Showing papers on "Recursively enumerable language published in 1976"

The Infinite Injury Priority Method

Abstract: One of the most important and distinctive tools in recursion theory has been the priority method whereby a recursively enumerable (r.e.) set A is constructed by stages to satisfy a sequence of conditions {R,},ncw called requirements. If n s be undone for the sake of R, thereby injuring Rm at stage t. The first priority method was invented by Friedberg [2] and Muchnik [11] to solve Post's problem and is characterized by the fact that each requirement is injured at most finitely often. Shoenfield [20, Lemma 1], and then independently Sacks [17] and Yates [25] invented a much more powerful method in which a requirement may be injured infinitely often, and the method was applied and refined by Sacks [15], [16], [17], [18], [19] and Yates [25], [26] to obtain many deep results on r.e. sets and their degrees. In spite of numerous simplifications and variations this infinite injury method has never been as well understood as the finite injury method because of its apparently greater complexity. The purpose of this paper is to reduce the Sacks method to two easily understood lemmas whose proofs are very similar to the finite injury case. Using these lemmas we can derive all the results of Sacks on r.e. degrees, and some by Yates and Robinson as well, in a manner accessible to the nonspecialist. The heart of the method is an ingenious observation of Lachlan [7] which is combined with a further simplification of our own. The reader need have no prior knowledge of priority arguments for in ?1 we review the finite injury method using a version invented by Sacks for his Splitting Theorem [15]. In ?2 we discuss the two principal obstacles in extending the strategy to the infinite injury case. We show how the obvious and well-known solution to the first obstacle has automatically solved the second and more fundamental one. We then prove the two main lemmas upon which all of the theorems depend, and from these we prove the Thickness Lemma of Shoenfield [21, p. 83]. In ?3 we apply the method to derive the Yates Index Set Theorem, and results of

Iterated a-NGSM maps and Γ systems*

Abstract: The notion of an iterated Γ map, that is, a nondeterministic generalized sequential machine with accepting states, is introduced, which leads to the notions of Γ systems and languages. It is shown that the family of Γ languages is equal to the family of recursively enumerable languages. A detailed study is presented of the properties of subfamilies of the family of L languages and their relationship with the various families of L languages is investigated. An example result is that the family of e -free context—sensitive languages is equal to the family of languages generated by extensions of propagating deterministic Γ systems with at most three Γ maps.

Types of simple α-recursively enumerable sets

Abstract: Introduction. One general program of a-recursion theory is to determine as much as possible of the lattice structure of W(a), the lattice of a-r.e. sets under inclusion. It is hoped that structure results will shed some light on whether or not the theory of W(a) is decidable with respect to a suitable language for lattice theory. Fix such a language S. Many of the basic results about the lattice structure involve various sorts of simple a -r.e. sets (we use definitions which are definable in S over W (a)). It is easy to see that simple sets exist for all admissible a. Chong and Lerman [1] have found some necessary and some sufficient conditions for the existence of hhsimple a-r.e. sets, although a complete determination of these conditions has not yet been made. Lerman and Simpson [9] have obtained some partial results concerning r-maximal a-r.e. sets. Lerman [6] has shown that maximal a r.e. sets exist iff a is a certain sort of constructibly countable ordinal. Lerman [5] has also investigated the congruence relations, filters, and ideals of W(a). Here various sorts of simple sets have also proved to be vital tools. The importance of simple a-r.e. sets to the study of the lattice structure of W(a) is hence obvious. Lerman [6, Q22] has posed the following problem: Find an admissible a for which all simple a-r.e. sets have the same 1-type with respect to the language S. The structure of W (a) for such an a would be much less complicated than that of W (w). Lerman [7] showed that such an a could not be a regular cardinal of L. We show that there is no such admissible a. This result is of interest for two reasons: it advances the general program for W(a) by revealing additional information about simple sets, and some new methodology is involved in the proof for the case o-2p(a) < o-2cf(a). The second author [10] developed a new way of assigning priorities in order to do the minimal pair construction in the case left open by Lerman and Sacks [8] when o-2p(a) < o-2cf(a), This priority assignment enables us to do the major subset construction whenever o-2p(a) < o-2cf(a). (The first author did the construction independently using a sequence of tame l2 projections converging to the given 12 projection defined on o-2p(a).) As a by-product of the construction we discover that if o-2p (a) < o-2cf (a), then o-2cf (a) = (o-2p (a))+. We assume that the reader is familiar with L and the basic notions of

Recursively Enumerable Sets

Abstract: In this chapter we shall deal in some detail with the set Σ1 of relations (see 5.24). Such relations are called recursively enumerable for reasons which will shortly become clear. The study of recursively enumerable relations is one of the main branches of recursive function theory. They play a large role in logic. In fact, for most theories the set of Godel numbers of theorems is recursively enumerable. Thus many of the concepts introduced in this section will have applications in our discussion of decidable and undecidable theories in Part III. Unless otherwise stated, the functions in this chapter are unary.

Types of Simple $\alpha$-Recursively Enumerable Sets

Abstract: Let α be an admissible ordinal, and let (α) denote the lattice of α-r.e. sets, ordered by set inclusion. An α-r.e. set A is α*- finite if it is α-finite and has ordertype less than α* (the Σ 1 projectum of α). An a-r.e. set S is simple if (the complement of S ) is not α*-finite, but all the α-r.e. subsets of are α*-finite. Fixing a first-order language ℒ suitable for lattice theory (see [2, §1] for such a language), and noting that the α*-finite sets are exactly those elements of (α), all of whose α-r.e. subsets have complements in (α) (see [4, p. 356]), we see that the class of simple α-r.e. sets is definable in ℒ over (α). In [4, §6, (Q22)], we asked whether an admissible ordinal α exists for which all simple α-r.e. sets have the same 1-type. We were particularly interested in this question for α = ℵ 1 L ( L is Godel's universe of constructible sets). We will show that for all α which are regular cardinals of L (ℵ 1 L is, of course, such an α), there are simple α-r.e. sets with different 1-types. The sentence exhibited which differentiates between simple α-r.e. sets is not the first one which comes to mind. Using α = ω for intuition, one would expect any of the sentences “ S is a maximal α-r.e. set”, “ S is an r -maximal α-r.e. set”, or “ S is a hyperhypersimple α-r.e. set” to differentiate between simple α-r.e. sets. However, if α > ω is a regular cardinal of L , there are no maximal, r -maximal, or hyperhypersimple α-r.e. sets (see [4, Theorem 4.11], [5, Theorem 5.1] and [1,Theorem 5.21] respectively). But another theorem of (ω) points the way.

An α-finite injury method of the unbounded type

Abstract: Let α be an admissible ordinal. In this paper we study the structure of the upper semilattice of α -recursively enumerable degrees. Various results about the structure which are of fundamental importance had been obtained during the past two years (Sacks-Simpson [7], Lerman [4], Shore [9]). In particular, the method of finite priority argument of Friedberg and Muchnik was successfully generalized in [7] to an α -finite priority argument to give a solution of Post's problem for all admissible ordinals. We refer the reader to [7] for background material, and we also follow closely the notations used there. Whereas [7] and [4] study priority arguments in which the number of injuries inflicted on a proper initial segment of requirements can be effectively bounded (Lemma 2.3 of [7]), we tackle here priority arguments in which no such bounds exist. To this end, we focus our attention on the fine structure of L α , much in the fashion of Jensen [2], and show that we can still use a priority argument on an indexing set of requirements just short enough to give us the necessary bounds we seek.

On homomorphic images of rational stochastic languages

Abstract: It is shown that a language is recursively enumerable if and only if it is a homomorphic image of a language belonging to a proper subfamily of all rational stochastic languages. Consequently, the homomorphic images of rational stochastic languages form the full principal AFL of all recursively enumerable languages. If only A-free homomorphisms are used, then the result is an intersection-closed AFL, the languages of which are accepted by deterministic linear bounded automata. Rational multistochastic automata and rational asynchronous stochastic sequential machines are machine models for these two AFLs.

Partially ordered sets of 1-degrees, contained in recursively enumerable m-degrees

Abstract: If A is a recursively enumerable (r.e.) nonrecursive set, then the family of all 1-degrees, contained in the m-degree of the set A, along with the natural relation ~ for the ldegrees, forms a partially ordered set L(A). In the present article, we investigate the structure of L(A) in terms of the properties of the set A. Let N={O,4... I" If A ~N, then f-N\A, and JA[ is the power of the set A. We denote by pf and ~f the domain of values and domain of definition of the one-place function f. We recall that, given any A, L(AI contains a maximal element. We will call aeL(A) a minimal element if it is not the least element in L(A), and, given any ~eL(A), ~ implies ~=a or that b is the least element of L(A). The concept of maximal element is defined in a similar way. We shall say that L(~) is dense if (~a)(~)(O,~EL(~)~a~==,~(3C)(a~c~). If L(A)con~ slats of a single element, then the m-degree of the set A is called irreducible. In this case, the m-degree of A contains only cylinders. It may be mentioned in this connection that simple and pseudosimple sets are not cylinders [i]. Some facts that are easily extracted from familiar results will be stated as propositions. Proposition I. If the m-degree of an r.e. set is not irreducible, then L(A) has no maximal elements and contains both an infinite chain and an infinite number of pairwise noncomparable elements. This follows from the results of [2, 3]. For, it was shown in [2] that, if A is not a cylinder, then A~A is likewise not a cylinder, and ~t~@A 9 When constructing the antichain, essential use was made of the recursive enumerability of the set A [3]. Proposition 2. If L(AJ contains minimal elements, then L(A) has a least element. This is obvious in the case when L(A} contains only one minimal element. If A ~ ~, ~m ~z, the 1-degrees of the sets Bz and B2 are minimal and noncomparable, and ~mBz by means of the general recursive function (g.r.f.) ~ , then let

On the congruence of the upper semilattices of recursively enumerable m-powers and tabular powers

Abstract: We have proved that in the upper semilattice of recursively enumerable tabular powers the set of minimal powers has an upper bound differing from the total power.