On minimal pairs of enumeration degrees

1985 ◽  
Vol 50 (4) ◽  
pp. 983-1001 ◽  
Author(s):  
Kevin McEvoy ◽  
S. Barry Cooper
Keyword(s):  
Minimal Pair ◽  
Natural Numbers ◽  
Enumeration Degrees ◽  
Minimal Pairs ◽  
Recursively Enumerable ◽  
Unpublished Work ◽  
Lower Case ◽  
Low Degree ◽  
Finite Set ◽  
Partial Orderings

For sets of natural numbers A and B, A is enumeration reducible to B if there is some effective algorithm which when given any enumeration of B will produce an enumeration of A. Gutteridge [5] has shown that in the upper semilattice of the enumeration degrees there are no minimal degrees (see Cooper [3]), and in this paper we study those pairs of degrees with gib 0. Case [1] constructed a minimal pair. This minimal pair construction can be relativised to any gib, and following a suggestion of Jockusch we can also fix one of the degrees and still construct the pair. These methods yield an easier proof of Case's exact pair theorem for countable ideals. 0″ is an upper bound for the minimal pair constructed in §1, and in §2 we improve this bound to any Σ2-high Δ2 degree. In contrast to this we show that every low degree c bounds a degree a which is not in any minimal pair bounded by c. The structure of the co-r.e. e-degrees is isomorphic to that of the r.e. Turing degrees, and Gutteridge has constructed co-r.e. degrees which form a minimal pair in the e-degrees. In §3 we show that if a, b is any minimal pair of co-r.e. degrees such that a is low then a, b is a minimal pair in the e-degrees (and so Gutteridge's result follows). As a corollary of this we can embed any countable distributive lattice and the two nondistributive five-element lattices in the e-degrees below 0′. However the lowness assumption is necessary, as we also prove that there is a minimal pair of (high) r.e. degrees which is not a minimal pair in the e-degrees (under the isomorphism). In §4 we present more concise proofs of some unpublished work of Lagemann on bounding incomparable pairs and embedding partial orderings.As usual, {Wi}i ∈ ω is the standard listing of the recursively enumerable sets, Du is the finite set with canonical index u and {‹ m, n ›}m, n ∈ ω is a recursive, one-to-one coding of the pairs of numbers onto the numbers. Capital italic letters will be variables over sets of natural numbers, and lower case boldface letters from the beginning of the alphabet will vary over degrees.

Bounding minimal pairs

1979 ◽  
Vol 44 (4) ◽  
pp. 626-642 ◽  
Author(s):  
A. H. Lachlan
Keyword(s):  
Minimal Pair ◽  
Minimal Pairs ◽  
Recursively Enumerable ◽  
The Many

A minimal pair of recursively enumerable (r.e.) degrees is a pair of degrees a, b of nonrecursive r.e. sets with the property that if c ≤ a and c ≤ b then c = 0. Lachlan [2] and Yates [4] independently proved the existence of minimal pairs. It was natural to ask whether for an arbitrary nonzero r.e. degree c there is a minimal pair a, b with a ≤ c and b ≤ c. In 1971 Lachlan and Ladner proved that a minimal pair below c cannot be obtained in a uniformly effective way from c for r.e. c ≠ 0. but the result was never published. More recently Cooper [1] showed that if c is r.e. and c′ = 0″ then there is a minimal pair below c.In this paper we prove two results:Theorem 1. There exists a nonzero r.e. degree with no minimal pair below it.Theorem 2. There exists a nonzero r.e. degree c such that, if d is r.e. and 0 < d ≤ c, then there is a minimal pair below d.The second theorem is a straightforward variation on the original minimal pair construction, but the proof of the first theorem has some novel features. After some preliminaries in §1, the first theorem is proved in §2 and the second in §3.I am grateful to Richard Ladner who collaborated with me during the first phase of work on this paper as witnessed by our joint abstract [3]. The many discussions we had about the construction required in Theorem 1 were of great help to me.

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).

Minimal pairs and high recursively enumerable degrees

1974 ◽  
Vol 39 (4) ◽  
pp. 655-660 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Recursive Function ◽  
Uniform Method ◽  
Minimal Pair ◽  
Minimal Pairs ◽  
Recursively Enumerable ◽  
Finite Approximations ◽  
Degree Bounds ◽  
Recursively Enumerable Degree ◽  
Image Position ◽  
High Set

A. H. Lachlan [2] and C. E. M. Yates [4] independently showed that minimal pairs of recursively enumerable (r.e.) degrees exist. Lachlan and Richard Ladner have shown (unpublished) that there is no uniform method for producing a minimal pair of r.e. degrees below a given nonzero r.e. degree. It is not known whether every nonzero r.e. degree bounds a r.e. minimal pair, but in the present paper it is shown (uniformly) that every high r.e. degree bounds a r.e. minimal pair. (A r.e. degree is said to be high if it contains a high set in the sense of Robert W. Robinson [3].)Theorem. Let a be a recursively enumerable degree for which a′ = 0″. Then there are recursively enumerable degrees b0 and b1 such that0 < bi < a for each i ≤ 1, and b0 ⋂ b1 = 0.The proof is based on the Lachlan minimal r.e. pair construction. For notation see Lachlan [2] or S. B. Cooper [1].By Robinson [3] we can choose a r.e. representative A of the degree a, with uniformly recursive tower {As, ∣ s ≥ 0} of finite approximations to A, such that CA dominates every recursive function whereWe define, stage by stage, finite sets Bi,s, i ≤ 1, s ≥ 0, in such a way that Bi, s + 1 ⊇ Bi,s for each i, s, and {Bi,s ∣ i ≤ 1, s ≥ 0} is uniformly recursive.

Bounding and nonbounding minimal pairs in the enumeration degrees

2005 ◽  
Vol 70 (3) ◽  
pp. 741-766 ◽  
Author(s):  
S. Barry Cooper ◽  
Angsheng Li ◽  
Andrea Sorbi ◽  
Yue Yang
Keyword(s):  
Minimal Pair ◽  
The Other ◽  
Enumeration Degrees ◽  
Minimal Pairs ◽  
Other Hand ◽  
Degree Bounds

AbstractWe show that every nonzero Δ20, e-degree bounds a minimal pair. On the other hand, there exist Σ20, e-degrees which bound no minimal pair.

Natural Partial Orders

1968 ◽  
Vol 20 ◽  
pp. 535-554 ◽  
Author(s):  
R. A. Dean ◽  
Gordon Keller
Keyword(s):  
Partial Orders ◽  
Partial Ordering ◽  
Finite Set ◽  
Partial Orderings

Let n be an ordinal. A partial ordering P of the ordinals T = T(n) = {w: w < n} is called natural if x P y implies x ⩽ y.A natural partial ordering, hereafter abbreviated NPO, of T(n) is thus a coarsening of the natural total ordering of the ordinals. Every partial ordering of a finite set 5 is isomorphic to a natural partial ordering. This is a consequence of the theorem of Szpielrajn (5) which states that every partial ordering of a set may be refined to a total ordering. In this paper we consider only natural partial orderings. In the first section we obtain theorems about the lattice of all NPO's of T(n).

On completely recursively enumerable classes and their key arrays

1956 ◽  
Vol 21 (3) ◽  
pp. 304-308 ◽  
Author(s):  
H. G. Rice
Keyword(s):  
Decision Procedure ◽  
Cardinal Number ◽  
Maximum Amount ◽  
Infinite Class ◽  
Finite Sets ◽  
Amount Of Information ◽  
Recursively Enumerable ◽  
Finite Set ◽  
Partial Recursive Functions ◽  
Set Up

The two results of this paper (a theorem and an example) are applications of a device described in section 1. Our notation is that of [4], with which we assume familiarity. It may be worth while to mention in particular the function Φ(n, x) which recursively enumerates the partial recursive functions of one variable, the Cantor enumerating functions J(x, y), K(x), L(x), and the classes F and Q of r.e. (recursively enumerable) and finite sets respectively.It is possible to “give” a finite set in a way which conveys the maximum amount of information; this may be called “giving explicitly”, and it requires that in addition to an effective enumeration or decision procedure for the set we give its cardinal number. It is sometimes desired to enumerate effectively an infinite class of finite sets, each given explicitly (e.g., [4] p. 360, or Dekker [1] p. 497), and we suggest here a device for doing this.We set up an effective one-to-one correspondence between the finite sets of non-negative integers and these integers themselves: the integer , corresponds to the set αi, = {a1, a2, …, an} and inversely. α0 is the empty set. Clearly i can be effectively computed from the elements of αi and its cardinal number.

scholarly journals A class of harmonically convergent sets

1975 ◽  
Vol 20 (3) ◽  
pp. 301-304
Author(s):  
Torleiv Kløve
Keyword(s):  
Natural Numbers ◽  
Lower Case ◽  
Image Position

Following Craven (1965) we say that a set M of natural numbers is harmonically convergent if converges, and we call μ(M) the harmonic sum of M. (Craven defined these concepts for sequences rather than sets, but we found it convenient to work with sets.) Throughout this paper, lower case italics denote non-negative integers.

scholarly journals AL-AKHTHA’ FII IKTISAB AL-TSUNAIYAT AL-SHUGHRA

2011 ◽  
Vol 5 (1) ◽  
Author(s):  
Miftahul Huda
Keyword(s):  
Saudi Arabia ◽  
Language Acquisition ◽  
Foreign Language ◽  
Error Analysis ◽  
Language Learning ◽  
Mother Tongue ◽  
Minimal Pair ◽  
Minimal Pairs ◽  
The Subject ◽  
Child Age

Language acquisition starts from the ability of listening basic letter(iktisab al-ashwat) since child age. The letter of a language is limited in number, and sometimes there is similarity of letters among languages. The similarity of letters in two languages make it easy to learn the language. On the contrary, the obstacle of language learning can be caused by different letters between two languages (mother tongue and second/foreign language). The problem may be caused by minimal pairs (tsunaiyat al-shughro). This research aims at finding out the error of minimal pair acquisition, with the subject of Indonesian students in Jami’ah Malik Saud Saudi Arabia, with the method of error analysis. The study concludes that in iktisab al-ashwat of minimal pairs, the error is around 3,3 %-58,3%. Second: the error on minimal pairs occurs on the letters shift ?? ?? ? to be ? , letter ? to be ? , letter ? to be ? , letter ? to be ? , letter ? to be ?? , and letter ? to be ?.

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