A direct construction of a minimal recursively enumerable truth-table degree

1990 ◽  
pp. 187-204 ◽  
Author(s):  
Peter A. Fejer ◽  
Richard A. Shore
Keyword(s):  
Truth Table ◽  
Direct Construction ◽  
Recursively Enumerable

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

scholarly journals Infima of recursively enumerable truth table degrees.

1988 ◽  
Vol 29 (3) ◽  
pp. 420-437 ◽  
Author(s):  
Peter A. Fejer ◽  
Richard A. Shore
Keyword(s):  
Truth Table ◽  
Recursively Enumerable

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]).

Degree theoretic definitions of the low2 recursively enumerable sets

1995 ◽  
Vol 60 (3) ◽  
pp. 727-756 ◽  
Author(s):  
Rod Downey ◽  
Richard A. Shore
Keyword(s):  
Recursion Theory ◽  
Truth Table ◽  
Major Theme ◽  
Syntactic Complexity ◽  
Access To Information ◽  
Turing Machines ◽  
Recursively Enumerable ◽  
Rates Of Growth ◽  
The Given ◽  
Effective Function

The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤TB, where arbitrary (Turing) machines, φe, can be used; access to information about (the oracle) B is unlimited and the lengths of computations are potentially unbounded. Many other interesting reducibilities result from restricitng one or more of these facets of the procedure. Thus, for example, the strongest notion considered is one-one reducibility on sets: A ≤1B iff there is a one-one recursive (= effective) function f such that x Є A ⇔ f(x) Є B. Many-one (≤m) reducibility simply allows f to be many-one. Other intermediate reducibilities include truth-table (≤tt) and weak truth-table (≤wtt). The latter imposes a recursive bound f(x) on the information about B that can be used to compute A(x). The former also bounds the length of computations by requiring that the computation of A(x) from B halt in at most f(x) many steps.Each such reducibility r defines a notion of degree, degr(A) = {B : A ≤rB ∧ B ≤rA}, and a corresponding structure of the r-degrees ordered by r-reducibility. (We typically denote the degree of A by a.) A major theme in recursion theory has been the investigation of the relation between a set's place in these orderings (the algebraic properties of its degree) and other algorithmic, set-theoretic or definability type notions of complexity. Important examples of such other notions include rates of growth of functions, the types of approximation procedures which converge to the given function or set and the (syntactic) complexity of defining the set (or function) in arithmetic or analysis.

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