43 research outputs found
A modal approach to dynamic ontology: modal mereotopology
In this paper we show how modal logic can be applied in the axiomatizations of some dynamic ontologies. As an example we consider the case of mereotopology, which is an extension of mereology with some relations of topological nature like contact relation. We show that in the modal extension of mereotopology we may define some new mereological and mereotopological relations with dynamic nature like stable part-of and stable contact. In some sense such “stable” relations can be considered as approximations of the “essential relations” in the domain of mereotopology
Elementary canonical formulae: extending Sahlqvist’s theorem
AbstractWe generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae. To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations à la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all inductive formulae are elementary canonical and thus extend Sahlqvist’s theorem over them. In particular, we give a simple example of an inductive formula which is not frame-equivalent to any Sahlqvist formula. Then, after a deeper analysis of the inductive formulae as set-theoretic operators in descriptive and Kripke frames, we establish a somewhat stronger model-theoretic characterization of these formulae in terms of a suitable equivalence to syntactically simpler formulae (‘primitive regular formulae’) in the extension of the language with reversive modalities. Lastly, we study and characterize the elementary canonical formulae in reversive languages with nominals, where the relevant notion of persistence is with respect to discrete frames
Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA
Modal formulae express monadic second-order properties on Kripke frames, but
in many important cases these have first-order equivalents. Computing such
equivalents is important for both logical and computational reasons. On the
other hand, canonicity of modal formulae is important, too, because it implies
frame-completeness of logics axiomatized with canonical formulae.
Computing a first-order equivalent of a modal formula amounts to elimination
of second-order quantifiers. Two algorithms have been developed for
second-order quantifier elimination: SCAN, based on constraint resolution, and
DLS, based on a logical equivalence established by Ackermann.
In this paper we introduce a new algorithm, SQEMA, for computing first-order
equivalents (using a modal version of Ackermann's lemma) and, moreover, for
proving canonicity of modal formulae. Unlike SCAN and DLS, it works directly on
modal formulae, thus avoiding Skolemization and the subsequent problem of
unskolemization. We present the core algorithm and illustrate it with some
examples. We then prove its correctness and the canonicity of all formulae on
which the algorithm succeeds. We show that it succeeds not only on all
Sahlqvist formulae, but also on the larger class of inductive formulae,
introduced in our earlier papers. Thus, we develop a purely algorithmic
approach to proving canonical completeness in modal logic and, in particular,
establish one of the most general completeness results in modal logic so far.Comment: 26 pages, no figures, to appear in the Logical Methods in Computer
Scienc
A system of relational syllogistic incorporating full Boolean reasoning
We present a system of relational syllogistic, based on classical
propositional logic, having primitives of the following form:
Some A are R-related to some B;
Some A are R-related to all B;
All A are R-related to some B;
All A are R-related to all B.
Such primitives formalize sentences from natural language like `All students
read some textbooks'. Here A and B denote arbitrary sets (of objects), and R
denotes an arbitrary binary relation between objects. The language of the logic
contains only variables denoting sets, determining the class of set terms, and
variables denoting binary relations between objects, determining the class of
relational terms. Both classes of terms are closed under the standard Boolean
operations. The set of relational terms is also closed under taking the
converse of a relation. The results of the paper are the completeness theorem
with respect to the intended semantics and the computational complexity of the
satisfiability problem.Comment: Available at
http://link.springer.com/article/10.1007/s10849-012-9165-
Modal Definability in Languages with a Finite Number of Propositional variables and a New Extension of the Sahlqvist's Class
this paper we will be interested only with the local version of modal de nability and from now on \modal de nability" will mean \local modal de nability"