A Minimal Substitution Basis for the Kalm\'ar Elementary Functions

TL;DR

This paper demonstrates that Kalmár elementary functions can be generated from addition, integer remainder, and base-two exponentiation, establishing a minimal substitution basis and exploring alternatives under arity constraints.

Contribution

It introduces a minimal substitution basis for Kalmár elementary functions and proves its minimality, improving upon previous results.

Findings

The class of Kalmár elementary functions can be generated from three basic operations.

The substitution basis composed of addition, remainder, and exponentiation is minimal.

Alternative bases under arity constraints are discussed.

Abstract

We show that the class of Kalm\'ar 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.