Contiguity and distributivity in the enumerable Turing degrees

1997 ◽  
Vol 62 (4) ◽  
pp. 1215-1240 ◽  
Author(s):  
Rodney G. Downey ◽  
Steffen Lempp
Keyword(s):  
Truth Table ◽  
Turing Degrees ◽  
Enumerable Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Degree

AbstractWe prove that a (recursively) enumerable degree is contiguous iff it is locally distributive. This settles a twenty-year old question going back to Ladner and Sasso. We also prove that strong contiguity and contiguity coincide, settling a question of the first author, and prove that no m-topped degree is contiguous, settling a question of the first author and Carl Jockusch [11]. Finally, we prove some results concerning local distributivity and relativized weak truth table reducibility.

Working below a high recursively enumerable degree

1993 ◽  
Vol 58 (3) ◽  
pp. 824-859 ◽  
Author(s):  
Richard A. Shore ◽  
Theodore A. Slaman
Keyword(s):  
Turing Degrees ◽  
Jump Operator ◽  
Recursively Enumerable Set ◽  
Enumerable Degree ◽  
Local Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Degree ◽  
Turing Reducibility ◽  
Definition Of ◽  
The Relationship

In recent work, Cooper [3, 1990] has extended results of Jockusch and Shore [6, 1984] to show that the Turing jump is definable in the structure given by the Turing degrees and the ordering of Turing reducibility. In his definition of x′ from x, Cooper identifies an order-theoretic property shared by all of the degrees that are recursively enumerable in x and above x. He then shows that x′ is the least upper bound of all the degrees with this property. Thus, the jump of x is identified by comparing the recursively enumerable degrees with other degrees which are not recursively enumerable. Of course, once the jump operator is known to be definable, the relation of jump equivalence x′ = y′ is also known to be a definable relation on x and y. If we consider how much of the global theory of the Turing degrees is sufficient for Cooper's methods, it is immediately clear that his methods can be implemented to show that the jump operator and its weakening to the relation of jump equivalence are definable in any ideal closed under the Turing jump. However, his methods do not localize to , the degrees, or to the recursively enumerable degrees.This paper fits, as do Shore and Slaman [16, 1990] and [17, to appear], within the general project to develop an understanding of the relationship between the local degree-theoretic properties of a recursively enumerable set A and its jump class. For an analysis of the possibility of defining jump equivalence in , consult Shore [15, to appear] who shows that the relation x(3) = y(3) is definable. In this paper, we will restrict our attention to definitions expressed completely in ℛ (Note: All sets and degrees discussed for the remainder of this paper will be recursively enumerable.) Ultimately, one would like to find some degree-theoretic properties definable in terms of the ordering of Turing reducibility and quantifiers over the recursively enumerable degrees that would define the relation of jump equivalence or define one or more of the jump classes Hn = {w∣ wn = 0n+1} or Ln = {w ∣ wn = 0n}. Such a result could very likely then be used as a springboard to other general definability results for the recursively enumerable degrees. It would be especially interesting to know whether every recursively enumerable degree is definable and whether every arithmetical degree-invariant property of the recursively enumerable sets is definable in .

Decision problems for tag systems

1971 ◽  
Vol 36 (2) ◽  
pp. 229-239 ◽  
Author(s):  
Stål Aanderaa ◽  
Dag Belsnes
Keyword(s):  
Related Problem ◽  
Decision Problems ◽  
Halting Problem ◽  
Enumerable Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Degree ◽  
Finite Word

The aim of this paper is to study tag systems as defined by Post [Post 1943, pp. 203–205 and Post, 1965, pp. 370–373]. The existence of a tag system with unsolvable halting problem was proved by Minsky by constructing a universal tag system [Minsky 1961, see also Cocke and Minsky 1964, Wang 1963, and Minsky 1967, pp. 267–273]. Hence the halting problem of a tag system can be of the complete degree 0′. We shall prove that the halting problem for a tag system can have an arbitrary (recursively enumerable) degree of undecidability (Corollary III).A related problem arises when we ask if there exists a uniform procedure for determining, given a tag system, whether or not there is any word on which the tag system does not halt, an “immortal” word in the system. The alternative, of course, being that the system eventually halts on every (finite) word. It is shown here that this problem, the immortality problem for tag systems, is recursively unsolvable of degree 0″ (Corollary II).

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

A theorem on minimal degrees

1997 ◽  
Vol 31 (4) ◽  
pp. 539-544 ◽  
Author(s):  
J. R. Shoenfield
Keyword(s):  
Enumerable Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Degree

In their original paper on degrees [3], Kleene and Post showed that there is a degree between 0 and 0′. Later, Friedberg [1] and Muchnik [4] showed that there is a recursively enumerable degree between 0 and 0′. Since then, this phenomenon has been repeated several times: a result has been proved for degrees, and then, after considerable additional effort, it has been proved for recursively enumerable degrees.There are some obvious respects in which that set of all degrees differs from the set of recursively enumerable degrees; e.g., the former is uncountable and has no largest member.

Contiguity and distributivity in the enumerable Turing degrees — Corrigendum

2002 ◽  
Vol 67 (4) ◽  
pp. 1579-1580
Author(s):  
Rodney G. Downey ◽  
Steffen Lempp
Keyword(s):  
Truth Table ◽  
Turing Degree ◽  
Turing Degrees ◽  
Computably Enumerable Set ◽  
Enumerable Degree ◽  
Computably Enumerable Sets ◽  
Image Position ◽  
Computably Enumerable Degree ◽  
Computably Enumerable

A computably enumerable Turing degree a is called contiguous iff it contains only a single computably enumerable weak truth table degree (Ladner and Sasso [2]). In [1], the authors proved that a nonzero computably enumerable degree a is contiguous iff it is locally distributive, that is, for all a1, a2, c with a1 ∪a2 = a and c ≤ a, there exist ci, ≤ ai with c1 ∪ c2 = c.To do this we supposed that W was a computably enumerable set and ∪ a computably set with a Turing functional Φ such that ΦW = U. Then we constructed computably enumerable sets A0, A1 and B together with functionals Γ0, Γ1, Γ, and Δ so thatand so as to satisfy all the requirements below.That is, we built a degree-theoretical splitting A0, A1 of W and a set B ≤TW such that if we cannot beat all possible degree-theoretical splittings V0, V1 of B then we were able to witness the fact that U ≤WW (via Λ).After the proof it was observed that the set U of the proof (page 1222, paragraph 4) needed only to be Δ20. It was then claimed that a consequence to the proof was that every contiguous computably enumerable degree was, in fact, strongly contiguous, in the sense that all (not necessarily computably enumerable) sets of the degree had the same weak truth table degree.

scholarly journals A recursively enumerable degree which will not split over all lesser ones

1976 ◽  
Vol 9 (4) ◽  
pp. 307-365 ◽  
Author(s):  
Alistair H. Lachlan
Keyword(s):  
Enumerable Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Degree

Pairs without infimum in the recursively enumerable weak truth table degrees

1986 ◽  
Vol 51 (1) ◽  
pp. 117-129 ◽  
Author(s):  
Paul Fischer
Keyword(s):  
Structural Difference ◽  
Truth Table ◽  
Turing Degrees ◽  
Recursively Enumerable ◽  
Weak Truth Table Degrees ◽  
Result Theorem ◽  
Turing Reducibility ◽  
Carry Over

wtt-reducibility has become of some importance in the last years through the works of Ladner and Sasso [1975], Stob [1983] and Ambos-Spies [1984]. It differs from Turing reducibility by a recursive bound on the use of the reduction. This makes some proofs easier in the wtt degrees than in the Turing degrees. Certain proofs carry over directly from the Turing to the wtt degrees, especially those based on permitting. But the converse is also possible. There are some r.e. Turing degrees which consist of a single r.e. wtt degree (the so-called contiguous degrees; see Ladner and Sasso [1975]). Thus it suffices to prove a result about contiguous wtt degrees using an easier construction, and it carries over to the corresponding Turing degrees.In this work we prove some results on pairs of r.e. wtt degrees which have no infimum. The existence of such a pair has been shown by Ladner and Sasso. Here we use a different technique of Jockusch [1981] to prove this result (Theorem 1) together with some stronger ones. We show that a pair without infimum exists above a given incomplete wtt degree (Theorem 5) and below any promptly simple wtt degree (Theorem 12). In Theorem 17 we prove, however, that there are r.e. wtt degrees such that any pair below them has an infimum. This shows that certain initial segments of the wtt degrees are lattices. Thus there is a structural difference between the wtt and Turing degrees where the pairs without infimum are dense (Ambos-Spies [1984]).

Lattice embeddings below a nonlow2 recursively enumerable degree

1996 ◽  
Vol 94 (1) ◽  
pp. 221-246 ◽  
Author(s):  
Rod Downey ◽  
Richard A. Shore
Keyword(s):  
Enumerable Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Degree

The undecidability of the Π4-theory for the r.e. wtt and Turing degrees

1995 ◽  
Vol 60 (4) ◽  
pp. 1118-1136 ◽  
Author(s):  
Steffen Lempp ◽  
André Nies
Keyword(s):  
Partial Order ◽  
Bipartite Graphs ◽  
Truth Table ◽  
Turing Degrees ◽  
Coding Scheme ◽  
Recursively Enumerable ◽  
Similar Fact ◽  
Weak Truth Table Degrees

AbstractWe show that the Π4-theory of the partial order of recursively enumerable weak truth-table degrees is undecidable, and give a new proof of the similar fact for r.e. T-degrees. This is accomplished by introducing a new coding scheme which consists in defining the class of finite bipartite graphs with parameters.

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