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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...
A Note on bttDegrees
, 1998
"... We prove that there are computable enumerable incomplete bttdegrees with the anticupping property. In fact no simple set is btt cuppable in the (not necessarily computable enumerable) bttdegrees. This solves a question of... ..."
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We prove that there are computable enumerable incomplete bttdegrees with the anticupping property. In fact no simple set is btt cuppable in the (not necessarily computable enumerable) bttdegrees. This solves a question of...
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. not cup ..."
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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 ...