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 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

Completeness in hybrid type theory

We show that basic hybridization (adding nominals and @ operators) makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret @i in propositional and first-order hybrid logic. This means: interpret @iαa, where αa is an expression of any type a, as an expression of type a that rigidly returns the value that αa receives at the i-world. The axiomatization and completeness proofs are generalizations of those found in propositional and first-order hybrid logic, and (as is usual in hybrid logic) we automatically obtain a wide range of completeness results for stronger logics and languages. Our approach is deliberately low-tech. We don’t, for example, make use of Montague’s intensional type s, or Fitting-style intensional models; we build, as simply as we can, hybrid logic over Henkin’s logic.submittedVersionFil: Areces, Carlos Eduardo. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Blackburn, Patrick. University of Roskilde. Centre for Culture and Identity. Department of Philosophy and Science Studies; Dinamarca.Fil: Huertas, Antonia. Universitat Oberta de Catalunya; España.Fil: Manzano, María. Universidad de Salamanca; España.Ciencias de la Computació

45 YEARS “ECONOMIC THOUGHT” JOURNAL: Statements

45 YEARS “ECONOMIC THOUGHT” JOURNAL: A discussion held on 18 may 2001 at the University of Economy - Varna. Statements.

Semi-qualitative reasoning about distances: a preliminary report

We introduce a family of languages intended for representing knowledge and reasoning about metric (and more general distance) spaces. While the simplest language can speak only about distances between individual objects and Boolean relations between sets, the more expressive ones are capable of capturing notions such as ‘somewhere in (or somewhere out of) the sphere of a certain radius’, ‘everywhere in a certain ring’, etc. The computational complexity of the satisfiability problem for formulas in our languages ranges from NP-completeness to undecidability and depends on the class of distance spaces in which they are interpreted. Besides the class of all metric spaces, we consider, for example, the spaces ℝ × ℝ and ℕ × ℕ with their natural metrics

Bilattice-based squares and triangles

In this paper, Ginsberg's/Fitting's theory of bilattices, and in particular the associated constructs of bilattice-based squares and triangles, is invoked as a natural accommodation and powerful generalization to both intuitionistic fuzzy sets (IFSs) and interval-valued fuzzy sets (IVFSs), serving on one hand to clarify the exact nature of the relationship between these two common extensions of fuzzy sets, and on the other hand providing a general and intuitively attractive framework for the representation of uncertain and potentially conflicting information. Close attention is also paid to the definition of adequate graded versions of basic logical connectives in this setting. Finally, application potential of the proposed framework is briefly illustrated in the context of preference modelling