VC-dimension: Combinatorics
VC-dimension: Model Theory
Vapnik-Chervonenkis Theory and the
Independence Property in Model Theory
Sílvia Reis
LisMath Seminar
Lisbon, March 27th , 2015
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
1 VC-dimension: Combinatorics
2 VC-dimension: Model Theory
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
1 VC-dimension: Combinatorics
2 VC-dimension: Model Theory
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
Let X be a set, let F be a family of subsets of X , and let A ⊆ X
be a set. Dene
F ∩ A := {F ∩ A : F ∈ F} .
Denition
We say that a family F shatters A if F ∩ A = P(A).
Remark: If B ⊆ A and F shatters A, then F shatters B.
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
Denition (VC-dimension of a family of sets)
Let F be a family of sets. The VC-dimension of F , VC (F), is
dened as:
• The greatest n such that there is a set of cardinality n
shattered by F , if such n exists;
• ∞, otherwise.
Remark: If F shatters a set of cardinality n and does not shatter
any set of cardinality n + 1, then VC (F) = n.
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
Denition (The shatter function)
Let X be a set, F ⊆ P(X ). We consider the function πF : N → N,
dened by:
πF (n) = max |F ∩ A| .
A⊆X
|A | ≤ n
Remark: VC (F) < n if and only if πF (n) < 2n .
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
Theorem (SauerShelah Lemma)
Let F be a family of sets with VC (F) ≤ k. Then, for n ≥ k,
π F (n ) ≤
k X
n
i =0
i
.
In particular, πF (n) = O (nk ).
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
1 VC-dimension: Combinatorics
2 VC-dimension: Model Theory
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
Denition
Let φ(x̄ , ȳ ) be a formula. We say that a set A is shattered by φ if
there is a family (b̄I )I ∈P(A) such that for every ā ∈ A,
C φ(ā, b̄I ) ⇔ ā ∈ I .
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
Denition
Let φ(x̄ , ȳ ) be a formula. We say that a set A is shattered by φ if
there is a family (b̄I )I ∈P(A) such that for every ā ∈ A,
C φ(ā, b̄I ) ⇔ ā ∈ I .
Denition (VC-dimension of a formula)
Let φ(x̄ , ȳ ) be a formula. The VC-dimension of φ is:
• The greatest natural number n such that there is a set of
cardinality n shattered by φ, if such n exists;
• ∞, otherwise.
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
Denition (Independence Property)
A formula φ(x̄ , ȳ ) is said to have the Independence Property if
VC (φ) is innite.
A formula φ(x̄ , ȳ ) is said to be NIP (or dependent) if φ does not
have the independence property.
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
Let φ(x̄ , ȳ ) be a formula. Given b̄ ∈ C, we dene
φC (x̄ , b̄) := {ā ∈ C : C φ(ā, b̄)} .
Remark
Let φ(x̄ , ȳ ) be a formula, and let F = {φC (x̄ , b̄) : b̄ ∈ C}. Then
VC (φ) = VC (F) .
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
Denition (NIP theory)
A theory T is said to be NIP if every formula of T is NIP.
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
Denition (NIP theory)
A theory T is said to be NIP if every formula of T is NIP.
Theorem (Shelah)
Let T be a theory, and assume every formula of T of the form
φ(x , ȳ ) is NIP. Then T is NIP.
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
Denition (o-minimal theory)
A theory T is said to be o-minimal if:
• Models of T are linearly ordered;
• For every formula φ(x , b̄ ) and every b̄ ∈ C, the set φC (x , b̄ ) is
a nite union of intervals.
Some examples of o-minimal theories:
• DLO;
• RCF;
• Th(R, +, −, ·, 0, 1, <, e x );
• ...
VC-Dimension and the Independence Property
Sílvia Reis
VC-dimension: Combinatorics
VC-dimension: Model Theory
References
1
2
3
4
5
H. Adler. Introduction to theories without the independence
property. Archive for Mathematical Logic, to appear.
N. Sauer. On the Density of Families of Sets. Journal of
Combinatorical Theory, Series A 13: 145-147, 1972.
S. Shelah. A Combinatorial Problem: Stability and Order for
Models and Theories in Innitary Languages. Pacic Journal
of Mathematics 41: 247-261, 1972.
P. Simon. A Guide to NIP Theories. Lecture Notes in Logic,
ASL and Cambridge University, to appear.
V. Vapnik and A. Chervonenkis. On the Uniform Convergence
of the Frequencies of Occurence of Events to Their
Probabilities. Theory of Probability and its Applications,
16(2): 264-280, 1971.
VC-Dimension and the Independence Property
Sílvia Reis