NIP (model theory)

Definition

Let T be a complete L-theory. An L-formula φ(x,y) is said to have the independence property (with respect to x, y) if in every model M of T there is, for each n = {0,1,…,n − 1} < ω, a family of tuples b₀,…,b_n−1 such that for each of the 2ⁿ subsets X of n there is a tuple a in M for which

The theory T is said to have the independence property if some formula has the independence property. If no L-formula has the independence property then T is called dependent, or said to satisfy NIP. An L-structure is said to have the independence property (respectively, NIP) if its theory has the independence theory (respectively, NIP). The terminology comes from the notion of independence in the sense of boolean algebras.

In the nomenclature of Vapnik–Chervonenkis theory, we may say that a collection S of subsets of X shatters a set B ⊆ X if every subset of B is of the form B ∩ S for some S ∈ S. Then T has the independence property if in some model M of T there is a definable family (S_a | a∈Mⁿ) ⊆ M^k that shatters arbitrarily large finite subsets of M^k. In other words, (S_a | a∈Mⁿ) has infinite Vapnik–Chervonenkis dimension.

Examples

Any complete theory T that has the independence property is unstable.

In arithmetic, i.e. the structure (N,+,·), the formula "y divides x" has the independence property. This formula is just

So, for any finite n we take the n 1-tuples b_i to be the first n prime numbers, and then for any subset X of {0,1,…,n − 1} we let a be the product of those b_i such that i is in X. Then b_i divides a if and only if i ∈ X.

Every o-minimal theory satisfies NIP. This fact has had unexpected applications to neural network learning.