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NIP and Distal Metric Structures

Aaron Anderson

TL;DR

The work develops a comprehensive framework for generalized VC-dimensions in fuzzy and real-valued contexts and leverages it to advance continuous logic. It establishes real-valued analogues of ε-approximations, ε-nets, and (p,q)-type results, and applies these to NIP metric structures, including the Shelah expansion and honest definitions. The distal side is developed with multiple equivalent characterizations—distal types, strong honest definitions, and distal cell decompositions—leading to a robust theory of distal metric structures. These foundations connect combinatorial geometry with continuous logic, offering tools for uniform definability, regularity, and future study of distal phenomena in analysis-like structures.

Abstract

Model theory, machine learning, and combinatorics each have generalizations of VC-dimension for fuzzy and real-valued versions of set systems. These different dimensions define a unique notion of a VC-class for both fuzzy sets and real-valued functions. We study these VC-classes, obtaining generalizations of certain combinatorial results from the discrete case. These include appropriate generalizations of $\varepsilon$-nets, the fractional Helly property and the $(p,q)$-theorem. We then apply these results to continuous logic. We prove that NIP for metric structures is equivalent to an appropriate generalization of honest definitions, which we use to study externally definable predicates and the Shelah expansion. We then examine distal metric structures, providing several equivalent characterizations, in terms of indiscernible sequences, distal types, strong honest definitions, and distal cell decompositions.

NIP and Distal Metric Structures

TL;DR

The work develops a comprehensive framework for generalized VC-dimensions in fuzzy and real-valued contexts and leverages it to advance continuous logic. It establishes real-valued analogues of ε-approximations, ε-nets, and (p,q)-type results, and applies these to NIP metric structures, including the Shelah expansion and honest definitions. The distal side is developed with multiple equivalent characterizations—distal types, strong honest definitions, and distal cell decompositions—leading to a robust theory of distal metric structures. These foundations connect combinatorial geometry with continuous logic, offering tools for uniform definability, regularity, and future study of distal phenomena in analysis-like structures.

Abstract

Model theory, machine learning, and combinatorics each have generalizations of VC-dimension for fuzzy and real-valued versions of set systems. These different dimensions define a unique notion of a VC-class for both fuzzy sets and real-valued functions. We study these VC-classes, obtaining generalizations of certain combinatorial results from the discrete case. These include appropriate generalizations of -nets, the fractional Helly property and the -theorem. We then apply these results to continuous logic. We prove that NIP for metric structures is equivalent to an appropriate generalization of honest definitions, which we use to study externally definable predicates and the Shelah expansion. We then examine distal metric structures, providing several equivalent characterizations, in terms of indiscernible sequences, distal types, strong honest definitions, and distal cell decompositions.
Paper Structure (16 sections, 49 theorems, 53 equations)

This paper contains 16 sections, 49 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Fuzzy Combinatorics
  3. Rademacher/Gaussian Complexity and $\varepsilon$-Approximations
  4. Covering Numbers and $\varepsilon$-Approximations
  5. Transversals and $\varepsilon$-nets
  6. Model-Theoretic Preliminaries and Notation
  7. Pairs
  8. Coding Tricks
  9. Other Facts
  10. NIP and Honest Definitions
  11. The Shelah Expansion
  12. Definitions of Distality
  13. Distal Types
  14. Strong Honest Definitions
  15. Distal Cell Decompositions
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $A$ be a closed subset of $M^x$ where $M \preceq \mathcal{U}$ and $(M,A)\preceq (M',A')$. Let $\phi(x;y)$ be a definable predicate. Then there exists a definable predicate $\psi(x;z)$, which we call a uniform honest definition for $\phi(x;y)$, such that for every $b \in M^y$, there exists $d \in

Theorems & Definitions (113)

  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['thm_distal']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.10
  • Lemma 2.11
  • ...and 103 more