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Degree theoretic definitions of the low_2 recursively enumerable sets
 J. SYMBOLIC LOGIC
, 1995
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Noncomputable Spectral Sets
, 2007
"... iii For my Mama, whose *minimal index is computable (because it’s finite). ..."
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iii For my Mama, whose *minimal index is computable (because it’s finite).
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 ..."
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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
Immunity and hyperimmunity for sets of minimal indices
 Notre Dame Journal of Formal Logic
"... We extend Meyer’s 1972 investigation of sets of minimal indices. Blum showed that minimal index sets are immune, and we show that they are also immune against high levels of the arithmetic hierarchy. We give optimal immunity results for sets of minimal indices with respect to the arithmetic hierarch ..."
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We extend Meyer’s 1972 investigation of sets of minimal indices. Blum showed that minimal index sets are immune, and we show that they are also immune against high levels of the arithmetic hierarchy. We give optimal immunity results for sets of minimal indices with respect to the arithmetic hierarchy, and we illustrate with an intuitive example that immunity is not simply a refinement of arithmetic complexity. Of particular note here are the fact that there are three minimal index sets located in Π3−Σ3 with distinct levels of immunity and that certain immunity properties depend on the choice of underlying acceptable numbering. We show that minimal index sets are never hyperimmune, however they can be immune against the arithmetic sets. Lastly, we investigate Turing degrees for sets of random strings defined with respect to Bagchi’s sizefunction s. 1 A short introduction to shortest programs The set of shortest programs is {e: (∀j < e) [ϕj 6 = ϕe]}. (1.1) In 1967, Blum [4] showed that one can enumerate at most finitely many shortest programs. Five years later, Meyer [13] formally initiated the investigation of minimal index sets with questions on the Turing and truthtable degrees of (1.1). Meyer’s research parallels inquiry from Kolmogorov complexity where one searches for shortest programs generating single numbers or strings. The clearest confluence
Low upper bounds of ideals
"... Abstract. We show that there is a low Tupper bound for the class of Ktrivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in ∆0 2 Tdegrees for which there is a low Tupper bound. 1. ..."
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Abstract. We show that there is a low Tupper bound for the class of Ktrivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in ∆0 2 Tdegrees for which there is a low Tupper bound. 1.
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 ..."
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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]},
Minimal upper bounds for sequences of recursively enumerable degrees
 J. London Math. Soc
"... G. E. Sacks [3; p. 171, q. 4] asked whether there is a uniformly recursively enumerable (r.e.) ascending sequence of r.e. degrees which has as one of its minimal upper bounds another r.e. degree. We show (see the Corollary below) that the answer is " yes ". a is said to be a string ..."
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G. E. Sacks [3; p. 171, q. 4] asked whether there is a uniformly recursively enumerable (r.e.) ascending sequence of r.e. degrees which has as one of its minimal upper bounds another r.e. degree. We show (see the Corollary below) that the answer is &quot; yes &quot;. a is said to be a string if it is the restriction A[n] of a characteristic function A to the first n +1 nonnegative integers for some number n. a is said to be a beginning of A of length n +1. We use (j> to denote the empty string. Let {<De} be a standard enumeration of the partial recursive functionals. We will need a uniformly recursive double sequence {Oc> s} of finite approximations to {Oc} such that for each e, s, and such that for each s, Oe s is empty for all but a finite number of e's. Define, for each e, Then {Fe} is a standard list of the partial recursive functions with suitably wellbehaved set of approximations {Fe< J. THEOREM. There is a sequence of simultaneously r.e. degrees {bj and a r.e. degree a such that a is recursive in no finite subset o/{b,}, b { is recursive in a for each i, and, for each c < a, c is not an upper bound for {bf}. Proof. We enumerate at stages 0, 1, 2,..., s,... finite sets As, Bt s,i ^ 0, such that, if A = (J As, B i = U Bi>s, then the following objectives are satisfied:
1 Introduction Degrees of Unsolvability
, 2006
"... Modern computability theory began with Turing [Turing, 1936], where he introduced ..."
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Modern computability theory began with Turing [Turing, 1936], where he introduced
Index Sets and Parametric Reductions
"... We investigate the index sets associated with the degree structures of computable sets under the parameterized reducibilities intorduced by the authors. We solve a question of the Peter Cholak and the rst author by proving the fundamental index sets associated with a computable set A, fe : W e ..."
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We investigate the index sets associated with the degree structures of computable sets under the parameterized reducibilities intorduced by the authors. We solve a question of the Peter Cholak and the rst author by proving the fundamental index sets associated with a computable set A, fe : W e q Ag for q 2 fm; Tg are 4 complete.