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Bounded Immunity and BttReductions
 MLQ Math. Log. Q
, 1999
"... We define and study a new notion called kimmunity that lies between immunity and hyperimmunity in strength. Our interest in kimmunity is justified by the result that # # does not ktt reduce to a kimmune set, which improves a previous result by Kobzev [7, 13]. We apply the result to show that ..."
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We define and study a new notion called kimmunity that lies between immunity and hyperimmunity in strength. Our interest in kimmunity is justified by the result that # # does not ktt reduce to a kimmune set, which improves a previous result by Kobzev [7, 13]. We apply the result to show that # # does not bttreduce to MIN, the set of minimal programs. Other applications include the set of Kolmogorov random strings, and retraceable and regressive sets. We also give a new characterization of e#ectively simple sets and show that simple sets are not bttcuppable. Keywords: Computability, Recursion Theory, bounded reducibilities, minimal programs, immunity, kimmune, regressive, retraceable, e#ectively simple, cuppable. 1 Introduction There seems to be a large gap between immunity and hyperimmunity (himmunity) that is waiting to be filled. What happens, one wonders if the disjoint strong arrays that try to witness that a set is not himmune are subjected to additional conditions...
Degree theoretic definitions of the low_2 recursively enumerable sets
 J. SYMBOLIC LOGIC
, 1995
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On the Universal Splitting Property
 Mathematical Logic Quarterly
, 1996
"... We prove that if an incomplete computably enumerable set has the the universal splitting property then it is low 2 . This solves a question from AmbosSpies and Fejer [1] and Downey and Stob [7]. Some technical improvements are discussed. 1 Introduction Two computably enumerable sets A 1 and A 2 ar ..."
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We prove that if an incomplete computably enumerable set has the the universal splitting property then it is low 2 . This solves a question from AmbosSpies and Fejer [1] and Downey and Stob [7]. Some technical improvements are discussed. 1 Introduction Two computably enumerable sets A 1 and A 2 are said to split A if A = A 1 [ A 2 and A 1 " A 2 = ;. We write A 1 t A 2 = A in the case that A 1 and A 2 split A. Splitting theorems for computably enumerable sets have played a central role in the history of classical computability theory. For instance, Sack's splitting theorm [14], demonstrated that every nonzero computably enumerable degree could be Downey's research supported by Cornell University, an IGC grant from Victoria University and the New Zealand Marsden Fund via grant 95VICMIS0698 under contract VIC509. Some of these results were obtained whilst Downey was a Visiting Professor at Cornell University in fall 1995. decomposed into a pair of incomparible nonzero computa...
Simple Sets Are Not BttCuppable
, 1997
"... We extend Post's result that a simple set cannot be bttcomplete by showing that in fact it cannot be bttcuppable, i.e. if the join of a c.e. set and a simple set is bttcomplete, then the nonsimple set is bttcomplete itself. The proof also yields that simple sets are not dcuppable (i.e. no ..."
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We extend Post's result that a simple set cannot be bttcomplete by showing that in fact it cannot be bttcuppable, i.e. if the join of a c.e. set and a simple set is bttcomplete, then the nonsimple set is bttcomplete itself. The proof also yields that simple sets are not dcuppable (i.e. not cuppable with regard to disjunctive reductions). Post showed that a simple set cannot be bttcomplete. In a paper by Stephen Fenner and the author [3] this was generalized to nonc.e. sets by isolating the immunity property which is responsible for the incompleteness. Another approach to the bttincompleteness of simple sets would have been through degrees. How incomplete are simple sets? Putting it dioeerently: can the join of a bttincomplete degree with a simple degree be bttcomplete? We will show that the answer is no. Deønition 1 A set A is called rcuppable, if there is a c.e. set B such that ; 0 r A \Phi B and ; 0 6 r B, where r is a class of reductions (like m, 1, btt, c, d, tt...
Simple Sets and Strong Reducibilities
"... We study connections between strong reducibilities and properties of computably enumerable sets such as simplicity. We call a class S of computably enumerable sets bounded iff there is an min complete computably enumerable set A such that every set in S is mreducible to A. For example, we show ..."
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We study connections between strong reducibilities and properties of computably enumerable sets such as simplicity. We call a class S of computably enumerable sets bounded iff there is an min complete computably enumerable set A such that every set in S is mreducible to A. For example, we show that the class of eectively simple sets is bounded; but the class of maximal sets is not. Furthermore, the class of computably enumerable sets Turing reducible to a computably enumerable set B is bounded iff B is low 2 . For r = bwtt, tt, wtt and T , there is a bounded class intersecting every computably enumerable rdegree; for r = c, d and p, no such class exists. AMS Classication: 03D30; 03D25 Keywords: Computably enumerable sets (= Recursively enumerable sets); Simple sets; mreducibility; Strong reducibilities; 3 classes; Ideals; Exact pairs 1 Introduction With a typical priority argument, one can show that for any simple set A, there is a simple set B such that B m A. Carl ...
Structural Properties and ... Enumeration Degrees
"... We prove that each \Sigma 2 0 set which is hypersimple relative to ; 0 is noncuppable in the structure of the \Sigma 0 2 enumeration degrees. This gives a connection between properties of \Sigma 0 2 sets under inclusion and and the \Sigma 0 2 enumeration degrees. We also prove that some ..."
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We prove that each \Sigma 2 0 set which is hypersimple relative to ; 0 is noncuppable in the structure of the \Sigma 0 2 enumeration degrees. This gives a connection between properties of \Sigma 0 2 sets under inclusion and and the \Sigma 0 2 enumeration degrees. We also prove that some low nonc.e. enumeration degree contains no set which is simple relative to ; 0 . 1 Introduction There is a wide range of theorems in computability theory asserting that, in a certain degree structure R r of computably enumerable (c.e. ) sets under a reducibility r , a simplicity property of a c.e. set A implies the incompleteness of the rdegree of A. (Here a simplicity property requires that in some sense the complement of A is sparse.) An example of such a result is that a simple set cannot be bttcomplete ([Pos44]). While a simple set may be ttcomplete, the stronger notion of hypersimplicity of A even implies wttincompleteness. Downey and Jockusch [DJ87] showed that the wttdeg...
Contiguity and Distributivity in the Enumerable Turing Degrees \Lambda
, 1996
"... Abstract We prove that a (recursively) enumerable degree is contiguous iff it is locally distributive. This settles a twentyyear old question going back to Ladner and Sasso. We also prove that strong contiguity and contiguity coincide, settling a question of the first author, and prove that no mtop ..."
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Abstract We prove that a (recursively) enumerable degree is contiguous iff it is locally distributive. This settles a twentyyear old question going back to Ladner and Sasso. We also prove that strong contiguity and contiguity coincide, settling a question of the first author, and prove that no mtopped degree is contiguous, settling a question of the first author and Carl Jockusch [11]. Finally, we prove some results concerning local distributivity and relativized weak truth table reducibility.
Notre Dame Journal of Formal Logic Cuppability of Simple and Hypersimple Sets
"... Abstract An incomplete degree is cuppable if it can be joined by an incomplete degree to a complete degree. For sets fulfilling some type of simplicity property one can now ask whether these sets are cuppable with respect to a certain type of reducibilities. Several such results are known. In this p ..."
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Abstract An incomplete degree is cuppable if it can be joined by an incomplete degree to a complete degree. For sets fulfilling some type of simplicity property one can now ask whether these sets are cuppable with respect to a certain type of reducibilities. Several such results are known. In this paper we settle all the remaining cases for the standard notions of simplicity and all the main strong reducibilities. There are two sides to every question.Protagoras, quoted in Diogenes Laertius, Lives of Eminent Philosophers. 1 Introduction In his approach to constructing an incomplete c.e. degree, Emil Post attempted to define structural properties of c.e. sets that would force their incompleteness. In his groundbreaking 1944 paper Recursively enumerablesets of positive integers and their decision problems ([24], reprinted in Davis's The Undecidable [1]) this goal led him to isolate many of the classical concepts of computability, including creativity, manyone reducibility, bounded and unbounded truthtable reducibility, simplicity, hypersimplicity, and hyperhypersimplicity.
Difference Splittings of Recursively Enumerable Sets
, 1997
"... We study here the degreetheoretic structure of settheoretical splittings of recursively enumerable (r.e.) sets into differences of r.e. sets. As a corollary we deduce that the ordering of wttdegrees of unsolvability of differences of r.e. sets is not a distributive semilattice and is not elemen ..."
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We study here the degreetheoretic structure of settheoretical splittings of recursively enumerable (r.e.) sets into differences of r.e. sets. As a corollary we deduce that the ordering of wttdegrees of unsolvability of differences of r.e. sets is not a distributive semilattice and is not elementarily equivalent to the ordering of r.e. wttdegrees of unsolvability.