Explicit affine formulas for distances between tuples in classical discrete structures

TL;DR

This paper provides an explicit affine formula with a logarithmic number of quantifier alternations for calculating distances between tuples in certain discrete structures, answering a specific open question.

Contribution

It introduces a novel method to construct affine formulas for tuple distances in $ ext{0,1}$-valued structures with a minimal number of quantifier alternations.

Findings

Explicit affine formulas with $ ext{O}( ext{log } n)$ quantifier alternations

Answer to an open question by Ben Yaacov, Ibarlucía, and Tsankov

Applicable to $ ext{0,1}$-valued $ ext{0,1}$-structures

Abstract

Answering a question of Ben Yaacov, Ibarluc\'ia, 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.