A Minimal Substitution Basis for the Kalmar Elementary Functions

TL;DR

This work identifies a minimal and explicit substitution basis for the Kalmar elementary functions, showing that $x+y$, $x \bmod y$, and $2^x$ suffice to generate the entire class $\mathcal{E}^3$ under substitution, thereby improving prior results that required squaring. It proves the minimality of this basis by demonstrating that each of the three operations cannot be derived from the other two, using growth and modular-exponential arguments. The paper also analyzes sets that fail to be substitution bases, and it surveys low-arity alternatives, including unary-only bases for univariate Kalmar functions and constructions that reduce the basis to a single binary operation, highlighting the structural landscape of substitution bases within $\mathcal{E}^3$. These results have practical implications for modeling Kalmar functions with compact operator sets and inform the design of finite bases under arity constraints.

Abstract

We show that the class of Kalmar elementary functions can be inductively generated from the addition, the integer remainder, and the base-two exponentiation, hence improving previous results by Marchenkov and Mazzanti. We also prove that the substitution basis defined by these three operations is minimal. Furthermore, we discuss alternative substitution bases under arity constraints.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• proof
• Corollary 1
• Theorem 3
• proof
• Theorem 4
• proof
• Corollary 2
• proof