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.

Further results on infinite valued predicate logic

Abstract: In a previous paper [1] Chang and the present author presented a system of infinite valued predicate logic, the truth values being the closed interval [0, 1] of real numbers. That paper was the result of an investigation attempting to establish the completeness of the system using the real number 1 as the sole designated value. In fact, we fell short of our mark and proved a weakened form of completeness utilizing positive segments, [0, a ], of linearly ordered abelian groups as admissible truth values. A result of Scarpellini [8], however, showing that the set of well-formed formulas of infinite valued logic valid (with respect to the sole designated real number 1) is not recursively enumerable indicates the above mentioned result is the best possible.

A decidable fragment of the elementary theory of the lattice of recursively enumerable sets

Abstract: 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. 0. Introduction. Much of the recent work dealing with S, the lattice of recursively enumerable sets, has dealt with global properties of S such as automorphisms and decidability, rather than local properties of S, i.e., properties of definable classes of recursively enumerable sets. Two of the major results are Lachlan's [3] decision procedure for a natural fragment of the elementary theory of S *, the quotient lattice of S by the ideal of finite sets, Soare's result [15] on the existence of automorphisms carrying any maximal set into any other maximal set. More recently, Shore [13] has determined the definable automorphism bases for £. These global results have inspired new local results, in that they have naturally led to the discovery of important new S-definable classes of recursively enumerable sets whose properties have been investigated. Lachlan's result led to the discovery of small sets, Soare's result led to the discovery of ¿/-simple sets, and Shore's result led to the discovery of nowhere simple sets; the first two of these classes have been studied by Lerman and Soare [8], and the third class by Shore [14]. The class of ¿-simple sets proved to be of particular importance, in that it led to the refutation of conjectures of Martin and Shoenfield which imply that the degrees of elements of any S-definable class can be characterized by a finite set of equalities and inequalities involving the jumps of those degrees. Evidence for these conjectures included Martin's result [9] that a is the degree of a maximal set if and only if a' = 0\", and results of Lachlan [2] and Shoenfield [12] that a is the degree of an atomless set if and only if a\" > 0\". In this paper, we give a decision procedure for a larger fragment of the Received by the editors April 14, 1978 and, in revised form, November 3, 1978. AMS (MOS) subject classifications (1970). Primary 02F25; Secondary 02G05.

An Axiomatization of a First-order Branching Time Temporal Logic

Abstract: We introduce a first-order temporal logic for reasoning about branching time. It is well known that the set of valid formulas is not recursively enumerable and there is no finitary axiomatization. We offer a sound and strongly complete axiomati- zation for the considered logic.

Abstract geometrical computation for Black hole computation (extended abstract)

Abstract: The Black hole model of computation provides super-Turing computing power since it offers the possibility to decide in finite (ob- server's) time any recursively enumerable (R.E.) problem. In this paper, we provide a geometric model of computation, conservative abstract geo- metrical computation, that, although being based on rational numbers, has the same property: it can simulate any Turing machine and can de- cide any R.E. problem through the creation of an accumulation. Finitely many signals can leave any accumulation, and it can be known whether anything leaves. This corresponds to a black hole effect.