VC-Dimensions of Short Presburger Formulas

Abstract

We study VC-dimensions of short formulas in Presburger Arithmetic, defined to have a bounded number of variables, quantifiers and atoms. We give both lower and upper bounds, which are tight up to a polynomial factor in the bit length of the formula.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson and S. Starchenko: Vapnik-Chervonenkis density in some theories without the inde-pendence property, I, Trans. AMS 368 (2016), 5889–5949.

    Article  Google Scholar 

  2. [2]

    A. Barvinok and K. Woods: Short rational generating functions for lattice point problems, Jour. AMS 16 (2003), 957–979.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    A. Chernikov: Models theory and combinatorics, course notes, UCLA; available electronically at https://tinyurl.com/y8ob6uyv.

  4. [4]

    M. J. Fischer and M. O. Rabin: Super-Exponential Complexity of Presburger Arithmetic, in: Proc. SIAM-AMS Symposium in Applied Mathematics, AMS, Providence, RI, 1974, 27–41.

    Google Scholar 

  5. [5]

    M. Karpinski and A. Macintyre: Polynomial bounds for VC dimension of sigmoidal and general Pfaffian neural networks, J. Comput. System Sci. 54 (1997), 169–176.

    MathSciNet  Article  Google Scholar 

  6. [6]

    M. Karpinski and A. Macintyre: Approximating volumes and integrals in o-minimal and p-minimal theories, in: Connections between model theory and algebraic and analytic geometry, Seconda Univ. Napoli, Caserta, 2000, 149–177.

    Google Scholar 

  7. [7]

    D. Nguyen and I. Pak: Enumeration of integer points in projections of unbounded polyhedra, SIAM J. Discrete Math. 32 (2018), 986–1002.

    MathSciNet  Article  Google Scholar 

  8. [8]

    D. Nguyen and I. Pak: Short Presburger Arithmetic is hard, in: Proc. 58th FOCS, IEEE, Los Alamitos, CA, 2017, 37–48.

    Google Scholar 

  9. [9]

    J. C. Lagarias and A. M. Odlyzko: Computing π(x): an analytic method, J. Al-gorithms 8 (1987), 173–191.

    MathSciNet  MATH  Google Scholar 

  10. [10]

    D. C. Oppen: A 222pn upper bound on the complexity of Presburger arithmetic, J. Comput. System Sci. 16 (1978), 323–332.

    MathSciNet  Article  Google Scholar 

  11. [11]

    N. Sauer: On the density of families of sets, J. Combin. Theory, Ser. A 13 (1972), 145–147.

    MathSciNet  Article  Google Scholar 

  12. [12]

    S. Shelah: A combinatorial problem; stability and order for models and theories in infinitary languages, Pacific J. Math. 41 (1972), 247–261.

    MathSciNet  Article  Google Scholar 

  13. [13]

    L. J. Stockmeyer and A. R. Meyer: Word problems requiring exponential time: preliminary report, in: Proc. Fifth STOC, ACM, New York, 1973, 1–9.

    Google Scholar 

  14. [14]

    T. Tao, E. Croot and H. Helfgott: Deterministic methods to find primes, Math. Comp. 81 (2012), 1233–1246.

    MathSciNet  Article  Google Scholar 

  15. [15]

    V. N. Vapnik and A. Ja. Červonenkis: The uniform convergence of frequencies of the appearance of events to their probabilitie, Theor. Probability Appl. 16 (1971), 264–280.

    MathSciNet  Article  Google Scholar 

  16. [16]

    V. N. Vapnik: Statistical learning theory, John Wiley, New York, 1998.

    MATH  Google Scholar 

  17. [17]

    V. D. Weispfenning: Complexity and uniformity of elimination in Presburger arithmetic, in: Proc. 1997 ISSAC, ACM, New York, 1997, 48–53.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Danny Nguyen.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nguyen, D., Pak, I. VC-Dimensions of Short Presburger Formulas. Combinatorica 39, 923–932 (2019). https://doi.org/10.1007/s00493-018-4004-x

Download citation

Mathematics Subject Classification (2010)

  • 03C45
  • 52C07