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A Friedberg enumeration of equivalence structures

Rodney G. Downey, Alexander G. Melnikov, Keng Meng Ng
2017 Journal of Mathematical Logic  
We also prove that there exists an e↵ective Friedberg enumeration of all isomorphism types of infinite computable equivalence structures.  ...  More specifically, we produce an e↵ective Friedberg (i.e., injective) enumeration of computable equivalence structures, up to isomorphism.  ...  There exists a Friedberg enumeration of the class of computable equivalence structures, up to isomorphism.  ... 
doi:10.1142/s0219061317500088 fatcat:zhvnxrd5jve3roofqu42sfx4lm

Intrinsic bounds on complexity and definability at limit levels

John Chisholm, Ekaterina B. Fokina, Sergey S. Goncharov, Valentina S. Harizanov, Julia F. Knight, Sara Quinn
2009 Journal of Symbolic Logic (JSL)  
We show that for every computable limit ordinal α, there is a computable structure that is categorical, but not relatively categorical (equivalently, it does not have a formally Scott family).  ...  We also show that for every computable limit ordinal α, there is a computable structure with an additional relation R that is intrinsically on , but not relatively intrinsically on (equivalently, it is  ...  S has a unique anti-Friedberg ω-f -enumeration, up to strong ∆ 0 ω -equivalence. Proof. We construct an anti-Friedberg ω-f -enumeration G of a family S.  ... 
doi:10.2178/jsl/1245158098 fatcat:swighvldknbpjnouhuj6ys2t4e

Page 4 of Mathematical Reviews Vol. , Issue 81K [page]

1981 Mathematical Reviews  
An enumeration y is a Gédel num- bering if y satisfies the enumeration and s-m-n theorems and y is a Friedberg numbering if is one-to-one. (R. M. Friedberg con- structed a Friedberg numbering [J.  ...  Regarding homo- geneity it is shown (Theorem 5.8) that if the ordering of degrees with a parameter for 0 is elementarily equivalent to the ordering of degrees >b with a corresponding parameter for b’,  ... 

Comparing Classes of Finite Structures

W. Calvert, D. Cummins, J. F. Knight, S. Miller
2004 Algebra and Logic  
Then (A n ) n∈ω is an infinite computable Friedberg enumeration of a subclass of K. Now, suppose that(A n ) n∈ω is an infinite computable Friedberg enumeration of a subclass of K.  ...  There is a class of finite structures K that has a computable enumeration but no computable Friedberg enumeration. Proof. We want to create a class K with a computable enumeration E.  ... 
doi:10.1023/b:allo.0000048827.30718.2c fatcat:x3rzqjvvfjdi7nu7fz6iqoaxzi

Comparing Classes of Finite Structures [article]

Wesley Calvert, Desmond Cummins, Sara Miller, Julia F. Knight
2008 arXiv   pre-print
We introduce a reducibility on classes of structures, essentially a uniform enumeration reducibility.  ...  The class of cyclic graphs and the class of finite prime fields are equivalent, and are properly below the class of arbitrary finite graphs.  ...  Then (A n ) n∈ω is an infinite computable Friedberg enumeration of a subclass of K. Now, suppose that(A n ) n∈ω is an infinite computable Friedberg enumeration of a subclass of K.  ... 
arXiv:0803.3291v1 fatcat:xxrl4kc5hnd43hn66zgh5knneu

Calibrating Computational Complexity via Definability: The Work of Julia F. Knight

Karen Lange
2022 Notices of the American Mathematical Society  
Determining whether the class of equivalence structures has a computable Friedberg enumeration proved challenging.  ...  However, Goncharov and Knight's conjecture was false-Downey, Melnikov, and Ng provided a Friedberg enumeration of all equivalence structures in 2017.  ... 
doi:10.1090/noti2436 fatcat:6g2qjctksnd7plspwz7ysetjpa

Page 4405 of Mathematical Reviews Vol. , Issue 2001G [page]

2001 Mathematical Reviews  
Call a pair of computably enumerable (c.e.) sets (Ap, A)) a Friedberg splitting of a c.e. set A if A is the disjoint union of Ao and A}, and for any c.e. set B, if Ag B or A; MB is complemented in the  ...  A congruence relation on the lattice of c.e. sets is an equivalence relation on # such that union and intersection are well defined on the equivalence classes.  ... 

Page 31 of Mathematical Reviews Vol. 58, Issue 1 [page]

1979 Mathematical Reviews  
Post’s problem about the existence of T-incomparable recursively enumerable sets was solved by R. Friedberg and A. Muénik by another method.  ...  From the introduction: “We fix a recursively enumerable (r.e.) field F with recursive structure, and let U be the vector space over F consisting of the ultimately vanishing elements of F° with the usual  ... 

Page 2284 of Mathematical Reviews Vol. , Issue 86f [page]

1986 Mathematical Reviews  
of recursively enumerable sets (pp. 21-32); Richard A.  ...  Kechris, Determinacy and the structure of L(R) (pp. 271- 283); Alain Louveau, Recursivity and capacity theory (pp. 285- 301); Donald A.  ... 

JSL volume 67 issue 2 Cover and Front matter

2002 Journal of Symbolic Logic (JSL)  
on computable structures in the presence of A-J isomorphisms, by DENIS R.  ...  FERNANDES and FERNANDO On orbits, pf prompt and low computably enumerable sets, by KEVIN WALD . 649 Definable incompleteness and Friedberg splittings, by RUSSELL MILLER 679 Degree spectra of relations  ... 
doi:10.1017/s0022481200009555 fatcat:jledthgadjenrmxcg6vxfdb4le

Isomorphisms of splits of computably enumerable sets

Peter A. Cholak, Leo A. Harrington
2003 Journal of Symbolic Logic (JSL)  
We show that if A and are automorphic via Φ then the structures (A) and () are Δ3 0-isomorphic via an isomorphism Ψ induced by Φ.  ...  Then we use this result to classify completely the orbits of hhsimple sets.  ...  Introduction We will work in the structure of the computably enumerable sets. The language is just inclusion, ⊆. This structure is called E. Our understanding of automorphisms of E is unique to E.  ... 
doi:10.2178/jsl/1058448453 fatcat:toj3wyhit5earei7woorv242be

Page 1844 of Mathematical Reviews Vol. , Issue 94d [page]

1994 Mathematical Reviews  
944:03082 94d:03082 03D25 Downey, Rod (NZ-VCTR; Wellington); Stob, Michael (1-CLVN; Grand Rapids, MI) Friedberg splittings of recursively enumerable sets.  ...  A splitting A; U A2 = A of an r.e. set A is called a Friedberg (or f-) splitting if for any r.e. set W with W \ A notr.e., W. Aj #2 for i= 1,2.  ... 

Algebraic aspects of the computably enumerable degrees

T. A. Slaman, R. I. Soare
1995 Proceedings of the National Academy of Sciences of the United States of America  
A set A of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements.  ...  The class of sets B which contain the same information as A under Turing computability (<T) is the (Turing) degree ofA, and a degree is c.e. if it contains a c.e. set.  ...  These results demonstrated that 2R has a more complicated global structure than anticipated by the results of the 1960s.  ... 
doi:10.1073/pnas.92.2.617 pmid:11607508 pmcid:PMC42793 fatcat:4a6nozmejjdlblqaee3wz2diwq

Friedberg numberings in the Ershov hierarchy

Serikzhan A. Badaev, Mustafa Manat, Andrea Sorbi
2014 Archive for Mathematical Logic  
We show that for every n ≥ 1, there exists a Σ −1 n -computable family which up to equivalence has exactly one Friedberg numbering which does not induce the least element of the corresponding Rogers semilattice  ...  Rogers semilattice R i a (A) of a family A ⊆ Σ i a is a quotient structure of all Σ i a -computable numberings of the family A modulo equivalence of the numberings ordered by the relation induced by reducibility  ...  For every k, if π k is a Friedberg numbering of A then g k is a total function and g k reduces π k to α. Proof. Suppose that π k is a Friedberg numbering of A.  ... 
doi:10.1007/s00153-014-0402-y fatcat:52zsc6vsevgmxcaeaguogpmzqq

Page 3557 of Mathematical Reviews Vol. , Issue 83i [page]

1983 Mathematical Reviews  
For the structures, he describes the closure of R under elementary equivalence, and the notions of projective relation (ARB then means that A and B are interpreted in a certain way within the same structure  ...  s book [Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967; MR 37 #61] for basic terminology). In 1956, R. M. Friedberg and A. A.  ... 
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