Don Pigozzi

Multi-valued logic programming semantics: an algebraic approach

Abstract In this paper we introduce the notion of join-irreducibility in the context of bilattices and present a procedural semantics for bilattice based logic programs which uses, as its basis, the join-irreducible elements of the knowledge part of the bilattice. The join-irreducible elements in a bilattice represent the “primitive bits” of information present within the system. In bilattices which have the descending chain property in their knowledge ordering, these elements provide a small representative set completely characterizing the bilattice. The overall complexity of the inference systems based on such bilattices can thus be reduced by restricting attention to the join-irreducible elements.

Lambda abstraction algebras: representation theorems

Abstract Lambda abstraction algebras (LAAs) are designed to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order predicate logic. Like combinatory algebras they can be defined by true identities and thus form a variety in the sense of universal algebra, but they differ from combinatory algebras in several important respects. The most natural LAAs are obtained by coordinatizing environment models of the lambda calculus. This gives rise to two classes of LAAs of functions of finite arity: functional LAAs (FLA) and point-relativized functional LAAs (RFA). It is shown that RFA is a variety and is the smallest variety including FLA. Dimension-complemented LAAs constitute the widest class of LAAs that can be represented as an algebra of functions and are known to have a natural intrinsic characterization. We prove that every dimension-complemented LAA is isomorphic to RFA. This is the crucial step in showing that RFA is a variety.

Fregean logics with the multiterm deduction theorem and their algebraization

AbstractA deductive system

Lambda Abstraction Algebras: Coordinatizing Models of Lambda Calculus

\mathcal{S}

Abstract Algebraic Logic

(in the sense of Tarski) is Fregean if the relation of interderivability, relative to any given theory T, i.e., the binary relation between formulas

Typed Homomorphic Relations Extended with Sybtypes

\{ \left\langle {\alpha ,\beta } \right\rangle :T,\alpha \vdash s \beta and T,\beta \vdash s \alpha \} ,