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Completing pseudojump operators

R. Coles, R. Downey, C. Jockusch, G. LaForte
2005 Annals of Pure and Applied Logic  
We introduce notions of nontriviality for such 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.  ... 
doi:10.1016/j.apal.2003.07.001 fatcat:q2ocnmxh75c25kvdm6vagnvoqi

Pseudojump inversion in special r. b. Π^0_1 classes [article]

Hayden R. Jananthan, Stephen G. Simpson
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 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 .  ... 
arXiv:2102.06135v1 fatcat:eoxc4k5mfzfqhg6xvkpavofzay

A remark on pseudojump operators

A. H. Lachlan, X. Yi
1989 Proceedings of the American Mathematical Society  
By choosing e such that Wg(X) = X © E(X) for all X ç co, we 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.  ... 
doi:10.1090/s0002-9939-1989-0937847-4 fatcat:ghr5dfrlvnd6rkbwljip6gg7me

Author Index

2005 Annals of Pure and Applied Logic  
., 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  ... 
doi:10.1016/s0168-0072(05)00122-3 fatcat:2sofulnvyvcplafvgoparymqni

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  ... 

On the Correctness of an Optimising Assembler for the Intel MCS-51 Microprocessor [chapter]

Dominic P. Mulligan, Claudio Sacerdoti Coen
2012 Lecture Notes in Computer Science  
This comes at the expense of assembler 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.  ... 
doi:10.1007/978-3-642-35308-6_7 fatcat:zxdnnozk5betrcxnrejgvcwdue

On resolutions of linearly ordered spaces

Agata Caserta, Alfio Giarlotta, Stephen Watson
2006 Applied General Topology  
(Note that the lexicographic 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.  ... 
doi:10.4995/agt.2006.1925 fatcat:k2c4uugapvhxhckdo4mqbsuzhe

STRONG JUMP-TRACEABILITY

NOAM GREENBERG, DAN TURETSKY
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.  ... 
doi:10.1017/bsl.2017.38 fatcat:w2ufaz6uffg6xlc4jn7yv4m56a

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).  ... 

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.  ... 

Pseudojump operators. I. The r.e. case

Carl G. Jockusch, Richard A. Shore
1983 Transactions of the American Mathematical Society  
Thus the (Turing) jump 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.  ... 
doi:10.1090/s0002-9947-1983-0682720-1 fatcat:ucebmoplubd2zm2pe35nohg4r4

Degree Structures: Local and Global Investigations

Richard A. Shore
2006 Bulletin of Symbolic Logic  
Jockusch and Shore [1984] then analyzed the notion of pseudojumps or iterated REA operators (e.g.  ...  We now have essentially complete classi…cations. Theorem 2.12.  ... 
doi:10.2178/bsl/1154698739 fatcat:ct6b4ko77zhyfctmtfooxbk4ua

The strength of Turing determinacy within second order arithmetic

Antonio Montalb\'an, Richard A. Shore
2015 Fundamenta Mathematicae  
By 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  ... 
doi:10.4064/fm27-12-2015 fatcat:gljkd4ob5bdh7hemgugeofdbxi

HTP-complete rings of rational numbers [article]

Russell Miller
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-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  ... 
arXiv:1907.03147v2 fatcat:oid3mzpb4rc35k2hm3l7vhs47a

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

Richard A. Shore
2007 Bulletin of Symbolic Logic  
Jockusch and Shore [1984] then introduced and analyzed the notion of 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.  ... 
doi:10.2178/bsl/1185803806 fatcat:6s4wggdofjdrzbk4op7yr27t5q
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