VC-Dimensions of Short Presburger Formulas

Abstract

We study VC-dimension 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.1.2. NIP theories and bounds on VC-dimension/density. It is of interest to distinguish the first-order theories in which VC-dimension and VC-density behave nicely. Let L be a first-order language and M be an L-structure. Consider a partitioned L-formula

Geometric decision procedures and the VC dimension of linear arithmetic theories

“…In the proof of Theorem 6.8, the main difficulty lies in handling the discrete behaviour of the various hybrid linear sets. Here, we employ a simple yet effective approach that consists of establishing an upper bound, firstly, on the VC dimension of a hybrid linear set with a proper period set, and then on the VC dimension of a semilinear set with the help of Proposition 3.3 together with the following proposition by Sauer and Shelah [39,43] (see also [30,Proof of Thm. 4]).…”

“…This line of research has led to deep results in model theory, for instance, that every quasi-o-minimal structure has linear VC density [1]. Recently, D. Nguyen and I. Pak established polynomial upper bounds on the VC dimension of LIA with a fixed number of variables [30,Thm. 1.4], building upon a geometric approach for inferring properties of Boolean operations on semilinear sets [29].…”

“…1.4], building upon a geometric approach for inferring properties of Boolean operations on semilinear sets [29]. The authors of [30] conjecture that it should be possible to establish a doubly exponential upper bound on the VC dimension of full LIA, but also remark that this is unlikely to be achieved by analysing quantifier elimination procedures because of known lower bounds on the growth of formula sizes in such procedures [45]. The second main contribution of our paper is to prove this conjecture, establishing singly and doubly exponential upper bounds on the VC dimension of LRA and LIA, respectively.…”

Therapeutic Review on Nanocurcumin

“…Nguyen et al . 10 Composite sponges using 10% curcumin, chitosan and gelatin were prepared Curcumin -loaded NPs have been prepared by an ionic gelation method using chitosan (Chi) and pluronic R F -127 (PF) as carriers to deliver curcumin to the target cancer cells.…”