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.

An introduction to formal languages and machine computation

Abstract: This book provides an elementary introduction to formal languages and machine computation. The materials covered include computation-oriented mathematics, finite automata and regular languages, push-down automata and context-free languages, Turing machines and recursively enumerable languages, and computability and complexity. As integers are important in mathematics and computer science, the book also contains a chapter on number-theoretic computation. The book is intended for university computing and mathematics students and computing professionals.

A Limit on Relative Genericity in the Recursively Enumerable Sets

Abstract: Work in the setting of the recursively enumerable sets and their Turing degrees. A set X is low if X′, its Turing jump, is recursive in ∅′ and high if X′ computes ∅″. Attempting to find a property between being low and being recursive, Bickford and Mills produced the following definition. W is deep, if for each recursively enumerable set A, the jump of A ⊕ W is recursive in the jump of A. We prove that there are no deep degrees other than the recursive one.Given a set W, we enumerate a set A and approximate its jump. The construction of A is governed by strategies, indexed by the Turing functionals Φ. Simplifying the situation, a typical strategy converts a failure to recursively compute W into a constraint on the enumeration of A, so that (W ⊕ A)′ is forced to disagree with Φ(−;A′). The conversion has some ambiguity; in particular, A cannot be found uniformly from W.We also show that there is a “moderately” deep degree: There is a low nonzero degree whose join with any other low degree is not high.

Partial results regarding word problems and recursively enumerable degrees of unsolvability

Abstract: Introduction and summary of results. With the settling of the Problem of Post by Friedburg and Mucnik a question that naturally presents itself is whether or not unsolvability results about word problems and related problems can be paralleled for arbitrary recursively enumerable degrees of unsolvability, i.e., for any such degree, D, does there exist a problem of such-and-such a kind having degree D? The present results furnish a partial answer to this general question. Throughout our statement of results, existence is intended in the strong sense of the exhibition of a uniform procedure for constructing. Corresponding to a well-known unsolvability result of Markov [12] and Post [20] we have the following. RESULT A. For any recursively enumerable degree of unsolvability, D, there exists a Thue system, XD, such that the word problem for XD is of degree D. Corresponding to an unsolvability result noted by Marshall Hall [7], we have the following. RESULT B. Result A may be strengthened to require that XD be a Thue system on two symbols. The following result corresponds more closely to the unsolvability result of WP, §36, than to the unsolvability of the word problem for groups as usually formulated. RESULT C. For any recursively enumerable degree of unsolvability, D, there exists a group presentation, XD, consisting of a finite number of generators and an infinite but recursive set of defining relations, such that the word problem for XD is of degree D. As elsewhere noted, \"arbitrary degree\" analogues of the Markov1 The author is an Associate Member of the Center for Advanced Study, University of Illinois. This research was supported earlier by the John Simon Guggenheim Memorial Foundation and the United States Office of Naval Research. 2 E. L. Post, Bull. Amer. Math. Soc. 50 (1944), 314, lines 17-22; R. M. Friedburg, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 236-238; A. A. Mucnik, Dokl. Akad. Nauk SSSR (N.S.) 108 (1956), 194-197. 3 «WP\" indicates The Word Problem, Ann. of Math. 70 (1959), 207-265 and numbers in square brackets refer to the bibliography of WP. 4 Meeting of the Association for Symbolic Logic, Leeds, August 1962. Result B and related results were presented to this meeting. Result A, to the International Congress of Mathematicians, Stockholm, August 1962. Result C, to the Internationales Kolloquium Über Endliche Gruppen, Oberwolfach, June, 1960. Result C was discovered independently by C. R. Clapham.

Universal unification and regular equational ACFM theories

Abstract: A Universal minimal type conformal matching algorithm and a universal minimal unification algorithm based on [FA79], [SL74], [LB79], [HU80] are presented for a restricted class of equational theories (the Regular ACFM Theories), i.e. it is shown that the set of most general unifiers is recursively enumerable for this class. The class of Regular ACFM Theories is wide enough to contain most special cases of unification algorithms that have been investigated so far. This paper is a (very) abbreviated version of [SS81C], all proofs and most of the technical material are omitted for lack of space. For reasons of consistency with the original paper, the numbering of definitions, lemmas and theorems has been retained.