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.

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]).

scholarly journals The weak truth table degrees of recursively enumerable sets

1975 ◽  
Vol 8 (4) ◽  
pp. 429-448 ◽  
Author(s):  
Richard E. Ladner ◽  
Leonard P. Sasso
Keyword(s):  
Truth Table ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Weak Truth Table Degrees

Minimal Weak Truth Table Degrees and Computably Enumerable Turing Degrees

2020 ◽  
Vol 265 (1284) ◽  
pp. 0-0
Author(s):  
Rodney Downey ◽  
Keng Meng Ng ◽  
Reed Solomon
Keyword(s):  
Truth Table ◽  
Turing Degrees ◽  
Weak Truth Table Degrees ◽  
Computably Enumerable

Infimum Properties Differ in the Weak Truth-table Degrees and the Turing Degrees

2004 ◽  
Vol 20 (1) ◽  
pp. 163-168
Author(s):  
Liang Yu ◽  
De Cheng Ding
Keyword(s):  
Truth Table ◽  
Turing Degrees ◽  
Weak Truth Table Degrees

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

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].

Promptness does not imply superlow cuppability

2009 ◽  
Vol 74 (4) ◽  
pp. 1264-1272 ◽  
Author(s):  
David Diamondstone
Keyword(s):  
Explicit Construction ◽  
Negative Answer ◽  
Related Question ◽  
Truth Table ◽  
Turing Degrees ◽  
Classical Theorem ◽  
Halting Problem ◽  
Simple Set ◽  
Complete Set

AbstractA classical theorem in computability is that every promptly simple set can be cupped in the Turing degrees to some complete set by a low c.e. set. A related question due to A. Nies is whether every promptly simple set can be cupped by a superlow c.e. set, i.e. one whose Turing jump is truth-table reducible to the halting problem ∅′. A negative answer to this question is provided by giving an explicit construction of a promptly simple set that is not superlow cuppable. This problem relates to effective randomness and various lowness notions.

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