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Computable isomorphisms, degree spectra of relations, and Scott families

Bakhadyr Khoussainov, Richard A. Shore
1998 Annals of Pure and Applied Logic  
Of course, the finite components are again isomorphic by the existence of isomorphisms at each stage. 2 Claim 7. The relation P is computable. Proof of Claim 7.  ...  A 0 and A 1 are isomorphic. Proof of Claim 6. It is immediate from the construction that A 0,t and A 1,t are isomorphic for every t.  ...  There has been an extensive study of the degree spectra of relations on computably structures and related results.  ... 
doi:10.1016/s0168-0072(97)00059-6 fatcat:iu3nfk3tjfbp7bohtv6hpochlu

Automorphism Groups of Substructure Lattices of Vector Spaces in Computable Algebra [chapter]

Rumen Dimitrov, Valentina Harizanov, Andrei Morozov
2016 Lecture Notes in Computer Science  
Harizanov, and A. Shlapentokh, "Turing degrees of isomorphism types of geometric objects," Computability 3 (2014), pp. 105-134. 54. E. Fokina, V. Harizanov, and A.  ...  Harizanov, and D. Turetsky, "Computability-theoretic categoricity and Scott families," 25 pages, submitted. 61. J. Chubb, M. Dabkowski, and V.  ...  Harizanov, "Degree spectra of the successor relation on computable linear orderings," Archive for Mathematical Logic 48 (2009), pp. 7-13. 38. D. Cenzer, V. Harizanov, and J.  ... 
doi:10.1007/978-3-319-40189-8_26 fatcat:2p5s2kf5xzfwfdf7zakfx43mpy

Page 4450 of Mathematical Reviews Vol. , Issue 99g [page]

1999 Mathematical Reviews  
M.] (1-CRNL; Ithaca, NY); Shore, Richard A. (1-CRNL; Ithaca, NY) Computable isomorphisms, degree spectra of relations, and Scott families. (English summary) Computability theory. Ann. Pure Appl.  ...  Summary: “The spectrum of a relation R on a computable struc- ture W is the set of Turing degrees of the image of R under all isomorphisms between W and any other computable structure &.  ... 

Degrees of categoricity on a cone [article]

Barbara Csima, Matthew Harrison-Trainor
2015 arXiv   pre-print
We investigate the complexity of isomorphisms of computable structures on cones in the Turing degrees.  ...  We also show that, for any fixed computable structure, there is an ordinal α and a cone in the Turing degrees such that the exact complexity of computing an isomorphism between the given structure and  ...  Let e be such that: (1) A is e-computable, and e computes a Scott family for A in which each tuple satisfies a unique formula and also computes, for tuples in A, which formula in the Scott family they  ... 
arXiv:1506.03066v1 fatcat:fq3sau4ubngd7cxonokut44mra

DEGREES OF CATEGORICITY ON A CONE VIA η-SYSTEMS

BARBARA F. CSIMA, MATTHEW HARRISON-TRAINOR
2017 Journal of Symbolic Logic (JSL)  
We investigate the complexity of isomorphisms of computable structures on cones in the Turing degrees.  ...  We also show that, for any fixed computable structure, there is an ordinal α and a cone in the Turing degrees such that the exact complexity of computing an isomorphism between the given structure and  ...  Let e be such that: (i) A and η are e-computable, and e computes a Scott family for A in which each tuple satisfies a unique formula and also computes, for tuples in A, which formula in the Scott family  ... 
doi:10.1017/jsl.2016.43 fatcat:tc7ogdc67bg6hggr6lcpcptgzu

Page 3186 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
(NZ-VCTR-SMC; Wellington) Degree spectra of relations on computable structures in the presence of A isomorphisms. (English summary) J. Symbolic Logic 67 (2002), no. 2, 697-720.  ...  The computable dimension of a computable structure is the num- ber of different computable isomorphism classes within the clas- sical isomorphism class (that is, the number of non-isomorphic computable  ... 

Theory spectra and classes of theories

Uri Andrews, Mingzhong Cai, David Diamondstone, Steffen Lempp, Joseph S. Miller
2017 Transactions of the American Mathematical Society  
We analyze the spectra of theories that are ω-stable, theories whose spectra include almost every degree, and theories with uniformly arithmetical n-quantifier fragments.  ...  In addition, we give examples of theory spectra that contain almost every degree, including ones that are known not to be structure spectra.  ...  of Scott sets containing every arithmetical degree.  ... 
doi:10.1090/tran/6917 fatcat:a426jwuld5bizpjhuwfbxxwpj4

An introduction to the Scott complexity of countable structures and a survey of recent results [article]

Matthew Harrison-Trainor
2020 arXiv   pre-print
Such a sentence is called a Scott sentence, and can be thought of as a description of the structure.  ...  The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure.  ...  Let F G be the class of free groups.  ... 
arXiv:2011.03923v1 fatcat:ebrlfwondnfhrp35ebqetymggi

An Introduction to the Scott Complexity of Countable Structures and a Survey of Recent Results

Matthew Harrison-Trainor
2021 Bulletin of Symbolic Logic  
Such a sentence is called a Scott sentence, and can be thought of as a description of the structure.  ...  The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure.  ...  Scott spectra The results of this section were proved using the notion of Scott rank defined using symmetric back-and-forth relations (as in Definition 2.5) except with arbitrary tuples, replacing (2)  ... 
doi:10.1017/bsl.2021.62 fatcat:miy6gyratjdlflwar52e2c7tyq

Computability-Theoretic Complexity of Countable Structures

Valentina S. Harizanov
2002 Bulletin of Symbolic Logic  
Computable model theory, also called effective or recursive model theory, studies algorithmic properties of mathematical structures, their relations, and isomorphisms.  ...  One of the major tasks of computable model theory is to obtain, whenever possible, computability-theoretic versions of various classical model-theoretic notions and results.  ...  Slaman [76] and Wehner [80] have independently constructed examples of Turing degree spectra of structures consisting of all nonzero Turing degrees. This is not always the case.  ... 
doi:10.2178/bsl/1182353917 fatcat:cymnib7nufcytmjl6hsynypz7m

Computability-Theoretic Complexity of Countable Structures

Valentina S. Harizanov
2002 Bulletin of Symbolic Logic  
Slaman [76] and Wehner [80] have independently constructed examples of Turing degree spectra of structures consisting of all nonzero Turing degrees. This is not always the case.  ...  Thus, a linear order that is not isomorphic to a computable one, does not have a degree of its isomorphism type.  ... 
doi:10.2307/797952 fatcat:lvanuu567rdefjo7zcepr4t3te

Page 576 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews  
.% and a new recursive relation R on A. By definition, the degree spectrum of R on &%, S(R), is the set of Turing degrees of the images of R under the isomorphisms that send . to recursive models.  ...  There is a natural definition of computability of a family of problems 2 = (Ro, Ri, ---) on M.  ... 

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

Karen Lange
2022 Notices of the American Mathematical Society  
The author is very grateful for Knight's ongoing mentorship, collaboration, and friendship, and her assistance in putting together this article.  ...  The second approach is syntactic, relating to Scott sentences and Scott families. In this approach, an effective characterization of 𝒦 is a computable infinitary definition of 𝒦 𝑐 , if any exists.  ...  In a standard proof of the Scott Isomorphism Theorem, one first shows that there is a Scott family for the given structure 𝒜 and an ordinal 𝛼 in which the definitions of all orbits of tuples in 𝒜 are  ... 
doi:10.1090/noti2436 fatcat:6g2qjctksnd7plspwz7ysetjpa

Degree spectra and computable dimensions in algebraic structures

Denis R. Hirschfeldt, Bakhadyr Khoussainov, Richard A. Shore, Arkadii M. Slinko
2002 Annals of Pure and Applied Logic  
This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we  ...  However, this can be an unnecessary duplication of effort, and lacks generality.  ...  and degree spectra of relations.  ... 
doi:10.1016/s0168-0072(01)00087-2 fatcat:sksasinkqrhihnq44on2wqq4dy

Point degree spectra of represented spaces [article]

Takayuki Kihara, Arno Pauly
2017 arXiv   pre-print
The notion of point degree spectrum creates a connection among various areas of mathematics including computability theory, descriptive set theory, infinite dimensional topology and Banach space theory  ...  We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees, and so on  ...  The authors are also grateful to Masahiro Kumabe, Joseph Miller, Luca Motto Ros, Philipp Schlicht, and Takamitsu Yamauchi for their insightful comments and discussions.  ... 
arXiv:1405.6866v4 fatcat:tdfa75abczedvnoczicrlxq6yy
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