Recursively enumerable language

About: Recursively enumerable language is a research topic. Over the lifetime, 1508 publications have been published within this topic receiving 32382 citations.

Recursion Theory on Orderings. I. A Model Theoretic Setting

Abstract: In [6], Metakides and Nerode introduced the study of the lattice of recursively enumerable substructures of a recursively presented model as a means to understand the recursive content of certain algebraic constructions. For example, the lattice of recursively enumerable subspaces, , of a recursively presented vector space V ∞ has been studied by Kalantari, Metakides and Nerode, Retzlaff, Remmel and Shore. Similar studies have been done by Remmel [12], [13] for Boolean algebras and by Metakides and Nerode [9] for algebraically closed fields. In all of these models, the algebraic closure of a set is nontrivial. (The formal definition of the algebraic closure of a set S , denoted cl( S ), is given in §1, however in vector spaces, cl( S ) is just the subspace generated by S , in Boolean algebras, cl( S ) is just the subalgebra generated by S , and in algebraically closed fields, cl( S ) is just the algebraically closed subfield generated by S .) In this paper, we give a general model theoretic setting (whose precise definition will be given in §1) in which we are able to give constructions which generalize many of the constructions of classical recursion theory. One of the main features of the models which we study is that the algebraic closure of set is just itself, i.e., cl( S ) = S. Examples of such models include the natural numbers under equality 〈 N , = 〉, the rational numbers under the usual ordering 〈 Q , ≤〉, and a large class of n -dimensional partial orderings.

A General Schema for Constructing One-Point Bases in the Lambda Calculus

Abstract: In this paper, we present a schema for constructing one-point bases for recursively enumerable sets of lambda terms. The novelty of the approach is that we make no assumptions about the terms for which the one-point basis is constructed: They need not be combinators and they may contain constants and free variables. The significance of the construction is twofold: In the context of the lambda calculus, it characterises one-point bases as ways of ``packaging'' sets of terms into a single term; And in the context of realistic programming languages, it implies that we can define a single procedure that generates any given recursively enumerable set of procedures, constants and free variables in a given programming language.

A characterization of effective topological spaces

Abstract: Starting with D. Scott's work on the mathematical foundations of programming language semantics, interest in topology has grown up in theoretical computer science, under the slogan "open sets are semidecidable properties". But whereas on Scott domains all such properties are also open, this is no longer true in general. In this paper we present a characterization of effectively given topological spaces that says which semidecidable sets are open. We consider countable topological To-spaces that satisfy certain additional topological and computational requirements which can be verified for a general class of Scott domains and metric spaces, and we show that the given topology is the recursively finest topology generated by semidecidable sets which is compatible with it. From this general result we derive the above mentioned theorem about the correspondence of the semidecidable properties with the Scott open sets. This theorem, in its turn, is a generalization of the Rice/Shapiro theorem on index sets of classes of recursively enumerable sets. Moreover, characterizations of the canonical topology of a recursively separable recursive metric space are derived. It is shown that it is the recursively finest effective T3-topology that can be generated by semidecidable sets the topological complement of which is also semidecidable.