Results 1  10
of
29
Parameter Definability in the Recursively Enumerable Degrees
"... The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly defin ..."
Abstract

Cited by 36 (13 self)
 Add to MetaCart
The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that...
Enumeration reducibility, nondeterministic computations and relative computability of partial functions
 in Recursion Theory Week, Proceedings Oberwolfach
, 1989
"... ..."
Noncomputable Spectral Sets
, 2007
"... iii For my Mama, whose *minimal index is computable (because it’s finite). ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
iii For my Mama, whose *minimal index is computable (because it’s finite).
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
"... ..."
Conjectures and Questions from Gerald Sacks’s Degrees of Unsolvability
 Archive for Mathematical Logic
, 1993
"... We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. Gerald Sacks has had a major influence on the development of logic, parti ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. Gerald Sacks has had a major influence on the development of logic, particularly recursion theory, over the past thirty years through his research, writing and teaching. Here, I would like to concentrate on just one instance of that influence that I feel has been of special significance to the study of the degrees of unsolvability in general and on my own work in particular the conjectures and questions posed at the end of the two editions of Sacks's first book, the classic monograph Degrees of Unsolvability (Annals
Double Jump Inversions and Strong Minimal Covers in the Turing Degrees
, 2004
"... Decidability problems for (fragments of) the theory of the structure D of Turing degrees, form a wide and interesting class, much of which is yet unsolved. Lachlan showed in 1968 that the first order theory of D with the Turing reducibility relation is undecidable. Later results concerned the decida ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Decidability problems for (fragments of) the theory of the structure D of Turing degrees, form a wide and interesting class, much of which is yet unsolved. Lachlan showed in 1968 that the first order theory of D with the Turing reducibility relation is undecidable. Later results concerned the decidability (or undecidability) of fragments of this theory, and of other theories obtained by extending the language (e.g. with 0 or with the Turing jump operator). Proofs of these results often hinge on the ability to embed certain classes of structures (lattices, jumphierarchies, etc.) in certain ways, into the structure of Turing degrees. The first part of the dissertation presents two results which concern embeddings onto initial segments of D with known double jumps, in other words a double jump inversion of certain degree structures onto initial segments. These results may prove to be useful tools in uncovering decidability results for (fragments of) the theory of the Turing degrees in languages containing the double jump operator. The second part of the dissertation relates to the problem of characterizing the Turing degrees which have a strong minimal cover, an issue first raised by Spector in 1956. Ishmukhametov solved the problem for the recursively enumerable degrees, by showing that those which have a strong minimal cover are exactly the r.e. weakly recursive degrees. Here we show that this characterization fails outside the r.e. degrees, and also construct a minimal degree below 0 ′ which is not weakly recursive, thereby answering a question from Ishmukhametov’s paper.
On the Turing Degrees of Minimal Index Sets
, 2007
"... We study generalizations of shortest programs as they pertain to Schaefer’s MIN ∗ problem. We identify sets of mminimal and Tminimal indices and characterize their truthtable and Turing degrees. In particular, we show MIN m ⊕ ∅ ′ ′ ≡T ∅ ′′ ′ , MIN T(n) ∅ (n+2) ≡T ∅ (n+4) , and that there exists ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
We study generalizations of shortest programs as they pertain to Schaefer’s MIN ∗ problem. We identify sets of mminimal and Tminimal indices and characterize their truthtable and Turing degrees. In particular, we show MIN m ⊕ ∅ ′ ′ ≡T ∅ ′′ ′ , MIN T(n) ∅ (n+2) ≡T ∅ (n+4) , and that there exists a Kolmogorov numbering ψ satisfying both MIN m ψ ≡tt ∅ ′′ ′ and MIN T(n) ψ ≡T ∅ (n+4). This Kolmogorov numbering also achieves maximal truthtable degree for other sets of minimal indices. Finally, we show that the set of shortest descriptions, SD, is 2c.e. but not co2c.e. Some open problems are left for the reader. 1 The MIN ∗ problem The set of shortest programs is fMIN: = {e: (∀j < e) [ϕj � = ϕe]}. In 1972, Meyer demonstrated that fMIN admits a neat Turing characterization, namely fMIN ≡T ∅ ′ ′ [10]. In Spring 1990 (according to the best recollection of the author), John Case issued a homework assignment with the following definition [1]: fMIN ∗: = {e: (∀j < e) [ϕj � = ∗ ϕe]}, 1 where = ∗ means equal except for a finite set. Case notes that fMIN ∗ is Σ2immune, although his assignment exclusively refers to the Σ2sets as “limr.e. ” sets. On October 1, 1996, six years after the initial homework assignment, Case introduced the set fMIN ∗ to Marcus Schaefer in an email. The following year, Schaefer published a master’s thesis on minimal indices [14], which became the first public account of fMIN ∗. In his survey thesis, Schaefer proved that fMIN ∗ ⊕ ∅ ′ ≡T ∅ ′′ ′ , leaving open the tantalizing question of whether or not fMIN ≡T ∅ ′′ ′. All that would be required to answer this question affirmatively is to show that fMIN ∗ ≥T ∅ ′ , care of Schaefer’s result. This is the “MIN ∗ problem. ” The reader is encouraged to attempt this reduction before proceeding. This concludes our historical remarks. Our approach in this paper is to study c.e. sets in place of p.c. functions. This allows us to consider equivalence relations other than = and = ∗ which are especially natural for sets, namely: Definition 1.1. For n ≥ 0: MIN: = {e: (∀j < e) [Wj � = We]},
On the jump classes of noncuppable enumeration degrees
 the Journal of Symbolic Logic
"... Abstract. We prove that for every Σ02 enumeration degree b there exists a noncuppable Σ02 degree a> 0e such that b ′ ≤e a ′ and a′′≤e b′′. This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Abstract. We prove that for every Σ02 enumeration degree b there exists a noncuppable Σ02 degree a> 0e such that b ′ ≤e a ′ and a′′≤e b′′. This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding ι: DT → De, that there exist Σ02 noncuppable enumeration degrees at every possible—i.e. above low1—level of the high/low jump hierarchy in the context of De. 1.
The ∀∃ theory of D(≤, ∨, ′ ) is undecidable
 In Proceedings of Logic Colloquium
, 2003
"... We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable. ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable.