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.

On Recursively Enumerable and Arithmetic Models of Set Theory

Abstract: In this note we shall prove a certain relative recursiveness lemma concerning countable models of set theory (Lemma 5). From this lemma will follow two results about special types of such models. Kreisel [5] and Mostowski [6] have shown that certain (finitely axiomatized) systems of set theory, formulated by means of the ϵ relation and certain additional non-logical constants, do not possess recursive models. Their purpose in doing this was to construct consistent sentences without recursive models. As a first corollary of Lemma 5, we obtain a very simple proof, not involving any formal constructions within the system of the notions of truth and satisfiability, of an extension of the Kreisel-Mostowski theorems. Namely, set theory with the single non-logical constant ϵ does not possess any recursively enumerable model. Thus we get, as a side product, an easy example of a consistent sentence containing a single binary relation which does not possess any recursively enumerable model; this sentence being the conjunction of the (finitely many) axioms of set theory.

Exponential Diophantine Representation of Recursively Enumerable Sets

Abstract: Publisher Summary This chapter discusses the exponential diophantine representation of recursively enumerable sets The chapter presents several theorems and states that in the case of certain particular recursive sets one may be able to delete a quantifier The chapter shows that this is the case for primes, Mersenne primes, perfect numbers, and certain other recursive sets occurring in classical number theory These sets are all particular examples of Kalmar Elementary Relations The results are essentially the same as those of Jones–Matijasevic

Turing degrees and the ershov hierarchy

Abstract: An n-r.e. set can be defined as the symmetric difference of n recursively enumerable sets. The classes of these sets form a natural hierarchy which became a well-studied topic in recursion theory. In a series of ground-breaking papers, Ershov generalized this hierarchy to transfinite levels based on Kleene’s notations of ordinals and this work lead to a fruitful study of these sets and their many-one and Turing degrees. The Ershov hierarchy is a natural measure of complexity of the sets below the halting problem. In this paper, we survey the early work by Ershov and others on this hierarchy and present the most fundamental results. We also provide some pointers to concurrent work in the field.

Closure properties for fuzzy recursively enumerable languages and fuzzy recursive languages

Abstract: There are several variations of fuzzy Turing machines in the literature, many of them require a t-norm in order to establish their accepted language. This paper generalize the concept of non-deterministic fuzzy Turing machine - NTFM, replacing the t-norm operator for several aggregation functions. We establish the languages accepted by these machines, called fuzzy recursively enumerable languages or simply LFRE and show, among other results, which classes of LFRE are closed under unions and intersections.

Embeddings of Heyting algebras

Abstract: In this paper we study embeddings of Heyting Algebras. It is pointed out that such embeddings are naturally connected with Derived Rules. We compare the Heyting Algebras embeddable in the Heyting Algebra of the Intuitionistic Propositional Calculus (IPC), i.e. the free Heyting Algebra on countably infinitely many generators, and those embeddable in the Heyting Algebra of Heyting's Arithmetic (HA). A partial result is obtained. We show that every recursively enumerable prime Heyting Algebra is embeddable -in the Heyting Algebra of HA*, a ‘natural’ extension of HA.