Finite canonization

TL;DR

This paper improves the bounds on the canonization theorem's parameters, showing the minimal number of exponentiations needed to compute ER(n,m) for fixed n is optimal, advancing understanding of finite canonization.

Contribution

It refines the bounds on ER(n,m), establishing the minimal exponentiation complexity for fixed n, thus optimizing the calculation process in finite canonization.

Findings

Improved the bound on ER(n,m) for fixed n.

Established the minimal number of exponentiations needed.

Optimized the calculation complexity for finite canonization.

Abstract

The canonization theorem says that for given m,n for some m^* (the first one is called ER(n;m)) we have: for every function f with domain [{1, ...,m^*}]^n, for some A in [{1, ...,m^*}]^m, the question of when the equality f({i_1, ...,i_n})=f({j_1, ...,j_n}) (where i_1< ... <i_n and j_1 < ... < j_n are from A) holds has the simplest answer: for some v subseteq {1, ...,n} the equality holds iff (for all l in v)(i_l = j_l). In this paper we improve the bound on ER(n,m) so that fixing n the number of exponentiation needed to calculate ER(n,m) is best possible.