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The Continuity of Cupping to 0
 Annals of Pure and Applied Logic
, 1993
"... It is shown that, if a, b are recursively enumerable degrees such that 0 ! a ! 0 0 and a [ b = 0 0 , then there exists a recursively enumerable degree c such that c ! a and c [ b = 0 0 . 1 Introduction By analogy with the notion of major subset in the context of the lattice of r.e. sets, the ..."
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It is shown that, if a, b are recursively enumerable degrees such that 0 ! a ! 0 0 and a [ b = 0 0 , then there exists a recursively enumerable degree c such that c ! a and c [ b = 0 0 . 1 Introduction By analogy with the notion of major subset in the context of the lattice of r.e. sets, the r.e. degree c is called a major subdegree of the r.e. degree a if c ! a and for every r.e. degree b a [ b = 0 0 ) c [ b = 0 0 : This paper represents modest progress towards answering the question: Does every r.e. degree which is neither 0 nor 0 0 have a major subdegree? This question was first posed by the second author in 1967, although it does not seem to have appeared in print. In the 70's and 80's efforts were made to answer the question but bore little fruit. In this paper we prove: The second author was supported by NSERC (Canada) Grant A3040, and the third author by National Science Foundation Grant DMS 8807389. The third author presented the results in this paper and tho...
Simple and Immune Relations on Countable Structures ∗
"... Let A be a computable structure and let R be a new relation on its domain. We establish a necessary and sufficient condition for the existence of a copy B of A in which the image of R (¬R, resp.) is simple (immune, resp.) relative to B. We also establish, under certain effectiveness conditions on A ..."
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Let A be a computable structure and let R be a new relation on its domain. We establish a necessary and sufficient condition for the existence of a copy B of A in which the image of R (¬R, resp.) is simple (immune, resp.) relative to B. We also establish, under certain effectiveness conditions on A and R, a necessary and sufficient condition for the existence of a computable copy B of A in which the image of R (¬R, resp.) is simple (immune, resp.). ∗The first three authors gratefully acknowledge support of the NSF Binational Grant DMS
Dynamic Properties of Computably Enumerable Sets
 In Computability, Enumerability, Unsolvability, volume 224 of London Math. Soc. Lecture Note Ser
, 1995
"... A set A ` ! is computably enumerable (c.e.), also called recursively enumerable, (r.e.), or simply enumerable, if there is a computable algorithm to list its members. Let E denote the structure of the c.e. sets under inclusion. Starting with Post [1944] there has been much interest in relating t ..."
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A set A ` ! is computably enumerable (c.e.), also called recursively enumerable, (r.e.), or simply enumerable, if there is a computable algorithm to list its members. Let E denote the structure of the c.e. sets under inclusion. Starting with Post [1944] there has been much interest in relating the denable (especially Edenable) properties of a c.e. set A to its iinformation contentj, namely its Turing degree, deg(A), under T , the usual Turing reducibility. [Turing 1939]. Recently, Harrington and Soare answered a question arising from Post's program by constructing a nonemptly Edenable property Q(A) which guarantees that A is incomplete (A !T K). The property Q(A) is of the form (9C)[A ae m C & Q \Gamma (A; C)], where A ae m C abbreviates that iA is a major subset of Cj, and Q \Gamma (A; C) contains the main ingredient for incompleteness. A dynamic property P (A), such as prompt simplicity, is one which is dened by considering how fast elements elements enter A relat...
Extensions, Automorphisms, and Definability
 CONTEMPORARY MATHEMATICS
"... This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to ..."
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This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to know the dynamical properties of a c.e. set We , not merely whether an element x is enumerated in We but when, relative to its appearance in other c.e. sets. We present here
Compositions of Permutations and Algorithmic Reducibilities
"... For wttreducibility or an arbitrary reducibility stronger then wtt reducibility, there exists a set F T 0 0 such that the set ff j f is a permutation of ! and the graph of f reduce to Fg is not closed under composition. Let A be an arbitrary set such that A T 0 0 . There exists a permutati ..."
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For wttreducibility or an arbitrary reducibility stronger then wtt reducibility, there exists a set F T 0 0 such that the set ff j f is a permutation of ! and the graph of f reduce to Fg is not closed under composition. Let A be an arbitrary set such that A T 0 0 . There exists a permutation f such that the graph of f is 2c.e. computable and the graph of f 2 is not wttreducible to A. All necessary denitions can be found in [3, 4]. Algorithmic properties of permutations on natural numbers are studied in many papers. A lot of authors studied groups G d = ff j f is a permutation of ! and f T dg where d is a Turing degree. It turns out that any group G d characterizes the degree d, i.e two such groups are isomorphic i the corresponding degrees coincide and embedding on such groups is equivalent to Turing reducibility on degrees (see [2]). In [2] A. S. Morozov has formulated the following problem: is it possible to extend these results to other reducibilities? In our pa...