Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Compression Schemes, Stable Definable Families, and o-Minimal Structures

  • 124 Accesses

  • 5 Citations

Abstract

We show that any family of sets uniformly definable in an o-minimal structure has an extended compression scheme of size equal to the number of parameters in the defining formula.

As a consequence, the combinatorial complexity (or density) of any definable family in a structure with a o-minimal theory is bounded by the number of parameters in the defining formula.

Extended compression schemes for uniformly definable families corresponding to stable formulas are also shown to exist.

References

  1. 1.

    Assouad, P.: Densité et dimension. Ann. Inst. Fourier 33(3), 233–282 (1983)

  2. 2.

    Basu, S.: Combinatorial complexity in o-minimal geometry. In: Proc. of the 39th Ann. ACM Symp. on Theory of Computing, pp. 47–56 (2007)

  3. 3.

    Ben-David, S., Litman, A.: Combinatorial variability of Vapnik–Chervonenkis classes with applications to sample compression schemes. Discrete Appl. Math. 86(1), 3–25 (1998)

  4. 4.

    Dudley, R.M.: Uniform Central Limit Theorems. Cambridge University Press, New York (1999)

  5. 5.

    Floyd, S., Warmuth, M.: Sample compression, learnability, and the Vapnik–Chervonenkis dimension. Mach. Learn. 21(3), 269–304 (1995)

  6. 6.

    Gabrielov, A.: Projections of semi-analytic sets. Funct. Anal. Appl. 2, 282–291 (1968)

  7. 7.

    Haussler, D., Welzl, E.: ε-Nets and simplex range queries. Discrete Comput. Geom. 2, 127–151 (1987)

  8. 8.

    Hovanskii, A.: On a class of systems of transcendental equations. Sov. Math. Dokl. 22, 762–765 (1980)

  9. 9.

    Laskowski, M.C.: Vapnik–Chervonenkis classes of definable sets. J. Lond. Math. Soc. 45(2), 377–384 (1992)

  10. 10.

    Littlestone, N., Warmuth, M.: Unpublished notes

  11. 11.

    Marchland, M., Shaw-Taylor, J.: The set covering machine. J. Mach. Learn. Res. 3, 723–746 (2002)

  12. 12.

    Marchland, M., Shaw-Taylor, J.: The decision list machine. Adv. Neural Inf. Process. Syst. 15, 921–928 (2003)

  13. 13.

    Marker, D.: Model theory and exponentiation. Not. Am. Math. Soc. 43, 753–759 (1996)

  14. 14.

    Pillay, A.: Geometric Stability Theory. Oxford University Press, Oxford (1996)

  15. 15.

    Shelah, S.: Classification Theory, 2nd edn. North-Holland, Amsterdam (1990)

  16. 16.

    van den Dries, L.: Tame Topology and O-minimal Structures. London Mathematical Society Lecture Notes Series, vol. 248. Cambridge University Press, Cambridge (1998)

  17. 17.

    Wilkie, A.J.: Model completeness results for expansions of the real field by restricted Pfaffian functions and the exponential function. J. Am. Math. Soc. 9, 1051–1094 (1996)

  18. 18.

    Wilkie, A.J.: A theorem of the complement and some new o-minimal structures. Sel. Math., New Ser. 5(4), 397–421 (1999)

Download references

Author information

Correspondence to H. R. Johnson.

Additional information

M.C. Laskowski partially supported by NSF grant DMS-0600217.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Johnson, H.R., Laskowski, M.C. Compression Schemes, Stable Definable Families, and o-Minimal Structures. Discrete Comput Geom 43, 914–926 (2010). https://doi.org/10.1007/s00454-009-9201-3

Download citation

Keywords

  • Compression scheme
  • VC dimension
  • NIP
  • Independence dimension
  • Dependence
  • Warmuth conjecture
  • Stable
  • o-minimal
  • Type definition
  • Definable type
  • Density
  • Combinatorial complexity
  • UFTD
  • UDTFS