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.

Three Theorems on Recursive Enumeration. I. Decomposition. II. Maximal Set. III. Enumeration Without Duplication

Abstract: In this paper we shall prove three theorems about recursively enumerable sets. The first two answer questions posed by Myhill [1].The three proofs are independent and can be presented in any order. Certain notations will be common to all three. We shall denote by “Re” the set enumerated by the procedure of which e is the Godel number. In describing the construction for each proof, we shall suppose that a clerk is carrying out the simultaneous enumeration of R0, R1, R2, …, in such a way that at each step only a finite number of sets have begun to be enumerated and only a finite number of the members of any set have been listed. (One plan the clerk can follow is to turn his attention at Step a to the enumeration of Re where e+1 is the number of prime factors of a. Then each Re receives his attention infinitely often.) We shall denote by “Rea” the set of numbers which, at or before Step a, the clerk has listed as members of Re. Obviously, all the Rea are finite sets, recursive uniformly in e and a. For any a we can determine effectively the highest e for which Rea is not empty, and for any a, e we can effectively find the highest member of Rea, just by scanning what the clerk has done by Step a. Additional notations will be introduced in the proofs to which they pertain.

On degrees of unsolvability

Abstract: In [4], Kleene and Post showed that there are degrees between 0 (the degree of recursive sets) and 0' (the highest degree of recursively enumerable sets). Friedberg [1] and Muchnik [5] showed that there are recursively enumerable degrees (i.e., degrees containing a recursively enumerable set) between 0 and 0'. The question then arises: are there degrees between 0 and 0' which are not recursively enumerable? Since the recursively enumerable sets can be enumerated by a function of degree 0', the existence of such degrees follows from the following theorem, which may be roughly stated for d = 0 as: if a > 0', then the degrees < a cannot be enumerated by a function of degree < a. DEFINITION. A sequence {a,} of functions is uniformly of degree < a if an(x), as a function of (n, x), is of degree < a.

Lower Bounds for Pairs of Recursively Enumerable Degrees

Abstract: The degrees of unsolvability have been extensively studied by Sacks in (4). This paper studies problems concerned with lower bounds of pairs of recursively enumerable (r.e.) degrees. It grew out of an unpublished paper written in June 1964 which presented a proof of the following conjecture of Sacks ((4) 170): there exist two r.e. degrees a, b whose greatest lower bound is 0. This result was first announced by Yates (6); the present author's proof is superficially at least quite different from that of Yates. The author is grateful to Yates for pointing out two errors in the original proof of Lemma 3, and for his careful reading of the whole of the earlier paper. The result already mentioned is Theorem 1 of this paper. As a by-product of the construction we can obtain a' = b' = 0'; Yates has made a similar observation regarding his construction. In Theorem 2 it is proved that there are r.e. degrees a, b whose greatest lower bound is 0 such that a, b are the degrees of maximal r.e. sets. Next, we show that only for some r.e. degrees c in 0 < c < 0' do there exist r.e. degrees a, b such that c is the greatest lower bound of a and b. Finally, we show that if a and b are non-recursive r.e. degrees such that a u b = 0' then there exists a non-recursive r.e. degree c such that c < a and c < b. As a corollary it is shown that the upper semi-lattice of the r.e. degrees is not a lattice, thus verifying another conjecture of Sacks ((4) 170). Before this proof was discovered Sacks himself was developing a proof of the same conjecture. I should like to thank the referee for helpful comments.

Generalized schema-mappings: from termination to tractability

Abstract: Data-Exchange is the problem of creating new databases according to a high-level specification called a schema-mapping while preserving the information encoded in a source database. This paper introduces a notion of generalized schema-mapping that enriches the standard schema-mappings (as defined by Fagin et al) with more expressive power. It then proposes a more general and arguably more intuitive notion of semantics that rely on three criteria: Soundness, Completeness and Laconicity (non-redundancy and minimal size). These semantics are shown to coincide precisely with the notion of cores of universal solutions in the framework of Fagin, Kolaitis and Popa. It is also well-defined and of interest for larger classes of schema-mappings and more expressive source databases (with null-values and equality constraints). After an investigation of the key properties of generalized schema-mappings and their semantics, a criterion called Termination of the Oblivious Chase (TOC) is identified that ensures polynomial data-complexity. This criterion strictly generalizes the previously known criterion of Weak-Acyclicity. To prove the tractability of TOC schema-mappings, a new polynomial time algorithm is provided that, unlike the algorithm of Gottlob and Nash from which it is inspired, does not rely on the syntactic property of Weak-Acyclicity. As the problem of deciding whether a Schema-mapping satisfies the TOC criterion is only recursively enumerable, a more restrictive criterion called Super-weak Acylicity (SwA) is identified that can be decided in Polynomial-time while generalizing substantially the notion of Weak-Acyclicity.