On VC-density over indiscernible sequences

Guingona, Vincent; Hill, Cameron Donnay

doi:10.48550/arxiv.1108.2554

Cited by 4 publications

(5 citation statements)

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“…We have recently learned that Vincent Guingona and Cameron Hill have investigated V C * -density (and other properties) over indiscernible sequences in much greater detail in [3].…”

Section: Now |S P(y)mentioning

confidence: 99%

Additivity of the dp-rank

Kaplan¹,

Onshuus

Usvyatsov

2013

Trans. Amer. Math. Soc.

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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],

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“…We have recently learned that Vincent Guingona and Cameron Hill have investigated V C * -density (and other properties) over indiscernible sequences in much greater detail in [3].…”

Section: Now |S P(y)mentioning

confidence: 99%

Additivity of the dp-rank

Kaplan¹,

Onshuus

Usvyatsov

2013

Trans. Amer. Math. Soc.

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“…We may summarize Theorem 4.7 by the statement max(n) = dpR(n) for all n ∈ ω. In particular, for any dp-minimal theory and any n ∈ ω, max(n) = n. If V C ind -density is defined as in [7], and ϕ * (ȳ; x) = ϕ(x; ȳ) is the dual formula, then for any ϕ(x; ȳ), max(ϕ * ) ≤ V C ind -density of ϕ. This can be seen by using Lemma 4.6 and applying Ramsey's Theorem.…”

Section: Relations To Other Notionsmentioning

confidence: 99%

dp-Rank and Forbidden Configurations

Johnson¹

2013

Notre Dame J. Formal Logic

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“…In the recent past there have been a number of papers relating various measures of the combinatorial structure of NIP theories to one another [1,5,7]. One fact which emerged from this is the close relation of dp-rank and VCdensity restricted to indiscernible sequences.…”

Section: Introductionmentioning

confidence: 99%

“…Guingona and Hill have used the term VC ind -density to describe VC-density restricted to indiscernibles. At the end of their paper [5], Guingona and Hill ask if there is a useful characterization of when a formula has VC ind -density equal to VC-density. We offer such a characterization below (Corollary 3.4), and use it to compute the exact VC ind -density of certain formulas.…”

Section: Introductionmentioning

confidence: 99%

Vapnik–Chervonenkis Density on Indiscernible Sequences, Stability, and the Maximum Property

Johnson¹

2015

Notre Dame J. Formal Logic

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This paper presents some finite combinatorics of set systems with applications to model theory, particularly the study of dependent theories. There are two main results. First, we give a way of producing lower bounds on VC ind -density, and use it to compute the exact VC inddensity of polynomial inequalities, and a variety of geometric set families. The main technical tool used is the notion of a maximum set system, which we juxtapose to indiscernibles. In the second part of the paper we give a maximum set system analogue to Shelah's characterization of stability using indiscernible sequences.1991 Mathematics Subject Classification. 03C45.

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On VC-density over indiscernible sequences

Cited by 4 publications

References 0 publications

Additivity of the dp-rank

Additivity of the dp-rank

dp-Rank and Forbidden Configurations

Vapnik–Chervonenkis Density on Indiscernible Sequences, Stability, and the Maximum Property

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