Counting the number of inequivalent arithmetic expressions on $n$ variables

TL;DR

This paper presents a theoretical analysis of arithmetic expression equivalence and introduces a quadratic-time algorithm to count inequivalent expressions on n variables.

Contribution

It provides new theoretical insights into expression equivalence and offers a novel efficient algorithm for counting inequivalent arithmetic expressions.

Findings

Established theoretical results on expression equivalence

Developed a $ heta(n^2)$ algorithm for counting inequivalent expressions

Demonstrated the algorithm's effectiveness for n variables

Abstract

An expression is any mathematical formula that contains certain formal variables and operations to be executed in a specified order. In computer science, it is usually convenient to represent each expression in the form of an expression tree. Here, we consider only arithmetic expressions, i.e., those that contain only the four standard arithmetic operations: addition, subtraction, multiplication and division, alongside additive inversion. We first provide certain theoretical results concerning the equivalence of such expressions and then disclose a $\Theta(n^2)$ algorithm that computes the number of inequivalent arithmetic expressions on $n$ distinct variables.