We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight.
We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure.We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.
We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.
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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