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]).

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.

The theory of the recursively enumerable weak truth-table degrees is undecidable

1992 ◽  
Vol 57 (3) ◽  
pp. 864-874 ◽  
Author(s):  
Klaus Ambos-Spies ◽  
André Nies ◽  
Richard A. Shore
Keyword(s):  
Partial Order ◽  
Open Problem ◽  
Truth Table ◽  
Recursively Enumerable ◽  
Undecidable Theory ◽  
Weak Truth Table Degrees

AbstractWe show that the partial order of -sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.

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.

scholarly journals Infima in the Recursively Enumerable Weak Truth Table Degrees

1997 ◽  
Vol 38 (3) ◽  
pp. 406-418
Author(s):  
Rich Blaylock ◽  
Rod Downey ◽  
Steffen Lempp
Keyword(s):  
Truth Table ◽  
Recursively Enumerable ◽  
Weak Truth Table Degrees

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 .

Decidability of the two-quantifier theory of the recursively enumerable weak truth-table degrees and other distributive upper semi-lattices

1996 ◽  
Vol 61 (3) ◽  
pp. 880-905 ◽  
Author(s):  
Klaus Ambos-Spies ◽  
Peter A. Fejer ◽  
Steffen Lempp ◽  
Manuel Lerman
Keyword(s):  
Decision Procedure ◽  
Finite Lattice ◽  
Truth Table ◽  
Recursively Enumerable ◽  
Finite Lattices ◽  
Recursively Enumerable Sets ◽  
Weak Truth Table Degrees ◽  
Quantifier Theory ◽  
The Ideal

AbstractWe give a decision procedure for the ∀∃-theory of the weak truth-table (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e.wtt-degrees by a map which preserves the least and greatest elements: a finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable elements are disjoint.We formulate general criteria that allow one to conclude that a distributive upper semi-lattice has a decidable two-quantifier theory. These criteria are applied not only to the weak truth-table degrees of the recursively enumerable sets but also to various substructures of the polynomial many-one (pm) degrees of the recursive sets. These applications to thepmdegrees require no new complexity-theoretic results. The fact that thepm-degrees of the recursive sets have a decidable two-quantifier theory answers a question raised by Shore and Slaman in [21].

Jump embeddings in the Turing degrees

1991 ◽  
Vol 56 (2) ◽  
pp. 563-591 ◽  
Author(s):  
Peter G. Hinman ◽  
Theodore A. Slaman
Keyword(s):  
Partial Ordering ◽  
Turing Degrees ◽  
First Order ◽  
Upper Semilattice ◽  
Recursively Enumerable ◽  
Locally Countable ◽  
Image Set ◽  
Turing Reducibility ◽  
Image Position ◽  
Order Structures

Since its introduction in [K1-Po], the upper semilattice of Turing degrees has been an object of fascination to practitioners of the recursion-theoretic art. Starting from relatively simple concepts and definitions, it has turned out to be a structure of enormous complexity and richness. This paper is a contribution to the ongoing study of this structure.Much of the work on Turing degrees may be formulated in terms of the embeddability of certain first-order structures in a structure whose universe is some set of degrees and whose relations, functions, and constants are natural degree-theoretic ones. Thus, for example, we know that if {P, ≤P) is a partial ordering of cardinality at most ℵ1 which is locally countable—each point has at most countably many predecessors—then there is an embeddingwhere D is the set of all Turing degrees and <T is Turing reducibility. If (P, ≤P) is a countable partial ordering, then the image of the embedding may be taken to be a subset of R, the set of recursively enumerable degrees. Without attempting to make the notion completely precise, we shall call embeddings of the first sort global, in contrast to local embeddings which impose some restrictions on the image set.

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