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Page 8388 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
In fact, Shore and Slaman extend the Jockusch and Shore result to a-REA operators and the a-jump, for all a.  ...  Shore and Slaman have shown that y(a) fails for all n-REA degrees a [“A splitting theorem for n-REA degrees”, Proc. Amer. Math.  ... 

Page 637 of Mathematical Reviews Vol. , Issue 95b [page]

1995 Mathematical Reviews  
Analogues of the Sacks jump theorem are also obtained for n-fold and wn-fold iterations of REA operators. The proofs involve clever applications of the recursion theorem. C. G.  ...  Shore extended a number of results about the Turing jump operator to REA operators, i.e. those of the form J,(A) = A@ WA for some e [see, e.g., J.  ... 

Page 5415 of Mathematical Reviews Vol. , Issue 87j [page]

1987 Mathematical Reviews  
The jump operator is an REA operator, up to recursive isomorphism. Existence proofs in the theory of r.e. degrees give many more examples of REA operators.  ...  An operator on “2 of the form J, for some e is called an REA operator or, sometimes, a pseudojump operator. (Here ‘REA’ stands for ‘r.e. in and above’.)  ... 

Degree Structures: Local and Global Investigations

Richard A. Shore
2006 Bulletin of Symbolic Logic  
Institutionally, it was an honor to serve as President of the Association and I want to thank my teachers and predecessors for guidance and advice and my fellow officers and our publisher for their work  ...  Appropriately enough, my first administrative job for the Association, some thirty years ago, was to serve on a committee to plan a reorganization of the reviews.  ...  Together with the join theorem for n-REA operators it gives the following de…nition of the Turing jump from that of the double jump.. Theorem 4.16.  ... 
doi:10.2178/bsl/1154698739 fatcat:ct6b4ko77zhyfctmtfooxbk4ua

The jump is definable in the structure of the degrees of unsolvability

S. Barry Cooper
1990 Bulletin of the American Mathematical Society  
Kleene and Post also considered the enriched structure 3 1 equipped with the "jump operator", denoted ', which is a canonical operation on degrees which takes each degree d to a strictly  ...  Richter [24] showed that each jump preserving automorphism of the degrees fixes all a > 0 (3) . • Although the join theorem is not known for arbitrary REA operators, we can apply the special cupping  ...  derivable for operators of the particular form got from H-r.e. operators.  ... 
doi:10.1090/s0273-0979-1990-15923-3 fatcat:d3g7bx5i6neufhcr5b5i32okie

Defining jump classes in the degrees below ${\bf 0}'$

Richard A. Shore
1988 Proceedings of the American Mathematical Society  
Thus for each r.e. degree r, Th(^(< r)) uniquely determines r(3).  ...  We prove that, for each degree c r.e. in and above O'3', the class of degrees x < 0' with x'3) = c is definable without parameters in 3{< 0'), the degrees below 0'.  ...  Defining the jump classes. THEOREM 2.1. For each r.e. degree r and each c rea in 0'3^ the class of degrees x < r such that x.^ = c is definable in 3(< r). PROOF.  ... 
doi:10.1090/s0002-9939-1988-0958085-4 fatcat:ktjqutmsxvasfeqphy6sbt3p6u

Defining the Turing Jump

Richard A. Shore, Theodore A. Slaman
1999 Mathematical Research Letters  
Now for the completeness theorem needed. Theorem 1.6 (Jockusch and Shore (1984) ) For any α−REA operator J and any C ≥ T 0 (α) there is an A such that J(A) ≡ T C.  ...  n-REA Operators and Kumabe-Slaman Forcing n-REA Operators and Kumabe-Slaman Forcing n-REA Operators and Kumabe-Slaman Forcing n-REA Operators and Kumabe-Slaman Forcing n-REA Operators  ...  The uniformity of this construction clearly provides a proof of the theorem for the ω − REA operators as well.  ... 
doi:10.4310/mrl.1999.v6.n6.a10 fatcat:hmel7rrvw5bcvokljzcjytw6yq

Local Definitions in Degree Structures: The Turing Jump, Hyperdegrees and Beyond

Richard A. Shore
2007 Bulletin of Symbolic Logic  
Analogous results hold for various coarser degree structures.  ...  There are Π5 formulas in the language of the Turing degrees, D, with ≤, ⋁ and ⋀, that define the relations x″ ≤ y″, x″ = y″ and so x ∈ L 2(y) = {x ≥ y ∣ x″ = y″} in any jump ideal containing 0 (ω).  ...  The join theorem for n-REA operators of Shore and Slaman [1999] then de…nes the Turing jump from that of the double jump: For any degree x, x 0 = maxfz T xj(8g T x)(z _ g 6 = g 00 )g, i.e. x 0 is the  ... 
doi:10.2178/bsl/1185803806 fatcat:6s4wggdofjdrzbk4op7yr27t5q

On the Jumps of the Degrees Below a Recursively Enumerable Degree

David R. Belanger, Richard A. Shore
2018 Notre Dame Journal of Formal Logic  
The implication (1 ⇒ 3) follows from the monotonicity of the jump operator and the fact that if X ≤ T A and Y is r.e. in X then Y is r.e. in A. The final implication (3 ⇒ 2) follows from Theorem 1.2.  ...  . * Both authors were partially supported by NSF Grant DMS-1161175. 1. y = x for some x ≤ a. 2. y = x for some r.e. x ≤ a. 3. y ≥ 0 and y is REA(a). y = deg(W Proof.  ...  Theorem 1.2 (Robinson Jump Interpolation) . If c, d, e ∈ R, e c < d, z ≥ c and is REA(d), then there is an f ∈ R with c < f < d, e f , and f = z.  ... 
doi:10.1215/00294527-2017-0014 fatcat:qmzj4i55ofgytjpoy4vajcr7iu

DIRECT AND LOCAL DEFINITIONS OF THE TURING JUMP

RICHARD A. SHORE
2007 Journal of Mathematical Logic  
For a lower bound on de...nability, we show that no 2 or 2 formula in the language with just de...nes L 2 or L 2 (y).  ...  We show that there are 5 formulas in the language of the Turing degrees, D, with ,_ and^, that de...ne the relations x 00 y 00 , x 00 = y 00 and so x 2 L 2 (y) = fx yjx 00 = y 00 g in any jump ideal containing  ...  The join theorem for n-REA operators of Shore and Slaman [1999] then de…nes the Turing jump from that of the double jump: For any degree x, x 0 = maxfz T xj(8g T x)(z _ g 6 = g 00 )g, i.e. x 0 is the  ... 
doi:10.1142/s0219061307000676 fatcat:ymhxmsnwjrbidpz3q3krj3emvi

Page 578 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews  
The authors prove, via a forcing argument, that every countable jump partial ordering is embeddable in (D, <r, jy), the Turing degrees with the jump operator.  ...  Another observation shows that, in fact, the e-degrees of the n-rea sets are dense for each n.  ... 

Abstract hierarchies and degrees

Ljubomir L. Ivanov
1989 Journal of Symbolic Logic (JSL)  
Theorem 3 . 3 11. For all a there is a degree a' called the jump of a s.t. a' ~REa and whenever b ~REa, then b ~a'.  ...  The following statement introduces a jump operator over D. Proof: Take a ~ E a, then take a' = dgi: (~+) by definition. It is immediate that a' < a. Suppoc;e that b ~RE a.  ... 
doi:10.2307/2275011 fatcat:jgpbh5gu6zb3lihl7n3j3ue6ve

Low Level Nondelegability Results: Domination and Recursive Enumeration

Mingzhong Cai, Richard A. Shore
2013 Journal of Symbolic Logic (JSL)  
We prove a variety of results, including, for example, that being array nonrecursive is not definable by a Σ1 or Π1 formula in the language (≤, REA) where REA stands for the "r.e. in and above" predicate  ...  We also show that the Σ1-theory of (, ≤, REA) is decidable.  ...  In addition to the basic language ( ), they suggest two extensions augmenting it by either the jump operator 0 or the relation RE for "recursively enumerable in."  ... 
doi:10.2178/jsl.7803180 fatcat:c26gbzzhfzahvde4pyt2g2yj3m

Page 6730 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews  
95k:47104 47 OPERATOR THEORY 6730 ity of the operator I’ there is an incomprehensible jump from the modulars py and py to py, and py, (p. 741, starting at line 6). (6) In the formulation of Theorems 1,  ...  +a, -1<Rea< 0}, which is bounded by the “critical parabola” (4) P = {A¢€ C: 4=a*?+a, Rea =0}.  ... 

Interpolating d-r.e. and REA degrees between r.e. degrees

Marat Arslanov, Steffen Lempp, Richard A. Shore
1996 Annals of Pure and Applied Logic  
Here the motivating ideas were the jump operator and relative recursive enumerability. Definition 1.2. We define the sets REA in X by induction. 1. X is 0-REA in X. 2.  ...  ., c is low and h is high, then there is an a < h which is REA in c but not r.e. Theorem 2.1.  ...  Then a generalization of both the Friedberg completeness theorem and the Posner-Robinson cupping theorem for aREA operators derived from a-r.e. ones proved in [22] is applied to see that every X 6 O(  ... 
doi:10.1016/0168-0072(96)00014-0 fatcat:7lxj22cs6nelfkblxlqdwklfxm
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