Showing papers on "Recursively enumerable language published in 1973"

Mitotic Recursively Enumerable Sets

Abstract: A recursively enumerable (r.e.) set is mitotic if it is the disjoint union of two r.e. sets both of the same degree of unsolvability. A. H. Lachlan has shown in [3] that there exists a nonmitotic r.e. set. In this paper we make an initial investigation into the class of mitotic sets. The following results are proved. (i) An r.e. set is mitotic if and only if it is autoreducible. (ii) There is a nonmitotic r.e. set of degree 0'. (iii) If d is an arbitrary nonrecursive r.e. degree then there exists a nonmitotic r.e. set of degree < d. (iv) There exists a maximal set which is mitotic and a maximal set which is nonmitotic. Albert R. Meyer had independently proved (ii) and (iii) for nonautoreducible sets before (i) was known. We mention one further result which is not included here but which will appear at a later date [10]. There exists a nonrecursive r.e. degree d such that every r.e. set of degree d is mitotic.

On Complexity Properties of Recursively Enumerable Sets

Abstract: An important goal of complexity theory, as we see it, is to characterize those partial recursive functions and recursively enumerable sets having some given complexity properties, and to do so in terms which do not involve the notion of complexity. As a contribution to this goal, we provide characterizations of the effectively speedable, speedable and levelable [2] sets in purely recursive theoretic terms. We introduce the notion of subcreativeness and show that every program for computing a partial recursive function f can be effectively speeded up on infinitely many integers if and only if the graph of f is subcreative. In addition, in order to cast some light on the concepts of effectively speedable, speedable and levelable sets we show that all maximal sets are levelable (and hence speedable) but not effectively speedable and we exhibit a set which is not levelable in a very strong sense but yet is effectively speedable.

On Recursive Unsolvability of Hilbert's Tenth Problem

Abstract: Publisher Summary This chapter discusses recursive unsolvability of Hilbert's tenth problem The tenth problem is the only one of the 23 problems that clearly has an algorithmical nature The problem proved to be rather difficult and only last year it was shown to be unsolvable There is no algorithm for determining whether an arbitrary diophantine equation has a solution The chapter discusses history of how the unsolvability of Hilbert's tenth problem was proved Every recursively enumerable (re) predicate is diophantine The classification of arithmetical formulas with bounded universal quantifiers is defined

Intersection-closed full AFL and the recursively enumerable languages*°

Abstract: A study is made of conditions on a language L which ensure that the smallest intersection-closed full AFL containing L (written ℱ ^ ∩ ( L ) ) does or does not contain all recursively enumerable languages. For example, it is shown that if L = {ani/j⩾0} and limi→∞ inf(ni+1/ni) > 1, then ℱ ^ ∩ ( L ) contains all recursively enumerable languages. On the other hand, it is shown that if L ⊆ a* and the ratio of the number of words in L of length less than n to n goes to 1 as n → ∞, then ℱ ^ ∩ ( L ) does not contain all recursively enumerable languages.

Combinatorial systems with axiom

Abstract: Introduction:* The purpose of this paper is to investigate the general decision problems associated with a number of combinatorial systems with axiom. In particular, we shall show the many-one equivalence of the general halting problem for Turing machines, the general decision problem for Thue systems with axiom, the general decision problem for semi-Thue systems with axiom, and the general decision problem for Post normal systems with axiom. This, combined with a recent result of Overbeek [5], shows that every recursively enumerable (r.e.) many-one degree (of unsolvability) is represented by each of these general problems for systems with axiom. Finally, this latter result is proven to be best possible in that it does not hold for every r.e. one-one degree. Historical Background: Semi-Thue systems, Thue systems and Post normal systems were defined by Post as proper subsets of canonical forms. Decision problems associated with these systems have been studied by various authors, e.g., [l], [2], [3], [4], [6], [7], and [8]. In particular, W. E. Singletary [7] has combined results of his own and those of others in such a way as to provide an effective proof of the (r.e.) equivalence of the general decision problems which are of concern to us here. The stronger results to be proven here were announced in [4] and form part of an extensive study into the equivalence of general combinatorial decision problems. Preliminary Definitions: A semi-Thue system T is a pair (Σ,R) where Σ is a finite alphabet and R is a finite set of productions of the form a —» β, for a and β words over Σ. T is said to be a Thue system if a -* β belongs to R implies β —> a is also in R. For any arbitrary pair of words W, W r over Σ, we say that Wr is an immediate successor of W in T, denoted T(W, W) if W = P a Q, W = P β Q, P and Q are words over Σ, and a — β is inR.

Post's problem and his hypersimple set

Abstract: A standard enumeration of the recursively enumerable (r.e.) sets is an acceptable numbering { W n } n ∈ N of the r.e. sets in the sense of Rogers [5, p. 41], together with a 1:1 recursive function f with range In his quest for nonrecursive incomplete r.e. sets Post [4] constructed a hypersimple set H f , relative to a fixed but unspecified standard enumeration f . Although it was later shown that hyper-simplicity does not guarantee incompleteness, the ironic possibility remained that Post's own particular hypersimple set might be incomplete. We settle the question by proving that H , may be either complete or incomplete depending upon which standard enumeration f is used. In contrast, D. A. Martin has shown [3] that Post's simple set S [4, p. 298] is complete for any standard enumeration. Furthermore, what most modern recursion theorists would regard as the “natural” construction of a hypersimple set (which we give in §1) is also complete for any standard enumeration. There are two conclusions to be drawn from these results. First, they substantiate the often repeated remark among recursion theorists that Post's hypersimple set construction is a precursor of priority constructions because priorities play a strong role, and because there is a great deal of “restraint” which tends to keep elements out of the set. Secondly, the results warn recursion theorists that more properties than might have been supposed depend upon which standard enumeration is chosen at the beginning of the construction of some r.e. set.

Existence of families without positive numerations

Abstract: An example is constructed of a computable family of recursively enumerable sets which does not have computable positive numerations but does possess a computable minimal numeration.

Truth tabular degrees of recursively enumerable sets

Abstract: The upper semilattice of truth tabular degrees of recursively enumerable (r.e.) sets is studied. It is shown that there exists an infinite set of pairwise tabularly incomparable truth tabular degrees higher than any tabularly incomplete r.e. truth tabular degree. A similar assertion holds also for r.e. m-degrees. Hence follows that a complete truth tabular degree contains an infinite antichain of r.e. m-degrees.

Solving Diophantine Equations

Abstract: Publisher Summary This chapter discusses diophantine equations. It is shown that every recursively enumerable set is diophantine and hence that there is no algorithm for telling whether an arbitrary diophantine equation has a solution. The set of solvable (in N) diophantine equations is recursively enumerable but not recursive, the problem is to find ways of proving equations are unsolvable and of characterizing classes of equations for which there is a decision method. By the usual fixed-point argument, other unsolvable equations from any given recursively enumerable set of unsolvable diophantine equations can be constructed. This method, at least, has no parallel in classical number theory.

Pointwise decomposable sets

Abstract: We show that, under the conditionala′<0″, every recursively enumerable (r.e.) Ae bia has a pointwise decomposable complement. If A ≤ TB, A and ¯B are r.e. co-retraceable sets, and f(x)=fB(x), then there exists a r.e. co-retraceable C, such thatA∼(c),B≡T C , (A n) (f(n)