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Completing pseudojump operators

2005
*
Annals of Pure and Applied Logic
*

We introduce notions of nontriviality for such

doi:10.1016/j.apal.2003.07.001
fatcat:q2ocnmxh75c25kvdm6vagnvoqi
*operators*, and use these to study which additional properties can be required of sets which can be*completed*to the jump by given*operators*of this kind. ... We investigate*operators*which take a set X to a set relatively computably enumerable in and above X by studying which such sets X can be so mapped into the Turing degree of K . ... We consider the existence of incomparable c.e. sets*completing**pseudojump**operators*, and the question of the existence of sets not of c.e. degree*completing**pseudojump**operators*. ...##
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Pseudojump inversion in special r. b. Π^0_1 classes
[article]

2021
*
arXiv
*
pre-print

analogous refinements of two other well-known theorems: the Join Theorem – for all reals A and Z such that A ≥_T Z ⊕ 0' and Z >_T 0, there is a real B such that A ≡_T B' ≡_T B ⊕ 0' ≡_T B ⊕ Z – and the

arXiv:2102.06135v1
fatcat:eoxc4k5mfzfqhg6xvkpavofzay
*Pseudojump*... The existence of such*pseudojump**operators*is well known [20, §VII.1]. One way to obtain such an*operator*is to combine part 4 of Theorem 6.2 with the R. E. ... To see that*Pseudojump*Inversion fails for Q, consider a*pseudojump**operator*J e with the property 4 that X < T J e (X) and (J e (X)) ′ ≡ T X ′ for all X ∈ N N . ...##
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A remark on pseudojump operators

1989
*
Proceedings of the American Mathematical Society
*

By choosing e such that Wg(X) = X © E(X) for all X ç co, we

doi:10.1090/s0002-9939-1989-0937847-4
fatcat:ghr5dfrlvnd6rkbwljip6gg7me
*complete*the proof of the lemma. For any F ç (<(02) x co and a Çrw 2, let F (a) denote {«: (x,n) e F ,x ç o}. At stage 0 we do nothing. ...##
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Author Index

2005
*
Annals of Pure and Applied Logic
*

.,

doi:10.1016/s0168-0072(05)00122-3
fatcat:2sofulnvyvcplafvgoparymqni
*Completing**pseudojump**operators*(3) 297-333 Downey, R., see Coles, R. (3) 297-333 Goncharov, S., Harizanov, V., Knight, J., McCoy, C., Miller, R. and Solomon, R., Enumerations in computable structure ...##
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Page 2834 of Mathematical Reviews Vol. , Issue 86g
[page]

1986
*
Mathematical Reviews
*

Kobzev (Tbilisi)
Jockusch, Carl G., Jr. (1-CRNL); Shore, Richard A. (1-CRNL)

*Pseudojump**operators*. II. Transfinite iterations, hierarchies and minimal covers. J. ... An*operator*® is 1-REA if it is an r.e.*operator*. ® is n-REA for n > 1 if, for all sets X, there is an index e such that 6(X) = X @W, where {W; } is a standard enumeration of the r.e.*operators*and X is ...##
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On the Correctness of an Optimising Assembler for the Intel MCS-51 Microprocessor
[chapter]

2012
*
Lecture Notes in Computer Science
*

This comes at the expense of assembler

doi:10.1007/978-3-642-35308-6_7
fatcat:zxdnnozk5betrcxnrejgvcwdue
*completeness*as the generated program may be too large for code memory, there being a trade-off between the*completeness*of the assembler and the efficiency of the ... The*complete*development is spread across 29 files with around 20,000 lines of Matita source. ...##
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On resolutions of linearly ordered spaces

2006
*
Applied General Topology
*

(Note that the lexicographic

doi:10.4995/agt.2006.1925
fatcat:k2c4uugapvhxhckdo4mqbsuzhe
*completion*of a family of chains is obtained by inserting a jump per*pseudojump*, thus eliminating all*pseudojumps*in the representation of the sum.) ... ., the isomorphic chains (0, 1) (which lacks*pseudojumps*), (0, 1/2) ⊕ [1/2, 1) (which has exactly one*pseudojumps*) and n∈ω 1 n+3 , 1 n+2 ⊕ (1/2, 1) (which has countably many*pseudojumps*). ... This*completes*the definition of ψ in case (ii). It is immediate to check that ψ is injective and order-preserving. Thus ψ is a TO-embedding by Lemma 3.2. ...##
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STRONG JUMP-TRACEABILITY

2018
*
Bulletin of Symbolic Logic
*

*Pseudojump*inversion. A

*pseudojump*

*operator*is an increasing enumeration

*operator*: an

*operator*which takes a set X to a set J X ě T X which is c.e. in X. ... In particular, they introduced the technique of

*pseudojump*inversion, showing for example that if J is a strictly increasing

*pseudojump*

*operator*then 0 1 contains J X for some c.e. set X. ...

##
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Page 3159 of Mathematical Reviews Vol. , Issue 90F
[page]

1990
*
Mathematical Reviews
*

H. (3-SFR); Yi, X. (3-SFR)
A remark on

*pseudojump**operators*. Proc. Amer. Math. Soc. 106 (1989), no. 2, 489-491. Summary: “Let {W,}new be an enumeration of the recursively enumerable sets. ... A degree a is*completely*mitotic if every r.e. set in the degree is mitotic.*Completely*mi- totic degrees exist (Ladner), but not all r.e. degrees are*completely*mitotic (Ingrassia). ...##
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Page 5415 of Mathematical Reviews Vol. , Issue 87j
[page]

1987
*
Mathematical Reviews
*

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’.) ... Analogues of the Friedberg*completeness*criterion and the existence of a nonrecursive r.e. set A with A’ =; K hold with the jump*operator*replaced by an arbitrary REA*operator*. ...##
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Pseudojump operators. I. The r.e. case

1983
*
Transactions of the American Mathematical Society
*

Thus the (Turing) jump

doi:10.1090/s0002-9947-1983-0682720-1
fatcat:ucebmoplubd2zm2pe35nohg4r4
*operator*is a pseudo jump*operator*, and any existence proof in the theory of re. degrees yields, when relativized, one or more pseudo jump*operators*. ... Call an*operator*/ on the power set of w a pseudo jump*operator*if J(A) is uniformly recursively enumerable in A and A is recursive in J(A) for all subsets A of u. ... This*completes*the proof. ...##
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Degree Structures: Local and Global Investigations

2006
*
Bulletin of Symbolic Logic
*

Jockusch and Shore [1984] then analyzed the notion of

doi:10.2178/bsl/1154698739
fatcat:ct6b4ko77zhyfctmtfooxbk4ua
*pseudojumps*or iterated REA*operators*(e.g. ... We now have essentially*complete*classi…cations. Theorem 2.12. ...##
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The strength of Turing determinacy within second order arithmetic

2015
*
Fundamenta Mathematicae
*

By

doi:10.4064/fm27-12-2015
fatcat:gljkd4ob5bdh7hemgugeofdbxi
*pseudojump*inversion for REA*operators*(Jockusch and Shore [1984] ), which can also easily be proven in ACA 0 , there is an Z with Z > T Y such that W Z ≡ T Y . ... P = {X | ∃T [T ≡ T W X &T is a*complete*extension of T whose term model M I is an ω-model & ∀T (T ≡ T X &T is a*complete*extension of T whose term model M II is an ω-model → conditions R I new or R I 3 ...##
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HTP-complete rings of rational numbers
[article]

2021
*
arXiv
*
pre-print

In contrast, we show that every Turing degree contains a set W for which such a 1-reduction does hold: these W are said to be "HTP-

arXiv:1907.03147v2
fatcat:oid3mzpb4rc35k2hm3l7vhs47a
*complete*." ... We view HTP as an enumeration*operator*, mapping each set W of prime numbers to HTP(ℤ[W^-1]), which is naturally viewed as a set of polynomials in ℤ[X_1,X_2,...]. ... The jump*operator*J, mapping each A to A ′ , is the prototype of the functions called*pseudojump**operators*by Jocksuch and Shore in [5, 6] , whose output can be enumerated uniformly when we are given ...##
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Local Definitions in Degree Structures: The Turing Jump, Hyperdegrees and Beyond

2007
*
Bulletin of Symbolic Logic
*

Jockusch and Shore [1984] then introduced and analyzed the notion of

doi:10.2178/bsl/1185803806
fatcat:6s4wggdofjdrzbk4op7yr27t5q
*pseudojumps*or iterated REA*operators*(e.g. J e (A) = A W A e and then iterations of such*operators*into the trans…nite). ... The …rst was a version of a cone-avoiding join and*completeness*theorem for those 2-REA*operators*that correspond to constructions of d-r.e. sets, i.e. ones of the form A B for A and B r.e. ...
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