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Computable Categoricity of Graphs with Finite Components

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Logic and Theory of Algorithms (CiE 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5028))

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Abstract

A computable graph is computably categorical if any two computable presentations of the graph are computably isomorphic. In this paper we investigate the class of computably categorical graphs. We restrict ourselves to strongly locally finite graphs; these are the graphs all of whose components are finite. We present a necessary and sufficient condition for certain classes of strongly locally finite graphs to be computably categorical. We prove that if there exists an infinite \(\Delta_2^0\)-set of components that can be properly embedded into infinitely many components of the graph then the graph is not computably categorical. We outline the construction of a strongly locally finite computably categorical graph with an infinite chain of properly embedded components.

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References

  1. Cholak, P., Goncharov, S., Khoussainov, B., Shore, R.A.: Computably categorical structures and expansions by constants. J. Symbolic Logic 64(1), 13–37 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cholak, P., Shore, R.A., Solomon, R.: A computably stable structure with no Scott family of finitary formulas. Arch. Math. Logic 45(5), 519–538 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Crossley, J.N. (ed.): Aspects of effective algebra, Vic., 1981. Upside Down A Book Co. Yarra Glen.

    Google Scholar 

  4. Dzgoev, V.D., Gončarov, S.S.: Autostability of models. Algebra i Logika 19, 45–58 (1980)

    MathSciNet  Google Scholar 

  5. Yu., L., Ershov, S.S., Goncharov, A., Nerode, J.B.: Handbook of recursive mathematics. Studies in Logic and the Foundations of Mathematics, vol. 1, 138. North-Holland, Amsterdam (1998); Recursive model theory

    Google Scholar 

  6. Ershov, Y.L., Goncharov, S.S., Nerode, A., Remmel, J.B., Marek, V.W. (eds.): Handbook of recursive mathematics. Studies in Logic and the Foundations of Mathematics, vol. 2, 139, pp. 621–1372. North-Holland, Amsterdam (1998); Recursive algebra, analysis and combinatorics

    Google Scholar 

  7. Ershov, Y.L., Goncharov, S.S.: Constructive models. Siberian School of Algebra and Logic. Consultants Bureau, New York (2000)

    Google Scholar 

  8. Gončarov, S.S.: Autostability of models and abelian groups. Algebra i Logika 19(1), 23–44 (1980)

    MathSciNet  Google Scholar 

  9. Gončarov, S.S.: The problem of the number of nonautoequivalent constructivizations. Algebra i Logika 19(6), 621–639, 745 (1980)

    MathSciNet  Google Scholar 

  10. Goncharov, S.S.: Limit equivalent constructivizations. In: Mathematical logic and the theory of algorithms, “Nauka” Sibirsk. Otdel., Novosibirsk. Trudy Inst. Mat., vol. 2, pp. 4–12 (1982)

    Google Scholar 

  11. Goncharov, S.S.: Computability and computable models. In: Mathematical problems from applied logic. II. Int. Math. Ser (N. Y.), vol. 5, pp. 99–216. Springer, New York (2007)

    Chapter  Google Scholar 

  12. Goncharov, S.S., Lempp, S., Solomon, R.: The computable dimension of ordered abelian groups. Adv. Math. 175(1), 102–143 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hirschfeldt, D.R.: Degree spectra of relations on structures of finite computable dimension. Ann. Pure Appl. Logic 115(1-3), 233–277 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lempp, S., McCoy, C., Miller, R., Solomon, R.: Computable categoricity of trees of finite height. J. Symbolic Logic 70(1), 151–215 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Malćev, A.I.: Constructive algebras. I. Uspehi Mat. Nauk, 16(3 (99)), 3–60 (1961)

    Google Scholar 

  16. Rabin, M.O.: Computable algebra, general theory and theory of computable fields. Trans. Amer. Math. Soc. 95, 341–360 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  17. Remmel, J.B.: Recursively categorical linear orderings. Proc. Amer. Math. Soc. 83, 387–391 (1981)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo,  

    Barbara F. Csima

  2. Department of Computer Science, University of Auckland,  

    Bakhadyr Khoussainov & Jiamou Liu

Authors
  1. Barbara F. Csima
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  2. Bakhadyr Khoussainov
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  3. Jiamou Liu
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Editor information

Arnold Beckmann Costas Dimitracopoulos Benedikt Löwe

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© 2008 Springer-Verlag Berlin Heidelberg

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Csima, B.F., Khoussainov, B., Liu, J. (2008). Computable Categoricity of Graphs with Finite Components. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds) Logic and Theory of Algorithms. CiE 2008. Lecture Notes in Computer Science, vol 5028. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69407-6_15

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  • DOI: https://doi.org/10.1007/978-3-540-69407-6_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-69405-2

  • Online ISBN: 978-3-540-69407-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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