On characterizations of recursively enumerable languages (original) (raw)

Characterizations of recursively enumerable languages by using copy languages

Revue Roumaine des Mathematiques Pures et Appliquees

An insertion grammar is based on pure rules of the form uu + lc~v (the string x is inserted in the context (u,u)). A strict subfamily of the context-sensitive family is obtained, incomparable with the family of linear languages. We prove here that each recursively enumerable language can be written as the weak coding of the image by an inverse morphism of a language generated by an insertion grammar (with the maximal length of stings u, u as above equal to seven). This result is rather surprising in view of some closure properties established earlier in the literature. Some consequences of this result are also stated. When also erasing rules of the form uxu + uu are present (the string x is erased from the context (u,u)), then a much easier representation of recursively enumerable languages is obtained, as the intersection with V* of a language generated by an insertion grammar with erased strings (having the maximal length of strings u, u as above equal to two).

A Decidable Fragment of the Elementary Theory of the Lattice of Recursively Enumerable Sets

Transactions of the American Mathematical Society, 1980

A natural class of sentences about the lattice of recursively enumerable sets modulo finite sets is shown to be decidable. This class properly contains the class of sentences previously shown to be decidable by Lachlan. New structure results about the lattice of recursively enumerable sets are proved which play an important role in the decision procedure.

Characterizations of Recursively Enumerable Languages by Means of Insertion Grammars

Theoretical Computer Science, 1998

An insertion grammar is based on pure rules of the form uv → uxv (the string x is inserted in the context (u, v)). A strict subfamily of the context-sensitive family is obtained, incomparable with the family of linear languages. We prove here that each recursively enumerable language can be written as the weak coding of the image by an

Characterizations of recursively enumerable languages starting from internal contextual languages

Definability of r. e. sets in a class of recursion theoretic structures

Journal of Symbolic Logic, 1983

It is known [4, Theorem ll-X(b)] that there is only one acceptable universal function up to recursive isomorphism. It follows from this that sets definable in terms of a universal function alone are specified uniquely up to recursive isomorphism. (An example is the set K, which consists of all n such that {n}{n) is defined, where An, m{n}(m) is an acceptable universal function.) Many of the interesting sets constructed and studied by recursion theorists, however, have definitions which involve additional notions, such as a specific enumeration of the graph of a universal function. In particular, many of these definitions make use of the interplay between the purely number-theoretic properties of indices of partial recursive functions and their purely recursion-theoretic properties. This paper concerns r.e. sets that can be defined using only a universal function and some purely number-theoretic concepts. In particular, we would like to know when certain recursion-theoretic properties of r.e. sets definable in this way are independent of the particular choice of universal function (equivalently, independent of the particular way in which godel numbers are identified with natural numbers). We will first develop a suitable model-theoretic framework for discussing this question. This will enable us to classify the formulas defining r.e. sets by their logical complexity. (We use the number of alternations of quantifiers in the prenex form of a formula as a measure of logical complexity.) We will then be able to examine the question at each level. This work is an approach to the question of when the recursion-theoretic properties of an r.e. set are independent of the particular parameters used in its construction. As such, it does not apply directly to the construction techniques most commonly used at this time for defining particular r.e. sets, e.g., the priority method. A more direct attack on this question for these techniques is represented by such works as [3] and [5]. However, the present work should be of independent interest to the logician interested in recursion theory. Furthermore, it may be that a final answer to this question will involve techniques similar to those of the present work. §1. Preliminaries. An acceptable structure is a structure {N, *), where N is the

An Isomorphism Between Monoids of External Embeddings: About Definability in Arithmetic

Journal of Symbolic Logic, 2002

We use a new version of the Definability Theorem of Beth in order to unify classical Theorems of Yuri Matiyasevich and Jan Denef in one structural statement. We give similar forms for other important definability results from Arithmetic and Number Theory. A.M.S. Classification: Primary 03B99; Secondary 11D99. There is a relation with exponential increment

Enumeration Reducibility and Computable Structure Theory

2017

The relationship between enumeration degrees and abstract models of computability inspires a new direction in the field of computable structure theory. Computable structure theory uses the notions and methods of computability theory in order to find the effective contents of some mathematical problems and constructions. The paper is a survey on the computable structure theory from the point of view of enumeration reducibility.

On characterizations of recursively enumerable languages (original) (raw)

Related papers