Shoham. We also analyze modal
logics based on the five RCC5 relations, with similar results regarding the
expressive power, but weaker results regarding the complexity
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