Additivity of the dp-rank

Abstract

The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of its dp-minimal types, and discuss the possible relations between dp-rank and VC-density. introductionThis paper grew out of discussions that the authors had during a meeting in Oberwolfach in January 2010, following a talk of Deirdre Haskell, and conversations with Sergei Starchenko, on their recent joint work with Aschenbrenner, Dolich and Macpherson [2]. Haskell's talk made it apparent to us that the notion of VC-density (Vapnik-Chervonenkis density), investigated in [2], is closely related to "dependence rank" (dp-rank) introduced by the third author in [14]. Discussions with Starchenko helped us realize that certain questions, such as additivity, which were (and still are, to our knowledge) open for VC-density, may be approached more easily in the context of dp-rank. This paper is the first step in the program of investigating basic properties of dp-rank and its connections with VC-density.Whereas dp-rank is a relatively new notion, VC-density and related concepts have been studied for quite some time in the frameworks of machine learning, computational geometry, and other branches of theoretical computer science. Recent developments point to a connection between VC-density and dp-rank, strengthening the bridge between model theory and these subjects. We believe that investigating properties of dp-rank is important for discovering the nature of this connection. Furthermore, once this relation is better understood, theorems about dp-rank are likely to prove useful in the study of finite and infinite combinatorics related to VC-classes.Dp-rank was originally defined in [14] as an attempt to capture how far a certain type (or a theory) is from having the independence property. It also helped us to isolate a minimality notion of dependence for types and theories (that is, having rank 1). We called this notion dp-minimality and investigated it in [7]. Both dprank and dp-minimality were simplifications of Shelah's various ranks from [12],

Forking in VC-minimal theories

“…First, consider any tuple a ∈ U, and let n = l(a). By Theorem 2.7 of [9], the dp-rank of tp(a/A) is at most n. It follows from Proposition 2.3 of [9] that for any family I 0 , . .…”

Profinite groups with NIP theory andp-adic analytic groups

“…Let M be the disjoint union of the rings Z an pi (viewed in a language in which the rings have formally disjoint languages). Then M has finite dp-rank by [11,Theorem 4.8] and hence is strongly NIP. By Proposition 1.2, each P i is interpretable in Z an pi when viewed as a full 2-sorted profinite group.…”

Pseudo real closed fields, pseudo p-adically closed fields and NTP2

“…In particular RCF is strong. Since dp-rank is sub-additive (Corollary 4.2 of [31]) the burden satisfies the following: if r ∈ N and p(x 1 , . . .…”