Elementary classes of finite VC-dimension
Domenico Zambella
TL;DR
This paper investigates the properties of elementary classes with finite VC-dimension, establishing that such classes are externally definable, which links VC-theoretic properties to model-theoretic definability.
Contribution
It proves that if the class e(D) has finite VC-dimension, then D is externally definable, connecting VC-dimension with external definability in model theory.
Findings
Finite VC-dimension implies external definability of D.
e(D) classes with finite VC-dimension are characterized.
Links between VC-theory and model-theoretic definability are established.
Abstract
Let U be a monster model and let D be a subset of U. Let (U,D) denote theexpansion of U with a new predicate for D. Write e(D) for the collection of all subsets C of U such that (U,C) is elementary equivalent to (U,D). We prove that if e(D) has finite VC-dimension then D is externally definable (i.e. it is the trace on U of a set definable in an elementary superstructure of U).
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.