Top

Published in:

2017 | OriginalPaper | Chapter

Revisiting Uniform Computable Categoricity: For the Sixtieth Birthday of Prof. Rod Downey

Author : Russell Miller

Published in: Computability and Complexity

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

Inspired by recent work of Csima and Harrison-Trainor and of Montalbán in relativizing the notion of degrees of categoricity, we return to uniform computable categoricity, as described in work of Downey, Hirschfeldt and Khoussainov. Our attempt to integrate these notions together leads to certain new questions about relativizing the concept of the jump of a structure, as well as to an idea of the structural information content of a countable structure, i.e., that information which can be recovered uniformly from copies of the structure.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Irreducibles and Primes in Computable Integral Domains

next chapter Enumeration Reducibility and Computable Structure Theory

Ash, C.J., Knight, J.F., Manasse, M.S., Slaman, T.A.: Generic copies of countable structures. Ann. Pure Appl. Log. 42, 195–205 (1989)MathSciNetCrossRefMATH

Chisholm, J.: On intrisically \(1\)-computable trees. Unpublished MS

Csima, B.F., Franklin, J.N.Y., Shore, R.A.: Degrees of categoricity and the hyperarithmetic hierarchy. Notre Dame J. Form. Log. 54(2), 215–231 (2013)MathSciNetCrossRefMATH

Csima, B.F., Harrison-Trainor, M.: Degrees of categoricity on a cone. Submitted for publication

Downey, R.G., Hirschfeldt, D.R., Khoussainov, B.: Uniformity in computable structure theory. Algebra Log. 42, 318–332 (2003)CrossRefMATH

Ershov, Y.L.: Theorie der Numerierungen. Zeits. Math. Logik Grund. Math. 23, 289–371 (1977)MathSciNetCrossRefMATH

Fokina, E., Kalimullin, I., Miller, R.G.: Degrees of categoricity of computable structures. Arch. Math. Log. 49(1), 51–67 (2010)MathSciNetCrossRefMATH

Knight, J.F.: Degrees coded in jumps of orderings. J. Symb. Log. 51, 1034–1042 (1986)MathSciNetCrossRefMATH

Kudinov, O.V.: An autostable \(1\)-decidable model without a computable Scott family of \(\exists \)-formulas. Algebra Log. 35, 255–260 (1996)MathSciNetCrossRefMATH

10.

Kudinov, O.V.: A description of autostable models. Algebra Log. 36, 16–22 (1997)CrossRefMATH

11.

Miller, R.G.: \(d\)-Computable categoricity for algebraic fields. J. Symb. Log. 74(4), 1325–1351 (2009)MathSciNetCrossRefMATH

12.

Montalbán, A.: Notes on the Jump of a Structure. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds.) CiE 2009. LNCS, vol. 5635, pp. 372–378. Springer Berlin Heidelberg, Berlin, Heidelberg (2009). doi:10.1007/978-3-642-03073-4_38 CrossRef

13.

Montalbán, A.: Rice sequences of relations. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys Eng. Sci. 370, 3464–3487 (2012)MathSciNetCrossRefMATH

14.

Montalbán, A.: A robuster Scott rank. Proc. Am. Math. Soc. 143, 5427–5436 (2015)MathSciNetCrossRefMATH

15.

Richter, L.J.: Degrees of structures. J. Symb. Log. 46, 723–731 (1981)MathSciNetCrossRefMATH

16.

Soskova, A.A., Soskov, I.N.: A jump inversion theorem for the degree spectra. J. Log. Comput. 19(1), 199–215 (2009)MathSciNetCrossRefMATH

17.

Ventsov, Y.G.: Effective choice for relations and reducibilities in classes of constructive and positive models. Algebra Log. 31, 63–73 (1992)MathSciNetCrossRefMATH

Title: Revisiting Uniform Computable Categoricity: For the Sixtieth Birthday of Prof. Rod Downey
Author: Russell Miller
Publisher: Springer International Publishing
Book: Computability and Complexity
Print ISBN: 978-3-319-50061-4

Electronic ISBN: 978-3-319-50062-1

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-50062-1_18

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner