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Explicit affine formulas for distances between tuples in classical discrete structures

Arthur Molina-Mounier

Abstract

Answering a question of Ben Yaacov, Ibarlucía, and Tsankov [5], we show an explicit way to construct an affine formula for the distance between two $n$-tuples in a $\{0,1\}$-valued $\varnothing$-structure, using $\lceil \log_2 n \rceil$ quantifier alternations.

Explicit affine formulas for distances between tuples in classical discrete structures

Abstract

Answering a question of Ben Yaacov, Ibarlucía, and Tsankov [5], we show an explicit way to construct an affine formula for the distance between two -tuples in a -valued -structure, using quantifier alternations.
Paper Structure (15 sections, 23 theorems, 39 equations)

This paper contains 15 sections, 23 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Notation
  4. Algorithmic construction
  5. Theoretical basis
  6. Algorithmic derivation
  7. Proof of \ref{['thm:main']}
  8. Alternate elementary construction
  9. Overview
  10. Constructibility in classical continuous theories
  11. Types of tuples
  12. Properties of constructible sets
  13. Proof of \ref{['thm:main_weak']}
  14. Quantifier alternation analysis
  15. Code listing for \ref{['sec:alg']}

Key Result

Proposition 1.1

[prop]prop:thetadef Let $\mathcal{L} = \varnothing$ be the empty language (save for the metric $d$), $\ell \in \mathbb{N}^* \cup \qty{\infty}$, $M_\ell$ the unique countable classical $\mathcal{L}$-structure with $\ell$ elements, and $C_\ell$ its theory. Then, modulo $C_\ell$, every continuous formu

Theorems & Definitions (56)

  • Proposition 1.1: EMDI, 26.5, 26.7
  • Theorem 1.2
  • Remark 1.1
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • ...and 46 more