A hierarchy of families of recursively enumerable degrees

1984 ◽  
Vol 49 (4) ◽  
pp. 1160-1170 ◽  
Author(s):  
Lawrence V. Welch
Keyword(s):  
Finite Sets ◽  
Recursively Enumerable ◽  
Similar Theorem ◽  
Recursively Enumerable Sets ◽  
Complete Set ◽  
Image Position

Certain investigations have been made concerning the nature of classes of recursively enumerable sets, and the relation of such classes to the recursively enumerable indices of their sets. For instance, a theorem of Rice [3, Theorem XIV(a), p. 324] states that if A is the complete set of indices for a class of recursively enumerable sets (that is, if there is a class of recursively enumerable sets such that and if A is recursive, then either A = ⌀ or A = ω. A relate theorem by Rice and Shapiro [3, Theorem XIV(b), p. 324] can be stated as follows:Let be a class of recursively enumerable sets, and let A be the complete set of indices for . Then A is r.e. if and only if there is an r.e. set D of canonical indices of finite sets Du, u ∈ D, such thatA somewhat similar theorem of Yates is the following: Let be a class of recursively enumerable sets which contains all finite sets. Let A be the complete set of indices for . Then there is a uniform recursive enumeration of the sets in if and only if A is recursively enumerable in 0(2)—that is, if and only if A is Σ3. A corollary of this is that if C is any r.e. set such that C(2)≡T⌀(2), there is a uniform recursive enumeration of all sets We such that We ≤TC [9, Theorem 9, p. 265].

Rice Theorems for ∑n-1 Sets

1977 ◽  
Vol 29 (4) ◽  
pp. 794-805 ◽  
Author(s):  
Nancy Johnson
Keyword(s):  
Finite Sets ◽  
Index Set ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
The Difference

In [3] Hay proves generalizations of Rice's Theorem and the Rice-Shapiro Theorem for differences of recursively enumerable sets (d.r.e. sets). The original Rice Theorem [5, p. 3G4, Corollary B] says that the index set of a class C of r.e. sets is recursive if and only if C is empty or C contains all r.e. sets. The Rice-Shapiro Theorem conjectured by Rice [5] and proved independently by McNaughton, Shapiro, and Myhill [4] says that the index set of a class C of r.e. sets is r.e. if and only if C is empty or C consists of all r.e. sets which extend some element of a canonically enumerable class of finite sets. Since a d.r.e. set is a difference of r.e. sets, a d.r.e. set has an index associated with it, namely, the pair of indices of the r.e. sets of which it is the difference.

Types of simple α-recursively enumerable sets

1976 ◽  
Vol 41 (2) ◽  
pp. 419-426
Author(s):  
Manuel Lerman
Keyword(s):  
Lattice Theory ◽  
Regular Cardinal ◽  
Finite Sets ◽  
First Order ◽  
Recursively Enumerable ◽  
Regular Cardinals ◽  
Constructible Sets ◽  
Set Inclusion ◽  
Order Language ◽  
Recursively Enumerable Sets

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 Σ1 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 = ℵ1L (L is Gödel's universe of constructible sets). We will show that for all α which are regular cardinals of L (ℵ1L 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.

Nowhere simple sets and the lattice of recursively enumerable sets

1978 ◽  
Vol 43 (2) ◽  
pp. 322-330 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Boolean Algebra ◽  
Classification Scheme ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
The Other ◽  
Finite Sets ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Simple Sets ◽  
Invariant Properties

Ever since Post [4] the structure of recursively enumerable sets and their classification has been an important area in recursion theory. It is also intimately connected with the study of the lattices and of r.e. sets and r.e. sets modulo finite sets respectively. (This lattice theoretic viewpoint was introduced by Myhill [3].) Key roles in both areas have been played by the lattice of r.e. supersets, , of an r.e. set A (along with the corresponding modulo finite sets) and more recently by the group of automorphisms of and . Thus for example we have Lachlan's deep result [1] that Post's notion of A being hyperhypersimple is equivalent to (or ) being a Boolean algebra. Indeed Lachlan even tells us which Boolean algebras appear as —precisely those with Σ3 representations. There are also many other simpler but still illuminating connections between the older typology of r.e. sets and their roles in the lattice . (r-maximal sets for example are just those with completely uncomplemented.) On the other hand, work on automorphisms by Martin and by Soare [8], [9] has shown that most other Post type conditions on r.e. sets such as hypersimplicity or creativeness which are not obviously lattice theoretic are in fact not invariant properties of .In general the program of analyzing and classifying r.e. sets has been directed at the simple sets. Thus the subtypes of simple sets studied abound — between ten and fifteen are mentioned in [5] and there are others — but there seems to be much less known about the nonsimple sets. The typologies introduced for the nonsimple sets begin with Post's notion of creativeness and add on a few variations. (See [5, §8.7] and the related exercises for some examples.) Although there is a classification scheme for r.e. sets along the simple to creative line (see [5, §8.7]) it is admitted to be somewhat artificial and arbitrary. Moreover there does not seem to have been much recent work on the nonsimple sets.

Congruence relations, filters, ideals, and definability in lattices of α-recursively enumerable sets

1976 ◽  
Vol 41 (2) ◽  
pp. 405-418
Author(s):  
Manuel Lerman
Keyword(s):  
Congruence Relation ◽  
Lattice Theory ◽  
Elementary Theory ◽  
Finite Sets ◽  
Recursively Enumerable ◽  
Set Inclusion ◽  
Recursively Enumerable Sets ◽  
Congruence Relations ◽  
Elementary Theories

Throughout this paper, α will denote an admissible ordinal. Let (α) denote the lattice of α-r.e. sets, i.e., the lattice whose elements are the α-r.e. sets, and whose ordering is given by set inclusion. Call a set A ∈ (α)α*-finite if it is α-finite and has ordertype < α* (the Σ1-projectum of α). The α*-finite sets form an ideal of (α), and factoring (α) by this ideal, we obtain the quotient lattice *(α).We will fix a language ℒ suitable for lattice theory, and discuss decidability in terms of this language. Two approaches have succeeded in making some progress towards determining the decidability of the elementary theory of (α). Each approach was first used by Lachlan for α = ω. The first approach is to relate the decidability of the elementary theory of (α) to that of a suitable quotient lattice of (α) by a congruence relation definable in ℒ. This technique was used by Lachlan [4, §1] to obtain the equidecidability of the elementary theories of (ω) and *(ω), and was generalized by us [6, Corollary 1.2] to yield the equidecidability of the elementary theories of (α) and *(α) for all α. Lachlan [3] then adopted a different approach.

Index sets in Ershov's hierarchy

1974 ◽  
Vol 39 (1) ◽  
pp. 97-104 ◽  
Author(s):  
Jacques Grassin
Keyword(s):  
Recursion Theory ◽  
Case Note ◽  
Boolean Combination ◽  
Open Set ◽  
Recursively Enumerable ◽  
Power Set ◽  
Index Sets ◽  
Recursively Enumerable Sets ◽  
Complete Set ◽  
Recursive Isomorphism

This work is an attempt to characterize the index sets of classes of recursively enumerable sets which are expressible in terms of open sets in the Baire topology on the power set of the set N of natural numbers, usual in recursion theory. Let be a class of subsets of N and be the set of indices of recursively enumerable sets Wх belonging to .A well-known theorem of Rice and Myhill (cf. [5, p. 324, Rice-Shapiro Theorem]) states that is recursively enumerable if and only ifis a r.e. open set. In this case, note that if is not empty and does not contain all recursively enumerable sets, is a complete set. This theorem will be partially extended to classes which are boolean combinations of open sets by the following:(i) There is a canonical boolean combination which represents, namely the shortest among boolean combinations which represent.(ii) The recursive isomorphism type of depends on the length n of this canonical boolean combination (and trivial properties of ); for instance, is recursively isomorphic (in the particular case where is a boolean combination of recursive open sets) to an elementary set combination Yn or Un, constructed from {х ∣ х Wх) and depending on the length n. We can say also that is a complete set in the sense of Ershov's hierarchy [1] (in this particular case).

Three theorems on recursive enumeration. I. Decomposition. II. Maximal set. III. Enumeration without duplication

1958 ◽  
Vol 23 (3) ◽  
pp. 309-316 ◽  
Author(s):  
Richard M. Friedberg
Keyword(s):  
Finite Number ◽  
Finite Sets ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Prime Factors

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 Gödel 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.

Partial degrees and the density problem. Part 2: The enumeration degrees of the Σ2 sets are dense

1984 ◽  
Vol 49 (2) ◽  
pp. 503-513 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Density Theorem ◽  
Turing Degrees ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Recursively Enumerable ◽  
Recursive Sequences ◽  
Recursively Enumerable Sets ◽  
Finite Set ◽  
Image Position ◽  
Density Problem

As in Rogers [3], we treat the partial degrees as notational variants of the enumeration degrees (that is, the partial degree of a function is identified with the enumeration degree of its graph). We showed in [1] that there are no minimal partial degrees. The purpose of this paper is to show that the partial degrees below 0′ (that is, the partial degrees of the Σ2 partial functions) are dense. From this we see that the Σ2 sets play an analagous role within the enumeration degrees to that played by the recursively enumerable sets within the Turing degrees. The techniques, of course, are very different to those required to prove the Sacks Density Theorem (see [4, p. 20]) for the recursively enumerable Turing degrees.Notation and terminology are similar to those of [1]. In particular, We, Dx, 〈m, n〉, ψe are, respectively, notations for the e th r.e. set in a given standard listing of the r.e. sets, the finite set whose canonical index is x, the recursive code for (m, n) and the e th enumeration operator (derived from We). Recursive approximations etc. are also defined as in [1].Theorem 1. If B and C are Σ2sets of numbers, and B ≰e C, then there is an e-operator Θ withProof. We enumerate an e-operator Θ so as to satisfy the list of conditions:Let {Bs ∣ s ≥ 0}, {Cs ∣ s ≥ 0} be recursive sequences of approximations to B, C respectively, for which, for each х, х ∈ B ⇔ (∃s*)(∀s ≥ s*)(х ∈ Bs) and х ∈ C ⇔ (∃s*)(∀s ≥ s*)(х ∈ Cs).

Some representations of Diophantine sets

1972 ◽  
Vol 37 (3) ◽  
pp. 572-578 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Universal Quantifier ◽  
Natural Numbers ◽  
Recursively Enumerable Set ◽  
Minor Error ◽  
Integer Coefficients ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Image Position ◽  
A Minor

A set D of natural numbers is called Diophantine if it can be defined in the formwhere P is a polynomial with integer coefficients. Recently, Ju. V. Matijasevič [2], [3] has shown that all recursively enumerable sets are Diophantine. From this, it follows that a bound for n may be given.We use throughout the logical symbols ∧ (and), ∨ (or), → (if … then …), ↔ (if and only if), ⋀ (for every), and ⋁ (there exists); negation does not occur explicitly. The variables range over the natural numbers 0,1,2,3, …, except as otherwise noted.It is the purpose of this paper to show that if we do not insist on prenex form, then every Diophantine set can be defined existentially by a formula in which not more than five existential quantifiers are nested. Besides existential quantifiers, only conjunctions are needed. By Matijasevič [2], [3], the representation extends to all recursively enumerable sets. Using this, we can find a bound for the number of conjuncts needed.Davis [1] proved that every recursively enumerable set of natural numbers can be represented in the formwhere P is a polynomial with integer coefficients. I showed in [5] that we can take λ = 4. (A minor error is corrected in an Appendix to this paper.) By the methods of the present paper, we can again obtain this result, and indeed in a stronger form, with the universal quantifier replaced by a conjunction.

MINIMAL COOPERATION IN SYMPORT/ANTIPORT TISSUE P SYSTEMS

2007 ◽  
Vol 18 (01) ◽  
pp. 163-179 ◽  
Author(s):  
ARTIOM ALHAZOV ◽  
YURII ROGOZHIN ◽  
SERGEY VERLAN
Keyword(s):  
P Systems ◽  
Finite Sets ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Deterministic Behavior

We investigate tissue P systems with symport/antiport with minimal cooperation, i.e., when only 2 objects may interact. We show that 2 cells are enough in order to generate all recursively enumerable sets of numbers. Moreover, constructed systems simulate register machines and have purely deterministic behavior. We also investigate systems with one cell and we show that they may generate only finite sets of numbers.

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