Skip to main content
SpringerLink
Log in

Effectively Existentially-Atomic Structures

  • Chapter
  • First Online:
Computability and Complexity

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10010))

Abstract

The notions we study in this paper, those of existentially-atomic structure and effectively existentially-atomic structure, are not really new. The objective of this paper is to single them out, survey their properties from a computability-theoretic viewpoint, and prove a few new results about them. These structures are the simplest ones around, and for that reason alone, it is worth analyzing them. As we will see, they are the simplest ones in terms of how complicated it is to find isomorphisms between different copies and in terms of the complexity of their descriptions. Despite their simplicity, they are very general in the following sense: every structure is existentially atomic if one takes enough jumps, the number of jumps being (essentially) the Scott rank of the structure. That balance between simplicity and generality is what makes them important.

The author was partially supported by the Packard Fellowship and NSF grant DMS-0901169.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Melnikov and the author [MM] proved the equivalence between (A2) and (A3) in a much more general setting, that of Polish groups (\(S_\infty \) in this case) acting continuously on Polish spaces (the space of presentations of structures in this case). Furthermore, they showed that the equivalence is an easy corollary of a theorem of Effros from 1965 [Eff65].

  2. 2.

    \(\mathcal A\) is computably categorical for decidable copies if every decidable copy of \(\mathcal A\) is computably isomorphic to \(\mathcal A\).

  3. 3.

    A structure is effectively atomic if it has a c.e. Scott family of elementary first-order formulas.

References

  1. Ash, C.J., Knight, J.: Computable Structures and the Hyperarithmetical Hierarchy. Elsevier Science, Amsterdam (2000)

    MATH  Google Scholar 

  2. Ash, C., Knight, J., Manasse, M., Slaman, T.: Generic copies of countable structures. Ann. Pure Appl. Log. 42(3), 195–205 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chisholm, J.: Effective model theory vs. recursive model theory. J. Symb. Log. 55(3), 1168–1191 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Calvert, W., Harizanov, V., Shlapentokh, A.: Turing degrees of isomorphism types of algebraic objects. J. Lond. Math. Soc. 75(2), 273–286 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cholak, P., McCoy, C.: Effective prime uniqueness. Submitted for publication

    Google Scholar 

  6. Downey, R., Hirschfeldt, D., Khoussainov, B.: Uniformity in the theory of computable structures. Algebra Log. 42(5), 566–593 (2003). 637

    Article  MathSciNet  MATH  Google Scholar 

  7. Effros, E.G.: Transformation groups and \(C^{\ast } \)-algebras. Ann. Math. 2(81), 38–55 (1965)

    Article  MATH  Google Scholar 

  8. Frolov, A., Kalimullin, I., Miller, R.: Spectra of algebraic fields and subfields. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds.) CiE 2009. LNCS, vol. 5635, pp. 232–241. Springer, Heidelberg (2009). doi:10.1007/978-3-642-03073-4_24

    Chapter  Google Scholar 

  9. Herwig, B., Lempp, S., Ziegler, M.: Constructive models of uncountably categorical theories. Proc. Am. Math. Soc. 127(12), 3711–3719 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Knight, J.F.: Degrees of models. In: Handbook of Recursive Mathematics, vol. 1. Studies in Logic and the Foundations of Mathematics, vol. 138, pp. 289–309, Amsterdam (1998)

    Google Scholar 

  11. Kudinov, O.V.: An autostable \(1\)-decidable model without a computable Scott family of \(\exists \)-formulas. Algebra i Log. 35(4), 458–467 (1996). 498

    MathSciNet  MATH  Google Scholar 

  12. Kudinov, O.V.: Some properties of autostable models. Algebra i Log. 35(6), 685–698 (1996). 752

    MathSciNet  MATH  Google Scholar 

  13. Kudinov, O.V.: The problem of describing autostable models. Algebra i Log. 36(1), 26–36 (1997). 117

    MathSciNet  Google Scholar 

  14. Melnikov, A., Montalbán, A.: Computable Polish group actions. Submitted for publication

    Google Scholar 

  15. Montalbán, A.: A robuster Scott rank. In: The Proceedings of the American Mathematical Society. To appear

    Google Scholar 

  16. Montalbán, A.: Rice sequences of relations. Philos. Trans. R. Soc. A 370, 3464–3487 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Montalbán, A.: A fixed point for the jump operator on structures. J. Symb. Log. 78(2), 425–438 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nurtazin, A.T.: Computable classes and algebraic criteria for autostability. PhD thesis, Institute of Mathematics and Mechanics, Alma-Ata (1974)

    Google Scholar 

  19. Pouzet, M.: Modéle uniforméement préhomogéne. C. R. Acad. Sci. Paris Sér 274, A695–A698 (1972)

    Google Scholar 

  20. Richter, L.J.: Degrees of structures. J. Symb. Log. 46(4), 723–731 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Scott, D.: Logic with denumerably long formulas and finite strings of quantifiers. In: Theory of Models (Proceedings of 1963 International Symposium at Berkeley), pp. 329–341 (1965)

    Google Scholar 

  22. Simmons, H.: Large and small existentially closed structures. J. Symb. Log. 41(2), 379–390 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  23. Steiner, R.M.: Effective algebraicity. Arch. Math. Log. 52(1–2), 91–112 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ventsov, Y.G.: The effective choice problem for relations and reducibilities inclasses of constructive and positive models. Algebra i Log. 31(2), 101–118 (1992). 220

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, USA

    Antonio Montalbán

Authors
  1. Antonio Montalbán
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonio Montalbán .

Editor information

Editors and Affiliations

  1. Victoria University of Wellington, Wellington, New Zealand

    Adam Day

  2. University of Bergen, Bergen, Norway

    Michael Fellows

  3. Victoria University of Wellington, Wellington, New Zealand

    Noam Greenberg

  4. University of Auckland, Auckland, New Zealand

    Bakhadyr Khoussainov

  5. Massey University, Auckland, New Zealand

    Alexander Melnikov

  6. University of Bergen, Bergen, Norway

    Frances Rosamond

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Montalbán, A. (2017). Effectively Existentially-Atomic Structures. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-50062-1_16

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50061-4

  • Online ISBN: 978-3-319-50062-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation