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Complementing below recursively enumerable degrees

1987
*
Annals of Pure and Applied Logic
*

., there is some

doi:10.1016/0168-0072(87)90039-x
fatcat:cvtjmbjgybcbvhnvli4m34i3re
*degree*m < a such that m U b = a and m is minimal. We prove*below*two theorems related to this. ... Epstein showed, in fact, that in the special case of 0 < a r.e. < 0', one can construct a minimal*degree**complement*for a. ...*Complementing**below*r.e.*degrees*21 This completes the verification of our claim. ...##
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Degrees of Weakly Computable Reals
[chapter]

2006
*
Lecture Notes in Computer Science
*

Then 0.χ C is left-computable but not computable. • (Soare) A real α is called strongly

doi:10.1007/11780342_43
fatcat:roqnxjutfvfhtn7k7ehqto2oyq
*recursively**enumerable*(s.r.e. for short) if there exists a*recursively**enumerable*set A ⊂ ω such that α = n∈A 2 ... The proofs use the existence of noncuppable*degrees*and the fact that every nonzero r.e.*degree*has a 1-generic*complement*(Slaman and Steel) and a minimal*complement*(Seetapun and Slaman). • (Wu, 2006 ... There is a 1-generic*degree**below*0 such that every nonzero*degree**below*it contains no weakly computable reals. 2. ...##
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Page 1367 of Mathematical Reviews Vol. , Issue 90C
[page]

1990
*
Mathematical Reviews
*

Lachlan, which asserts that there do not exist such

*degrees*a, b which are*recursively**enumerable*.) ... In the paper under review, a new proof that the*degrees*< 0’ are*complemented*is given. ...##
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Page 807 of Mathematical Reviews Vol. , Issue 2003B
[page]

2003
*
Mathematical Reviews
*

Any c.e. set with semilow

*complement*is automorphic to some c.e. set*below*any given promptly simple*degree*. ... A Turing*degree*is d.c.e. if it contains a set that is the difference of c.e. (computably*enumerable*) sets. A d.c.e.*degree*d is isolated by a c.e.*degree*a if a is the greatest c.e.*degree**below*d. ...##
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Enumeration reducibility and partial degrees

1971
*
Annals of Mathematical Logic
*

The following theorem

doi:10.1016/0003-4843(71)90003-9
fatcat:qw5uuzq3kjhothy5rkfrbn5tre
*complements*(2.7). Theorem 2.8. Every r.e. non-*recursive*Turing*degree*contains a set A ~'uch that A le -~ and A, A lie in total partial*degrees*. Proof. ... Following [ 5 ] and [6, p. 146] a formal definition is give~*below*. In this paper*enumeration*reducibility and a ge~eralization of*enumeration*reducibility, arithmetical*enumerability*, a~e studied. ...##
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Page 5456 of Mathematical Reviews Vol. , Issue 98I
[page]

1998
*
Mathematical Reviews
*

Downey using known results: Let ap and a; be computably

*enumerable**degrees*such that a, a;, and a @a;, are contiguous, ap does not bound half of a minimal pair and every computably*enumerable**degree**below*... {For the entire collection see MR 98b:00020. } Peter Cholak (1-NDM; Notre Dame, IN) 98i:03057 03D30 Kumabe, Masahiro Minimal*complementation**below*uniform upper bounds for the arithmetical*degrees*. ...##
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Page 5035 of Mathematical Reviews Vol. , Issue 82m
[page]

1982
*
Mathematical Reviews
*

Posner on the

*degrees**below*0 (the latest results on*complementing**below*h high b+ © contained in his article (“Minimal*degrees*and high*degrees*”, to appear]). Barry Cooper (Leeds) Mal'cev, An. ...*enumerable*sets there is a total*recursive*function f(i,j) such that TC W,, RCW,=f(i,j)€ W,U W,. ...##
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On the Degrees of Index Sets

1966
*
Transactions of the American Mathematical Society
*

The latter approach is not only more precise but also, as we show

doi:10.2307/1994481
fatcat:odxkbmn65jdqxkylncn2wgfx6q
*below*, provides an alternative method for solving certain problems on*recursively**enumerable*sets and their*degrees*of unsolvability. ... The main result of the present paper is the computation, for every*recursively**enumerable**degree*a, of the*degree*(in fact, isomorphism-type) of the index-set corresponding to the*recursively**enumerable*... . // b is any*recursively**enumerable**degree*, then the*degree*of Gib) is b(3\ Also G(fc)eS3(fe) -n3(fc). Proof. ...##
###
Hereditarily retraceable isols

1967
*
Bulletin of the American Mathematical Society
*

Moreover, if a has

doi:10.1090/s0002-9904-1967-11669-0
fatcat:dcx6ywhla5fcbdz6ndwqi7ecvm
*recursively**enumerable**complement*then we can satisfy the additional requirement that /3 have*recursively**enumerable**complement*. ... Such, then, are the objects referred to in the title of the note; the existence of a continuum of them follows from Theorem 1*below*. Our terminology is, in all other respects, that of [l], [2]. ... Moreover, if a has*recursively**enumerable**complement*then we can satisfy the additional requirement that /3 have*recursively**enumerable**complement*. ...##
###
On the degrees of index sets

1966
*
Transactions of the American Mathematical Society
*

The latter approach is not only more precise but also, as we show

doi:10.1090/s0002-9947-1966-0184855-9
fatcat:4u3meicnn5fxrkpansfpp2pygu
*below*, provides an alternative method for solving certain problems on*recursively**enumerable*sets and their*degrees*of unsolvability. ... The main result of the present paper is the computation, for every*recursively**enumerable**degree*a, of the*degree*(in fact, isomorphism-type) of the index-set corresponding to the*recursively**enumerable*... If sé is any many-one*degree*that contains an infinite*recursively**enumerable*set whose*complement*is nonempty, then the*degree*of Gísé) is 0(3). Also G0OeE3-n3. Proof. ...##
###
Page 4723 of Mathematical Reviews Vol. , Issue 87i
[page]

1987
*
Mathematical Reviews
*

(A Turing

*degree*a is said to be nearly*recursive*, or bi-immune free, if for any set A of*degree*< a, either A or its*complement*has an infinite*recursively**enumerable*subset.) ... For instance, it is shown that every e-*degree*b which is “D2-high” has the property that every “low” e-*degree*a*below*b is part of a minimal pair of e-*degrees**below*b. ...##
###
Page 464 of Mathematical Reviews Vol. , Issue 86b
[page]

1986
*
Mathematical Reviews
*

Press, Cam- bridge, 1980; MR 83i:03071], this paper presents a result with striking negative implications for the possible jumps of

*degrees**below*high*degrees*. ... Author’s summary (translated from the Russian): “We prove the existence of a*recursively**enumerable*independent subset in a con- structive Boolean algebra, which cannot be*complemented*.” ...##
###
Completeness, the Recursion Theorem, and Effectively Simple Sets

1966
*
Proceedings of the American Mathematical Society
*

., of the highest

doi:10.2307/2036264
fatcat:bgvxrodyorfbbhg7g4wit44ui4
*recursively**enumerable**degree*of unsolvability. ... A*recursively**enumerable*set of natural numbers is called simple if its*complement*, though infinite, possesses no infinite*recursive*subset. ...##
###
Completeness, the recursion theorem, and effectively simple sets

1966
*
Proceedings of the American Mathematical Society
*

., of the highest

doi:10.1090/s0002-9939-1966-0216950-5
fatcat:d3rvrarjozcvzmku3rximbel6q
*recursively**enumerable**degree*of unsolvability. ... A*recursively**enumerable*set of natural numbers is called simple if its*complement*, though infinite, possesses no infinite*recursive*subset. ...##
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On a Problem of G. E. Sacks

1965
*
Proceedings of the American Mathematical Society
*

A set F is called j-

doi:10.2307/2035595
fatcat:d7m7i2gukfdvzcq6hxo44rkmde
*recursive*if both F and its*complement*T arej-*enumerable*; such a set is to be specified by indices of F and F as /-*enumerable*sets. ... Further, their /-*recursiveness*is uniform in / and n in the sense that given j and n we can effectively find indices of AnJ, A\J and their*complements*as/-*enumerable*sets. ...
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