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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 defin ..."
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Cited by 37 (14 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...
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 6 (4 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.
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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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,
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).
RANDOM REALS, THE RAINBOW RAMSEY THEOREM, AND ARITHMETIC CONSERVATION
, 2012
"... We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts? ” Let 2RAN be the principle that for every real X there is a real R which is 2random relative to X. In Section 2, we observe that the arguments of Csima and Mileti [3] can be impleme ..."
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We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts? ” Let 2RAN be the principle that for every real X there is a real R which is 2random relative to X. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory RCA0 and so RCA0 + 2RAN implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over RCA0 for arithmetic sentences. Thus, from the CsimaMileti fact that the existence of random reals has infinitarycombinatorial consequences we can conclude that 2RAN has nontrivial arithmetic consequences. In Section 4, we show that 2RAN is conservative over RCA0 + BΣ2 for Π1 1sentences. Thus, the set of firstorder consequences of 2RAN is strictly stronger than P − + I Σ1 and no stronger than P − + BΣ2.
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.
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.
Undecidability Results for low complexity time classes
 J. Comput. System Sci
"... We prove that the theory of EXPTIME degrees with respect to polynomial time Turing and manyone reducibility is undecidable. To do so we use a coding method based on ideal lattices of Boolean algebras which was introduced in Nies [12]. The method can be applied in fact to all time classes given by a ..."
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We prove that the theory of EXPTIME degrees with respect to polynomial time Turing and manyone reducibility is undecidable. To do so we use a coding method based on ideal lattices of Boolean algebras which was introduced in Nies [12]. The method can be applied in fact to all time classes given by a time constructible function which dominates all polynomials. By a similar method, we construct an oracle U such that Th(NP U ; `) is undecidable. 1 Introduction If h is a time constructible function which dominates all polynomials, then, by the methods of the deterministic time hierarchy theorem, DTIME(h) properly contains PTIME. Therefore, a polynomial time reducibility like polynomial time manyone or Turing reducibility induces a nontrivial degree structure on DTIME(h), which is an uppersemilattice with a least element. By the methods of Ladner [10] (also see ([11] or [5, Chapter I.7]), this degree structure is dense. This was so far the only fact known to hold in general for all su...
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 ; : : : ; ..."
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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...
On Turing Reducibility
, 1994
"... We show that the transitivity of pointwise Turing reducibility on the recursively enumerable sets of integers cannot be proven in P  + I# 1 , first order arithmetic with induction limited to # 1 predicates. We produce a example of intransitivity in a nonstandard model of P  +I# 1 by a finite in ..."
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We show that the transitivity of pointwise Turing reducibility on the recursively enumerable sets of integers cannot be proven in P  + I# 1 , first order arithmetic with induction limited to # 1 predicates. We produce a example of intransitivity in a nonstandard model of P  +I# 1 by a finite injury priority construction.