Degrees of members of Π<sub>1</sub><sup>0</sup>classes

Jockusch, Carl G.; Soare, Robert I.

doi:10.2140/pjm.1972.40.605

Cited by 124 publications

(68 citation statements)

References 12 publications

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“…This confirmed Bennett's intuitive arguments by establishing a definite relationship between computational depth and computational usefulness. It also substantially extended Bennett's result on χ K by implying (in combination with known results of recursion theory [1,[8][9][10]) that all high Turing degrees and some low Turing degrees contain strongly deep sequences.…”

Section: Introductionmentioning

confidence: 63%

Weakly useful sequences

Fenner

Lutz

Mayordomo

1995

Automata, Languages and Programming

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An infinite binary sequence x is defined to be (i) strongly useful if there is a computable time bound within which every decidable sequence is Turing reducible to x; and (ii) weakly useful if there is a computable time bound within which all the sequences in a non-measure 0 subset of the set of decidable sequences are Turing reducible to x.Juedes, Lathrop, and Lutz (1994) proved that every weakly useful sequence is strongly deep in the sense of Bennett (1988) and asked whether there are sequences that are weakly useful but not strongly useful.The present paper answers this question affirmatively. The proof is a direct construction that combines the martingale diagonalization technique of Lutz (1994) with a new technique, namely, the construction of a sequence that is "computably deep" with respect to an arbitrary, given uniform reducibility. The abundance of such computably deep sequences is also proven and used to show that every weakly useful sequence is computably deep with respect to every uniform reducibility.

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Section: Introductionmentioning

confidence: 63%

Weakly useful sequences

Fenner

Lutz

Mayordomo

1995

Automata, Languages and Programming

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“…Properties (1) and (2) (4) and (5) (4) and (5) hold. Finally if o -(b, then j G Ak implies that R3Abo,bi) holds in C and j E Bk implies ->i?i(6o,i»i) holds in C by clause (B) of our definition of C so that (4) and (5) (4) and (5) of the definition of Km follows from the fact that in conditions (4) and (5) we referred to sets of indiscernibles with respect to Lm. Now consider (c).…”

Section: Indiscerniblesmentioning

confidence: 97%

Degrees of indiscernibles in decidable models

Kierstead¹,

Remmel²

1985

Trans. Amer. Math. Soc.

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ABSTRACT. We show that the problem of finding an infinite set of indiscernibles in an arbitrary decidable model of a first order theory is essentially equivalent to the problem of finding an infinite path through a recursive wbranching tree. Similarly, we show that the problem of finding an infinite set of indiscernibles in a decidable model of an u>-categorical theory with decidable atoms is essentially equivalent to finding an infinite path through a recursive binary tree. Introduction.Ehrenfeucht and Mostowski [2] introduced the notion of indiscernibles and proved that every first order theory has a model with an infinite set of order indiscernibles. In [6], we investigated the question of which decidable theories have decidable models with infinite recursive sets of indiscernibles.For example, we showed that every w-stable decidable theory and every stable theory which has a certain strong decidability property called BQ-decidability have such models. Moreover, we gave a series of examples of decidable theories which have no decidable models with infinite recursive sets of indiscernibles which show that the various hypotheses of our positive results are necessary.In this paper, we investigate the possible degrees of indiscernibles in a decidable model Ai. We shall show, in a sense to be made precise in §1, that the problem of finding a set of indiscernibles in an arbitrary decidable model M of a first order theory T is essentially equivalent to the problem of finding an infinite path through a recursive w-branching tree T. Similarly, we shall show that the problem of finding an infinite set of indiscernibles in a decidable model Ai of an w-categorical theory T with decidable atoms is essentially equivalent to finding an infinite path through

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“…We also remark that the fact (Jockusch and Soare [6]) that the leftmost path in a recursive tree T is of r.e. degree relativizes in a strong way.…”

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confidence: 88%

“…One (common in, for example, the study or Π 0 1 classes as in Jockusch and Soare [5] and [6]) defines a tree T as a subset of 2 <ω closed downward (i.e., under initial segments). The other (common in minimal degree constructions as in Lerman [9]) defines a "tree" F as a function from 2 <ω to 2 <ω that preserves both order (⊆) and nonorder ( ).…”

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confidence: 99%

Minimal degrees which are Σ₂⁰ but not Δ₂⁰

Shore¹

2003

Proc. Amer. Math. Soc.

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Abstract. We give a short proof of the existence of minimal Turing degrees which are Σ 0 2 but not ∆ 0 2 .Manaster [8] proved that there are minimal Turing degrees which are ∆ n+2 but not Σ n+1 for every n. A natural question then was whether there are ones which are Σ n+2 but not ∆ n+2 . For n > 0 we noticed long ago (as did Carl Jockusch) that there is a simple solution using Lachlan 1-trees [7]. Take such a function f : ω → 3 recursive in 0 with D = {n | f (n) = 2} infinite such that every function obtained by replacing each of the values 2 by either 0 or 1 (independently) gives a function of minimal degree. If D is listed in increasing order as d i and C ∈ Σ n+2 − ∆ n+2 , then the function g which equals f onD and C(i) on d i is of the desired degree when n > 2. The problem for n = 2 remained open. We present here a similarly simple solution for n = 2. After we found this proof and inquired about the status of the problem, S. B. Cooper informed us that he and Yue Yang had already recently found a direct (recursive in 0 ) construction of such a degree. Our proof should be noticeably simpler since it basically consists of pointing out how to combine two known types of tree constructions from different settings.There are two common notions of trees on 2 <ω , the set of binary sequences of finite length ordered by inclusion. One (common in, for example, the study or Π 0

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Degrees of members of Π₁⁰classes

Cited by 124 publications

References 12 publications

Weakly useful sequences

Weakly useful sequences

Degrees of indiscernibles in decidable models

Minimal degrees which are Σ₂⁰ but not Δ₂⁰

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Degrees of members of Π10classes

Cited by 124 publications

References 12 publications

Weakly useful sequences

Weakly useful sequences

Degrees of indiscernibles in decidable models

Minimal degrees which are Σ₂⁰ but not Δ₂⁰

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Degrees of members of Π₁⁰classes