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VAPNIK-CHERVONENKIS DENSITY IN SOME THEORIES WITHOUT THE INDEPENDENCE PROPERTY, I
 

Summary: VAPNIK-CHERVONENKIS DENSITY IN SOME THEORIES
WITHOUT THE INDEPENDENCE PROPERTY, I
MATTHIAS ASCHENBRENNER, ALF DOLICH, DEIRDRE HASKELL,
DUGALD MACPHERSON, AND SERGEI STARCHENKO
For Lou van den Dries, on his 60th birthday.
Abstract. We recast the problem of calculating Vapnik-Chervonenkis (VC) den-
sity into one of counting types, and thereby calculate bounds (often optimal) on the
VC density for some weakly o-minimal, weakly quasi-o-minimal, and P-minimal
theories.
Contents
1. Introduction 1
2. VC Density 7
3. The Model-Theoretic Context 15
4. Some VC Density Calculations 27
5. Theories with the VC d Property 35
6. Examples of VC d: Weakly O-minimal Theories and Variants 42
7. A Strengthening of VC d, and P-adic Examples 47
References 56
1. Introduction
The notion of VC dimension, which arose in probability theory in the work of Vapnik and

  

Source: Aschenbrenner, Matthias - Department of Mathematics, University of California at Los Angeles

 

Collections: Mathematics