Operator synthesis. I. Synthetic sets, bilattices and tensor algebras

The interplay between the invariant subspace theory and spectral synthesis for locally compact abelian group discovered by Arveson is extended to include other topics as harmonic analysis for Varopoulos algebras and approximation by projection-valued measures. We propose a ''coordinate'' approach which nevertheless does not use the technique of pseudo-integral operators, as well as a coordinate free one which allows to extend to non-separable spaces some important results and constructions of [W.Arveson, Operator Alegebras and Invariant subspaces, Ann. of Math. (2) 100 (1974)] and solve some problems posed there.Comment: 32 pages. to appear in Journal of Functional Analysi

Elimination of Cuts in First-order Finite-valued Logics

A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information

Generalized knowledge-based semantics for multi-valued logic programs

A generalized logic programming system is presented which uses bilattices as the underlying framework for the semantics of programs. The two orderings of the bilattice reflect the concepts of truth and knowledge. Programs are interpreted according to their knowledge content, resulting in a monotonic semantic operator even in the presence of negation. A special case, namely, logic programming based on the four-valued bilattice is carefully studied on its own right. In the four-valued case, a version of the Closed World Assumption is incorporated into the semantics. Soundness and Completeness results are given with and without the presence of the Closed World Assumption. The concepts studied in the four-valued case are then generalized to arbitrary bilattices. The resulting logic programming systems are well suited for representing incomplete or conflicting information. Depending on the choice of the underlying bilattice, the knowledge-based logic programming language can provide a general framework for other languages based on probabilistic logics, intuitionistic logics, modal logics based on the possible-worlds semantics, and other useful non-classical logics. A novel procedural semantics is given which extends SLDNF-resolution and can retrieve both negative and positive information about a particular goal in a uniform setting. The proposed procedural semantics is based on an AND-parallel computational model for logic programs. The concept of substitution unification is introduced and many of its properties are studied in the context of the proposed computational model. Some of these properties may be of independent interest, particularly in the implementation of parallel and distributed logic programs. Finally, soundness and completeness results are proved for the proposed logic programming system. It is further shown that for finite distributive bilattices (and, more generally, bilattices with the descending chain property), an alternate procedural semantics can be developed based on a small subset of special truth values which turn out to be the join irreducible elements of the knowledge part of the bilattice. The algebraic properties of these elements and their relevance to the corresponding logic programming system are extensively studied

Stone-type representations and dualities for varieties of bisemilattices

In this article we will focus our attention on the variety of distributive bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and involutive bisemilattices. After extending Balbes' representation theorem to bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn duality and introduce the categories of 2spaces and 2spaces$^{\star}$. The categories of 2spaces and 2spaces$^{\star}$ will play with respect to the categories of distributive bisemilattices and De Morgan bisemilattices, respectively, a role analogous to the category of Stone spaces with respect to the category of Boolean algebras. Actually, the aim of this work is to show that these categories are, in fact, dually equivalent