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11
EMBEDDING JUMP UPPER SEMILATTICES INTO THE TURING DEGREES
"... We prove that every countable jump upper semilattice can be embedded in D, where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and D is the jusl of Turing degrees. As a corollary we get that the existential ..."
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We prove that every countable jump upper semilattice can be embedded in D, where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and D is the jusl of Turing degrees. As a corollary we get that the existential theory of 〈D, ≤T, ∨, ′ 〉 is decidable. We also prove that this result is not true about jusls with 0, by proving that not every quantifier free 1type of jusl with 0 is realized in D. On the other hand, we show that every quantifier free 1type of jump partial ordering (jpo) with 0 is realized in D. Moreover, we show that if every quantifier free type, p(x1,..., xn), of jpo with 0, which contains the formula x1 ≤ 0 (m) &... & xn ≤ 0 (m) for some m, is realized in D, then every every quantifier free type of jpo with 0 is realized in D. We also study the question of whether every jusl with the c.p.p. and size κ ≤ 2 ℵ0 is embeddable in D. We show that for κ = 2 ℵ0 the answer is no, and that for κ = ℵ1 it is independent of ZFC. (It is true if MA(κ) holds.)
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. ..."
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We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable.
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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Cited by 2 (2 self)
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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
Minimal pair constructions and iterated trees of strategies, Logical methods, in honor of Anil Nerode's 60th birthday, Birkh auser
, 1993
"... 0. Introduction. We use the iterated trees of strategies approach developed in [LL1], [LL2] to prove some theorems about minimal pairs. In Sections 13, we show how to use these methods to prove the Minimal Pair Theorem of Lachlan [L] and Yates [Y]: Theorem 3.4 (Minimal Pair): There exist nonrecurs ..."
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0. Introduction. We use the iterated trees of strategies approach developed in [LL1], [LL2] to prove some theorems about minimal pairs. In Sections 13, we show how to use these methods to prove the Minimal Pair Theorem of Lachlan [L] and Yates [Y]: Theorem 3.4 (Minimal Pair): There exist nonrecursive r.e. degrees a and b such that aÙb = 0.
EMBEDDINGS INTO THE TURING DEGREES.
, 2007
"... The structure of the Turing degrees was introduced by Kleene and Post in 1954 [KP54]. Since then, its study has been central in the area of Computability Theory. One approach for analyzing the shape of this structure has been looking at the structures that can be ..."
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The structure of the Turing degrees was introduced by Kleene and Post in 1954 [KP54]. Since then, its study has been central in the area of Computability Theory. One approach for analyzing the shape of this structure has been looking at the structures that can be
Iterated Trees of Strategies and Priority Arguments
"... 0. Introduction. It is our intent, in this paper, to try to describe some of the key ideas developed in the series or papers by Lempp and Lerman [LL1, LL2, LL3, LL4], which use the iterated trees of strategies approach to priority arguments. This is an approach which provides a framework for carryin ..."
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0. Introduction. It is our intent, in this paper, to try to describe some of the key ideas developed in the series or papers by Lempp and Lerman [LL1, LL2, LL3, LL4], which use the iterated trees of strategies approach to priority arguments. This is an approach which provides a framework for carrying out priority arguments at all levels of the arithmetical hierarchy. The first attempt to find a framework for (finite injury) priority arguments was
Computability Theory, Algorithmic Randomness and Turing’s Anticipation
"... This article looks at the applications of Turing’s Legacy in computation, particularly to the theory of algorithmic randomness, where classical mathematical concepts such as measure could be made computational. It also traces Turing’s anticipation of this theory in an early manuscript. ..."
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This article looks at the applications of Turing’s Legacy in computation, particularly to the theory of algorithmic randomness, where classical mathematical concepts such as measure could be made computational. It also traces Turing’s anticipation of this theory in an early manuscript.
Immunity and hyperimmunity
, 2007
"... Abstract 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 arithm ..."
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Abstract 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 \Pi 3\Sigma 3 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: (8j < e) ['j 6 = 'e]}. (1.1) In 1967, Blum [2] showed that one can enumerate at most finitely many shortest programs. Five years later, Meyer [11] 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