VC-Dimensions of Short Presburger Formulas
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.
Mathematics Subject Classification (2010)03C45 52C07
Unable to display preview. Download preview PDF.
- A. Chernikov: Models theory and combinatorics, course notes, UCLA; available electronically at https://tinyurl.com/y8ob6uyv.Google Scholar
- 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
- 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
- D. Nguyen and I. Pak: Short Presburger Arithmetic is hard, in: Proc. 58th FOCS, IEEE, Los Alamitos, CA, 2017, 37–48.Google Scholar
- 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
- V. D. Weispfenning: Complexity and uniformity of elimination in Presburger arithmetic, in: Proc. 1997 ISSAC, ACM, New York, 1997, 48–53.Google Scholar