A jump class of noncappable degrees

1989 ◽  
Vol 54 (2) ◽  
pp. 324-353 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Minimal Pair ◽  
Recursively Enumerable ◽  
Image Position

Friedberg [3] showed that every degree of unsolvability above 0′ is the jump of some degree, and Sacks [9] showed that the degrees above 0′ which are recursively enumerable (r.e.) in 0′ are the jumps of the r.e. degrees.In this paper we examine the extent to which the Sacks jump theorem can be combined with the minimal pair theorem of Lachlan [4] and Yates [13]. We prove below that there is a degree c > 0′ which is r.e. in 0′ but which is not the jump of half a minimal pair of r.e. degrees.This extends Yates' result [13] proving the existence of noncappable degrees (that is, r.e. degrees a < 0′ for which there is no corresponding r.e. b > 0 with a ∩ b = 0).It also throws more light on the class PS of promptly simple degrees. It was shown by Ambos-Spies, Jockusch, Shore and Soare [1] that PS coincides with the class NC of noncappable degrees, and with the class LC of all low-cuppable degrees, and (using earlier work of Maass, Shore and Stob [5]) that PS splits every class Hn or Ln, n ≥ 0, in the high-low hierarchy of r.e. degrees.If c > 0′, with c r.e. in 0′, letand call c−1 the jump class for c. It is easy to see that every jump class contains members of PS (= NC = LC). By Sacks [8] there exists a low a ∈ LC, where of course [a, 0′] (= {br.e. ∣a ≤ b ≤ 0′}) ⊆ LC = PS. But by Robinson [7] [a, 0′] intersects with every jump class.

An extension of the nondiamond theorem in classical and α-recursion theory

1984 ◽  
Vol 49 (2) ◽  
pp. 586-607 ◽  
Author(s):  
Klaus Ambos-Spies
Keyword(s):  
Boolean Algebra ◽  
Recursion Theory ◽  
Upper Bounds ◽  
Minimal Pair ◽  
Splitting Theorem ◽  
Recursively Enumerable ◽  
Interesting Corollary ◽  
Image Position

Lachlan's nondiamond theorem [7, Theorem 5] asserts that there is no embedding of the four-element Boolean algebra (diamond) in the recursively enumerable degrees which preserves infima, suprema, and least and greatest elements. Lachlan observed that, essentially by relativization, the theorem can be extended toUsing the Sacks splitting theorem he concluded that there exists a pair of r.e. degrees which does not have an infimum, thus showing that the r.e. degrees do not form a lattice.We will first prove the following extension of (1):where an r.e. degree a is non-b-cappable if . From (2) we obtain more information about pairs of r.e. degrees without infima: For every nonzero low r.e. degree there exists an incomparable one such that the two degrees do not have an infimum and there is an r.e. degree which is not half of a pair of incomparable r.e. degrees which has an infimum in the low r.e. degrees. Probably the most interesting corollary of (2) is that the join of any cappable r.e. degree (i.e. half of a minimal pair) and any low r.e. degree is incomplete. Consequently there is an incomplete noncappable degree above every incomplete r.e. degree. Cooper's result [3] that ascending sequences of uniformly r.e. degrees can have minimal upper bounds in the set R of r.e. degrees is another corollary of (2).

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.

Register machine proof of the theorem on exponential diophantine representation of enumerable sets

1984 ◽  
Vol 49 (3) ◽  
pp. 818-829 ◽  
Author(s):  
J. P. Jones ◽  
Y. V. Matijasevič
Keyword(s):  
Natural Number ◽  
Polynomial Time ◽  
Simple Proof ◽  
Diophantine Equations ◽  
Polynomial Equation ◽  
Exponential Diophantine Equations ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Exponential Relation

The purpose of the present paper is to give a new, simple proof of the theorem of M. Davis, H. Putnam and J. Robinson [1961], which states that every recursively enumerable relation A(a1, …, an) is exponential diophantine, i.e. can be represented in the formwhere a1 …, an, x1, …, xm range over natural numbers and R and S are functions built up from these variables and natural number constants by the operations of addition, A + B, multiplication, AB, and exponentiation, AB. We refer to the variables a1,…,an as parameters and the variables x1 …, xm as unknowns.Historically, the Davis, Putnam and Robinson theorem was one of the important steps in the eventual solution of Hilbert's tenth problem by the second author [1970], who proved that the exponential relation, a = bc, is diophantine, and hence that the right side of (1) can be replaced by a polynomial equation. But this part will not be reproved here. Readers wishing to read about the proof of that are directed to the papers of Y. Matijasevič [1971a], M. Davis [1973], Y. Matijasevič and J. Robinson [1975] or C. Smoryński [1972]. We concern ourselves here for the most part only with exponential diophantine equations until §5 where we mention a few consequences for the class NP of sets computable in nondeterministic polynomial time.

Reductions of Hilbert's tenth problem

1958 ◽  
Vol 23 (2) ◽  
pp. 183-187 ◽  
Author(s):  
Martin Davis ◽  
Hilary Putnam
Keyword(s):  
Diophantine Equation ◽  
Hilbert's Tenth Problem ◽  
Recursively Enumerable ◽  
Hilbert’S Tenth Problem ◽  
Essential Change ◽  
Recursive Predicate ◽  
Image Position ◽  
Positive Integers

Hilbert's tenth problem is to find an algorithm for determining whether or not a diophantine equation possesses solutions. A diophantine predicate (of positive integers) is defined to be one of the formwhere P is a polynomial with integral coefficients (positive, negative, or zero). Previous work has considered the variables as ranging over nonnegative integers; but we shall find it more useful here to restrict the range to positive integers, no essential change being thereby introduced. It is clear that the recursive unsolvability of Hilbert's tenth problem would follow if one could show that some non-recursive predicate were diophantine. In particular, it would suffice to show that every recursively enumerable predicate is diophantine. Actually, it would suffice to prove far less.

Solution of a problem of Tarski

1956 ◽  
Vol 21 (1) ◽  
pp. 49-51 ◽  
Author(s):  
John Myhill
Keyword(s):  
Unary Operation ◽  
Negative Answer ◽  
Operation Symbol ◽  
Counter Example ◽  
Recursively Enumerable ◽  
Consistent Extension ◽  
Axiomatizable Theory ◽  
Definition Of ◽  
Image Position ◽  
The Given

We presuppose the terminology of [1], and we give a negative answer to the following problem ([1], p. 19): Does every essentially undecidable axiomatizable theory have an essentially undecidable finitely axiomatizable subtheory?We use the following theorem of Kleene ([2], p. 311). There exist two recursively enumerable sets α and β such that (1) α and β are disjoint (2) there is no recursive set η for which α ⊂ η, β ⊂ η′. By the definition of recursive enumerability, there are recursive predicates Φ and Ψ for whichWe now specify a theory T which will afford a counter-example to the given problem of Tarski. The only non-logical constants of T are two binary predicates P and Q, one unary operation symbol S, and one individual constant 0. As in ([1], p. 52) we defineThe only non-logical axioms of T are the formulae P(Δm, Δn) for all pairs of integers m, n satisfying Δ(m, n); the formulae Q(Δm, Δn) for all pairs of integers m, n satisfying Ψ(m, n); and the formulaT is consistent, since it has a model. It remains to show that (1) every consistent extension of T is undecidable (2) if T1 is a finitely axiomatizable subtheory of T, there exists a consistent and decidable extension of T1 which has the same constants as T1.

Applications of Strict Π11 predicates to infinitary logic

1969 ◽  
Vol 34 (3) ◽  
pp. 409-423 ◽  
Author(s):  
Jon Barwise
Keyword(s):  
Normal Form ◽  
Infinitary Logic ◽  
Natural Numbers ◽  
Recursively Enumerable ◽  
Image Position

Consider the predicate of natural numbers defined by: where R is recursive. If, as usual, the variable ƒ ranges over ωω (the set of functions from natural numbers to natural numbers) then this is just the usual normal form for Π11 sets. If, however, ƒ ranges over 2ω (the set of functions from ω into {0, 1}) then every such predicate is recursively enumerable.3 Thus the second type of formula is generally ignored. However, the reduction just mentioned requires proof, and the proof uses some form of the Brower-König Infinity Lemma.

On a question of G. E. Sacks

1966 ◽  
Vol 31 (1) ◽  
pp. 66-69 ◽  
Author(s):  
Donald A. Martin
Keyword(s):  
Enumerable Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Degree ◽  
Image Position ◽  
Combinatorial Device

In [1], p. 171, Sacks asks (question (Q5)) whether there is a recursively enumerable degree of unsolvability d such that for all n ≧ 0. Sacks points out that the set of conditions which d must satisfy is not arithmetical. For this reason he suggests that a proof of (Q5) might require some new combinatorial device. The purpose of this note is to show how (Q5) may be proved simply by extending the methods of [l].2

scholarly journals ON THE INTERPLAY BETWEEN EFFECTIVE NOTIONS OF RANDOMNESS AND GENERICITY

2019 ◽  
Vol 84 (1) ◽  
pp. 393-407
Author(s):  
LAURENT BIENVENU ◽  
CHRISTOPHER P. PORTER
Keyword(s):  
Random Sequence ◽  
Complete Classification ◽  
Minimal Pair ◽  
Turing Degrees ◽  
Random Oracles ◽  
Image Position

AbstractIn this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence (as shown by Nies, Stephan, and Terwijn). We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence and that every Demuth random sequence forms a minimal pair with every pb-generic sequence (where pb-genericity is an effective notion of genericity that is strictly between 1-genericity and 2-genericity). Moreover, we prove that for every comeager${\cal G} \subseteq {2^\omega }$, there is some weakly 2-random sequenceXthat computes some$Y \in {\cal G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity.

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.

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.

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