Counting arithmetic formulas

TL;DR

This paper derives asymptotic formulas for counting arithmetic formulas involving only ones, addition, and multiplication, addressing a conjecture and analyzing encoding methods for such formulas.

Contribution

It provides the first asymptotic enumeration of these formulas and compares different encoding schemes for representing integers.

Findings

Asymptotic formula for the number of formulas evaluating to n

Asymptotic enumeration for formulas with exactly k multiplications

Comparison of encoding lengths and tradeoffs for representing integers

Abstract

An arithmetic formula is an expression involving only the constant $1$, and the binary operations of addition and multiplication, with multiplication by $1$ not allowed. We obtain an asymptotic formula for the number of arithmetic formulas evaluating to $n$ as $n$ goes to infinity, solving a conjecture of E. K. Gnang and D. Zeilberger. We give also an asymptotic formula for the number of arithmetic formulas evaluating to $n$ and using exactly $k$ multiplications. Finally we analyze three specific encodings for producing arithmetic formulas. For almost all integers $n$, we compare the lengths of the arithmetic formulas for $n$ that each encoding produces with the length of the shortest formula for $n$ (which we estimate from below). We briefly discuss the time-space tradeoff offered by each.