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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 definabl ..."
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Cited by 34 (13 self)
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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...
THE COMPLEXITY OF THE INDEX SETS OF ℵ0CATEGORICAL THEORIES AND OF EHRENFEUCHT THEORIES
, 2006
"... Abstract. We classify the computabilitytheoretic complexity of two index sets of classes of firstorder theories: We show that the property of being an ℵ0categorical theory is Π0 3complete; and the property of being an Ehrenfeucht theory Π1 1complete. We also show that the property of having con ..."
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Cited by 4 (0 self)
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Abstract. We classify the computabilitytheoretic complexity of two index sets of classes of firstorder theories: We show that the property of being an ℵ0categorical theory is Π0 3complete; and the property of being an Ehrenfeucht theory Π1 1complete. We also show that the property of having continuum many models is Σ1 1 hard. Finally, as a corollary, we note that the properties of having only decidable models, and of having only computable models, are both Π1 1complete. 1. The Main Theorem Measuring the complexity of mathematical notions is one of the main tasks of mathematical logic. Two of the main tools to classify complexity are provided by Kleene’s arithmetical and analytical hierarchy. These two hierarchies provide convenient ways to determine the exact complexity of properties by various notions of completeness, and to give lower bounds on the complexity by various notions of hardness. (See, e.g., Kleene [1], Soare [10] or Odifreddi [4, 5] for the definitions.) This paper will investigate the complexity of properties of a firstorder theory, more precisely, the complexity of a countable firstorder theory having a certain number of models. Recall that a theory is called ℵ0categorical if it has only one countable model up to isomorphism, and an Ehrenfeucht theory if it has more than one but only finitely many countable models up to isomorphism. In order to measure the complexity of these notions, we will use decidable firstorder theories,
Turing Incomparability in Scott Sets
 Proceedings of the American Mathematical Society
"... Abstract. For every Scott set F and every nonrecursive set X in F, there is a Y ∈ F such that X and Y are Turing incomparable. 1. ..."
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Abstract. For every Scott set F and every nonrecursive set X in F, there is a Y ∈ F such that X and Y are Turing incomparable. 1.
On the Query Complexity of Sets
, 1996
"... . There has been much research over the last eleven years that considers the number of queries needed to compute a function as a measure of its complexity. We are interested in the complexity of certain sets in this context. We study the sets ODD A n = f(x1 ; : : : ; xn) : jA " fx1 ; : : : ; xn gj ..."
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Cited by 3 (3 self)
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. There has been much research over the last eleven years that considers the number of queries needed to compute a function as a measure of its complexity. We are interested in the complexity of certain sets in this context. We study the sets ODD A n = f(x1 ; : : : ; xn) : jA " fx1 ; : : : ; xn gj is oddg and WMOD(m) A n = f(x1 ; : : : ; xn) : jA " fx1 ; : : : ; xn gj 6j 0 (mod m)g. If A = K or A is semirecursive, we obtain tight bounds on the query complexity of ODD A n and WMOD(m) A n . We obtain lower bounds for A r.e. The lower bounds for A r.e. are derived from the lower bounds for A semirecursive. We obtain that every ttdegree has a set A such that ODD A n requires n parallel queries to A, and a set B such that ODD B n can be decided with one query to B. Hence for boundedquery complexity, how information is packaged is more important than Turing degree. We investigate when extra queries add power. We show that, for several nonrecursive sets A, the more queries you can...
The Complexity of Finding SUBSEQ(A)
"... Higman showed that if A is any language then SUBSEQ(A) is regular. His proof wasnonconstructive. We show that the result cannot be made constructive. In particular we show that if f takes as input an index e of a total Turing Machine Me, and outputs a DFA forSUBSEQ(L(M e)), then;00 ^T f (f is \Sig ..."
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Higman showed that if A is any language then SUBSEQ(A) is regular. His proof wasnonconstructive. We show that the result cannot be made constructive. In particular we show that if f takes as input an index e of a total Turing Machine Me, and outputs a DFA forSUBSEQ(L(M e)), then;00 ^T f (f is \Sigma 2hard). We also study the complexity of going from Ato SUBSEQ(A) for several representations of A and SUBSEQ(A).
Ramsey’s Theorem and cone avoidance
 JOURNAL
"... It was shown by Cholak, Jockusch, and Slaman that every computable 2coloring of pairs admits an infinite low2 homogeneous set H. We answer a question of the same authors by showing that H may be chosen to satisfy in addition C ̸≤T H, where C is a given noncomputable set. This is shown by analyzing ..."
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Cited by 3 (3 self)
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It was shown by Cholak, Jockusch, and Slaman that every computable 2coloring of pairs admits an infinite low2 homogeneous set H. We answer a question of the same authors by showing that H may be chosen to satisfy in addition C ̸≤T H, where C is a given noncomputable set. This is shown by analyzing a new and simplified proof of Seetapun’s cone avoidance theorem for Ramsey’s theorem. We then extend the result to show that every computable 2coloring of pairs admits a pair of low2 infinite homogeneous sets whose degrees form a minimal pair.
Bounding and nonbounding minimal pairs in the enumeration degrees
 J. Symbolic Logic
"... Abstract. We show that every nonzero ∆ 0 2 edegree bounds a minimal pair. On the other hand, there exist Σ 0 2 edegrees which bound no minimal pair. 1. ..."
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Abstract. We show that every nonzero ∆ 0 2 edegree bounds a minimal pair. On the other hand, there exist Σ 0 2 edegrees which bound no minimal pair. 1.
LIMITS ON JUMP INVERSION FOR STRONG REDUCIBILITIES
"... Abstract. We show that Sacks ’ and Shoenfield’s analogs of jump inversion fail for both tt and wttreducibilities in a strong way. In particular we show that there is a ∆0 2 set B>tt ∅ ′ such that there is no c.e. set A with A ′ ≡wtt B. We also show that there is a Σ0 2 set C>tt ∅ ′ such that there ..."
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Abstract. We show that Sacks ’ and Shoenfield’s analogs of jump inversion fail for both tt and wttreducibilities in a strong way. In particular we show that there is a ∆0 2 set B>tt ∅ ′ such that there is no c.e. set A with A ′ ≡wtt B. We also show that there is a Σ0 2 set C>tt ∅ ′ such that there is no ∆0 2 set D with D ′ ≡wtt C. 1.
Interpreting N in the computably enumerable weak truth table degrees
"... We give a firstorder coding without parameters of a copy of (N;+; \Theta) in the computably enumerable weak truth table degrees. As a tool,we develop a theory of parameter definable subsets. Given a degree structure from computability theory, once the undecidability of its theory is known, an im ..."
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We give a firstorder coding without parameters of a copy of (N;+; \Theta) in the computably enumerable weak truth table degrees. As a tool,we develop a theory of parameter definable subsets. Given a degree structure from computability theory, once the undecidability of its theory is known, an important further problem is the question of the actual complexity of the theory. If the structure is arithmetical, then its theory can be interpreted in true arithmetic, i.e. Th(N; +; \Theta). Thus an upper bound is ; (!) , the complexity of Th(N; +; \Theta). Here an interpretation of theories is a manyone reduction based on a computable map defined on sentences in some natural way. An example of an arithmetical structure is D T ( ; 0 ), the Turing degrees of \Delta 0 2 sets. Shore [16] proved that true arithmetic can be interpreted in Th(D T ( ; 0 )). A stronger result is interpretability without parameters of a copy of (N; +; \Theta) in the structure (interpretability of struc...