Showing papers on "Recursively enumerable language published in 1971"

Interpolation and Embedding in the Recursively Enumerable Degrees

Abstract: By a degree is meant a degree of recursive unsolvability. A degree is recursively enumerable (r.e.) just if it contains a recursively enumerable subset of N, the set of non-negative integers. Two infinite injury priority arguments are presented, in ? 2 and ? 3, which are generalizations of parts of Sacks' proof that the r.e. degrees are dense [9]. These results are in a form sufficiently flexible to admit a variety of applications. It is found, for example, that a degree which is a minimal r.e.-uniform upper bound for a sequence of degrees must be the join of a finite subsequence. By way of contrast, any non-recursive r.e. degree is a minimal upper bound for some strictly ascending sequence of r.e. degrees. It is also found that if a, b are r.e. and a < b, then any countable partial ordering is embeddable in the r.e. degrees between a and b with joins preserved whenever they exist. In ? 4 it is shown that if a' = 0' then there is always such an embedding which also preserves greatest and least elements when they exist. The proof of the latter result is a finite injury priority argument which is more closely related to two splitting theorems of Sacks [8, Th. 2 of ? 5 and Th. 2 of ? 61 than to the preceding results. Our basic notation follows Kleene [4], Kleene and Post [5], and Sacks [8]. A convenient property of Kleene's T-predicates is that if Tn(e, x1, * * * x ,nq y) or Tn(p e xi, ...,xny) holds then U(y)

Ramsey's Theorem does not Hold in Recursive Set Theory

Abstract: Publisher Summary This chapter discusses Ramsey's theorem that does not hold in recursive set theory. The theorem is described, which states that there exists a recursive binary symmetric relation (R) on natural numbers (N) such that no recursively enumerable infinit subset of N is R-homogeneous. The proof of the theorem is based on the existence of two recursively enumerable sets of incomparable degrees of insolvability. The proofs of Ramsey's theorem show that there exist arithmetical R-homogeneous sets for recursive relations R. The existence of an infinite recursively enumerable R -homogeneous set implies that either S 1 is recursive in S 2 or S 2 is recursive in S 1 . There exist recursively enumerable sets of incomparable degrees.

Word Problems and Recursively Enumerable Degrees of Unsolvability

Abstract: In [2] it was shown that for every r.e. (i.e., recursively enumerable) Turing degree of unsolvability D there exists a Thue system with word problem of degree D. In [3] the analogous result was shown for finite presentations of groups. Except for minor slips set aright at the end, this corrective note is concerned only with footnote 8, p. 523 of [2] and footnote 4, p. 50 of [3]. In these footnotes it is claimed that, in effect, somewhat sharper results had been proved, viz., the corresponding results for tt (i.e., unbounded truth-table) degrees instead of Turing degrees. Two points should be made: these footnotes do not bear upon the validity of the body of [2] and [3] as they were not used, indeed not even referred to, elsewhere in these papers; moreover, these stronger results about tt degrees are in fact valid, and for presentations having the particular form discussed. What is seriously amiss is the implied claim that both these tt results could be verified by the "interested reader" once he had understood [2] and [3]. For the weaker result about Thue systems and tt degrees this still seems possible' to the author; but certainly not for the stronger result about finite presentations of groups and tt degrees no matter how dedicated the reader! (In a letter to the present author, A. A. Fridman has explained that his methods of [6] and [7] do not settle the question about finitely presented groups and truth-tables either.) Fortunately all the required arguments have now been supplied by Donald J. Collins in DJC.2' Remark (1) of footnote 8 of [2] claims that Thue system Zs i.e., Z, of [2], is tt-reducible to the r.e. set S of natural numbers. From this follows the existence of a Thue system with word problem of arbitrarily preassigned r.e. tt degree. Remark (1) is correct and can be verified by a straightforward systematic reworking of [2], much like writing a computer program; this

A Minimal Pair of $\Pi^0_1$ Classes

Abstract: A pair of sets (A0, A1) forms a minimal pair if A0 and A1 are nonrecursive, and if whenever a set B is recursive both in A0 and in A1 then B is recursive. C. E. M. Yates [8] and independently A. H. Lachlan [4] proved the existence of a minima] pair of recursively enumerable (r.e.) sets thereby establishing a conjecture of G. E. Sacks [6]. We simplify Lachlan's construction, and then generalize this result by constructing two disjoint pairs of r.e. sets (A0, B0) and (A1B1) such that if C0 separates (A0, A1 and C1 separates (B0, B1), then C0 and C1 form a minimal pair. (We say that C separates (A0, A1) if A0 ⊂ C and C ∩ = ∅.) The question arose in our study of (Turing) degrees of members of certain classes, where we proved the weaker result [2, Theorem 4.1] that the above pairs may be chosen so that C0 and C2 are merely Turing incomparable. (There we used a variation of the weaker result to improve a result of Scott and Tennenbaum that no complete extension of Peano arithmetic has minimal degree.)

The generative capacity of transformational grammars of ginsburg and partee

Abstract: Any recursively enumerable language is generated by a transformational grammar with a type 3 base of trees and with no ordering or control device for the transformations. Furthermore, the set of transformation rules depends on the alphabet alone.

Some Theorems on $R$-Maximal Sets and Major Subsets of Recursively Enumerable Sets

Abstract: Introduction. In [5], we studied the relational systems 1/J obtained from the recursive functions of one variable by identifying two such functions if they are equal for all but finitely many x E A, where A is an r-cohesive set. The relational systems R/A with addition and multiplication defined pointwise on them, were once thought to be potential candidates for nonstandard models of arithmetic. This, however, turned out not to be the case, as was shown by Feferman, Scott, and Tennenbaum [1]. We showed, letting A and B be r-maximal sets, and letting X denote the complement of X, that 1/A. and .?/A are elementarily equivalent (9/A-_= Mfg) if there are r-maximal supersets C and D of A and B respectively such that C and D have the same many-one degree (C =m D). In fact, if A and B are maximal sets, R/T _=/A if, and only if, A =m B. We wish to study the relationship between the elementary equivalence of f/.T and s/A, and the Turing degrees of A and B. If we restrict A and B to be maximal sets, it is clear from the above-mentioned results that if Rf/X and ?!/A are elementarily equivalent, the A and B have the same many-one degree, and hence the same Turing degree. The converse, however, is not true, as we showed in [6]; there we proved that if a is the Turing degree of a maximal set, then there are infinitely many maximal sets M1, M2, . of Turing degree a but of pairwise distinct many-one degrees. Hence they generate relational systems R/M1, i = 1, 2, . . . with pairwise distinct elementary theories. The question thus arises as to what happens when we restrict A and B to rmaximal, nonmaximal sets. Such sets fall into two classes; r-maximal sets with a maximal superset, and r-maximal sets with no maximal superset. We show that for any Turing degrees a and b of r-maximal sets, we can get r-maximal sets A and B restricted to either of these classes, such that A has Turing degree a, B has Turing degree b, and J/. -s/A. Also, for a fixed Turing degree a of an r-maximal set with a maximal superset, we show that there are infinitely many r-maximal sets with maximal supersets AO, A1, such that if Al # AS, then. A/A1 # .1ji. Finally, we show that if a is any Turing degree of an r-maximal set, there are infinitely many r-maximal sets with no maximal supersets, all of Turing degree a, but all having different many-one degrees. For the sake of completeness, we will define some of the terms we have used above, or will be using later on. Let T1 be the predicate with the same name as defined in Kleene [3]. A set A is cohesive (r-cohesive) if it is infinite, and there is no recursively enumerable (recursive) set B such that A n B and A n B are both

Languages for defining sets in arbitrary algebras

Abstract: This paper presents a self-contained and more elementary treatment of our mathematical theory of the syntax and semantics of language developed in [W-1] and [ W-2]. It applies this theory to the definition of subsets, and operators on subsets of the carrier of algebras. We show how regular and context-free sets of strings, recognizable sets of trees, and recursively enumerable (r.e.) sets of natural numbers or strings can be defined in a "natural" algebraic manner which defines "similar" types of sets for arbitrary algebras. We employ our mathematical framework to develop semantic and syntactic normal form theorems which explicate the relationship between different languages which define the same classes of sets and operators. We also investigate the relationship between our languages and the earlier work of Mezei, Eilenberg and Wright [M-W], [E-W] and the work of Eilenberg and Elgot [E-E].

Intersection-closed full AFL and the recursively enumerable languages

Abstract: @ a* and the ratio of the number of words in L of length less than n to n goes to 1 as [equation], then [equation] does not contain all recursively enumerable languages.

Decision problems for tag systems

Abstract: The aim of this paper is to study tag systems as defined by Post [Post 1943, pp. 203–205 and Post, 1965, pp. 370–373]. The existence of a tag system with unsolvable halting problem was proved by Minsky by constructing a universal tag system [Minsky 1961, see also Cocke and Minsky 1964, Wang 1963, and Minsky 1967, pp. 267–273]. Hence the halting problem of a tag system can be of the complete degree 0′. We shall prove that the halting problem for a tag system can have an arbitrary (recursively enumerable) degree of undecidability (Corollary III).A related problem arises when we ask if there exists a uniform procedure for determining, given a tag system, whether or not there is any word on which the tag system does not halt, an “immortal” word in the system. The alternative, of course, being that the system eventually halts on every (finite) word. It is shown here that this problem, the immortality problem for tag systems, is recursively unsolvable of degree 0″ (Corollary II).

The family of all recursively enumerable classes of finite sets

Abstract: We prove that if P(x) is any first-order arithmetical predicate which enumerates the family Fin of all r.e. classes of finite sets, then P(x) must reside in a level of the Kleene hierarchy at least as high as II3 -O. (It is more easily established that some of the predicates P(x) which enumerate Fin do lie in HO -EO.)

A Note on Arithmetical Sets of Indiscernibles

Abstract: Publisher Summary This chapter provides a note on arithmetical sets of indiscernibles. It has been proved that there is a recursive partition (of the set N of natural numbers) that possesses no infinite recursively enumerable (Σ 0 1 ) set of indiscernibles; the existence of some infinite set of indiscernibles is the familiar theorem of Ramsey. The chapter presents an entirely different proof based on the theory of retraceable sets. A set is called Σ 0 n , Π 0 n , Δ 0 n if it is definable in prenex normal form with n alternating quantifiers where the first quantifier can be chosen to be respectively existential, universal or both. This general procedure is referred to as the priority method, which could be more accurately replaced by the approximation method or the trial-and error method.