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A non-splitting theorem for d.r.e. sets

Xiaoding Yi
1996 Annals of Pure and Applied Logic  
Sacks showed that for each non-recursive recursively enumerable set A there are disjoint recursively enumerable sets B, C which cover A such that A is recursive in neither A n B nor A n C.  ...  Abstract A set of natural numbers is called d.r.e.  ...  There exists a properly d.r.e. set D such that for all r.e. sets AO, A1 with A0 n A1 = 0, D~A O U A~~[ D <~A O~D V D <~A~~D ] . r 6 3 ' 3 J, for all j E {jo,.  ... 
doi:10.1016/0168-0072(95)00070-4 fatcat:enkgzixxwreqbawb7n66q3a3fq

Page 5961 of Mathematical Reviews Vol. , Issue 97J [page]

1997 Mathematical Reviews  
This is in contrast to Sacks’s splitting theorem for re. sets.  ...  Peter Cholak (1-NDM; Notre Dame, IN) 03D Recursion theory 97}:03089 97j:03086 03D30 03D25 Yi, Xiaoding (1-CT; Storrs, CT) A non-splitting theorem for d.r.e. sets. (English summary) Ann. Pure Appl.  ... 

Recursively enumerable sets and degrees

Robert I. Soare
1978 Bulletin of the American Mathematical Society  
The structure, automorphisms, and elementary theory of the r.e. sets. 6. Basic facts and splitting theorems. 7. Hh-simple sets. 8. Major subsets and r-maximal sets. 9. Automorphisms of &. 10.  ...  A subset A of <o (the set of nonnegative integers) is recursive (also called decidable or computable) if there is an algorithm for determining whether a  ...  SETS 6. Basic facts and splitting theorems.  ... 
doi:10.1090/s0002-9904-1978-14552-2 fatcat:arxp4btvhzfzjeoufybmnemt2u

Master index to volumes 61-70

1994 Annals of Pure and Applied Logic  
Jin, R. and Shelah, S., Essential Kurepa trees versus essential Jech-Kunen trees Joyal, A. and Moerdijk, I., A completeness theorem for open maps Kaddah, D., Infima in the d.r.e. degrees Kagan, V., Nerode  ...  with non-constructive p-operator.  ... 
doi:10.1016/s0168-0072(94)90013-2 fatcat:jcewrks7xbf4peme3526yoypre

d-simple sets, small sets, and degree classes

Manuel Lerman, Robert Soare
1980 Pacific Journal of Mathematics  
Thus ^iCΰ and D splits L u and likewise for D replaced by N, the degrees containing non-small sets. We do not know whether D^L λ or whether N -D. THEOREM 4.1.  ...  The construction is very similar to the usual construction [18, Theorem 4.1] of a low simple set A except that A must now intersect certain infinite d.r.e. sets instead of certain infinite r.e. sets.  ... 
doi:10.2140/pjm.1980.87.135 fatcat:xilieu56zbh7fi57s2wg3bbyxa

The index set of uncountably categorical theories

Uri Andrews, Tamvana Makuluni
2013 Israel Journal of Mathematics  
We show that this index set surprisingly falls at the intermediate stage of being complete for intersections of Π 2 sets with Σ 2 sets.  ...  We classify the complexity of the index set of uncountably categorical theories.  ...  Surprisingly, uncountable categoricity is arithmetical and lies at an intermediate level, being complete for intersections of a Π 2 set and a Σ 2 set (also known as 0 -d.r.e. sets).  ... 
doi:10.1007/s11856-013-0011-1 fatcat:3ficgnd3v5dy5lq3siqgy7miwi

On α- and β-recursively enumerable degrees

Wolfgang Maass
1979 Annals of Mathematical Logic  
In particular by applying the splitting theorem to many local structures we get a global splitting theorem for all regular /3-r.e. degrees.  ...  It turns out that routine precautions are sufficient to make the proof work for all admissible structures. Call a set D 9l-d.r.e. if D = A -B for some ')l-r e. sets A. B.  ... 
doi:10.1016/0003-4843(79)90002-0 fatcat:kaew2ei5gjgb7p4g76ba277uly

On the Turing Degrees of Weakly Computable Real Numbers

X. Zheng
2003 Journal of Logic and Computation  
The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust.  ...  Therefore, every non-computable c.e. degree contains both semi-computable and non-semi-computable real numbers. Theorem 1.7 For any non-computable c.e.  ...  By Sacks' Splitting Theorem [14] there exist two incomparable c.e. degrees b 0 , b 1 such that a = b 0 ∪ b 1 . Choose two c.e. sets B 0 ∈ b 0 and B 1 ∈ b 1 and define set A =: B 0 ⊕ B 1 .  ... 
doi:10.1093/logcom/13.2.159 fatcat:4lp67kl5vvh2tpphr53gxt4w6e

Page 96 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
theorem for d.r.e. sets.  ...  (English summary) 971:03026 — (with Sui, Yuefei) An extended Lachlan splitting theorem. (English summary) 97b:03055 — see also Arsianov, M.  ... 

Page 104 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
(English summary) 98c:03092 Kontostathis, Kyriakos The combinatorics of the splitting theorem. 98j:03059 Kumabe, Masahiro Minimal complementation below uniform upper bounds for the arithmetical degrees  ...  03D25 03D25 Recursively enumerable sets and degrees Cichon, E. A. (with Weiermann, Andreas) Term rewriting theory for the primitive recursive functions. (English summary) 98b:03057 Cooper, S.  ... 


2016 Bulletin of Symbolic Logic  
, or a set of assignments.  ...  In set theory, a reflection principle says that a proposition true in the universe holds already in a smaller set.  ...  We prove that for some sequence of tautologies ϕn the proof steps and the proof sizes in  ... 
doi:10.1017/bsl.2016.22 fatcat:cm4a5yyvgfajzm4qvlszviblni

Page 110 of Mathematical Reviews Vol. 32, Issue Index [page]

Mathematical Reviews  
Maximal sets and the jump operator. 2000a:03065 Ding, Decheng see Lu, Hong et al., 2000f:03123 and 2000m:03100 Downey, Rodney G. (with Shore, Richard A.) Splitting theorems and the jump operator.  ...  Arslanov, Asat Difference splittings of recursively enumerable sets.  ... 

Page 90 of Mathematical Reviews Vol. 27, Issue Index [page]

Mathematical Reviews  
(English summary) 95h:03103 Splitting theorems in recursion theory. (English Highness and bounding minimal pairs. with Kummer, Martin Walter) Diagonals and ®-maximal sets.  ...  (with Yi, Xiaoding) Jump theorems for REA operators. (English summary) 95b:03048 Morales-Luna, Guillermo Degrees of complexity of recursive functions and degrees of unsolvability. (Spanish.  ... 

Page 1638 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
(Peter Cholak) 97e:03063 03D25 — A non-splitting theorem for d.r.e. sets. (English summary) Ann. Pure Appl. Logic 82 (1996), no. 1, 17-96.  ...  (see 97a:90003) 93E20 (90B70, 90C40) — (with Zhang, Qing') A central limit theorem for singularly perturbed nonstationary finite state Markov chains. (English summary) Ann. Appl.  ...