Internal Structure of Addition Chains: Well-Ordering

TL;DR

This paper investigates the structure of addition chains by analyzing the addition chain defect, showing it forms a well-ordered set with a specific order type, and extends these results to various restricted chain lengths.

Contribution

It introduces the concept of addition chain defect and proves that its set of values is well-ordered with order type ω^ω, extending to restricted chain length variants.

Findings

The set of addition chain defects is well-ordered with order type ω^ω.

For any n, the addition chain length stabilizes after a certain power of two.

Similar well-ordering results hold for restricted addition chain lengths.

Abstract

An addition chain for $n$ is defined to be a sequence $(a_0,a_1,\ldots,a_r)$ such that $a_0=1$, $a_r=n$, and, for any $1\le k\le r$, there exist $0\le i, j<k$ such that $a_k = a_i + a_j$; the number $r$ is called the length of the addition chain. The shortest length among addition chains for $n$, called the addition chain length of $n$, is denoted $\ell(n)$. The number $\ell(n)$ is always at least $\log_2 n$; in this paper we consider the difference $\delta^\ell(n):=\ell(n)-\log_2 n$, which we call the addition chain defect. First we use this notion to show that for any $n$, there exists $K$ such that for any $k\ge K$, we have $\ell(2^k n)=\ell(2^K n)+(k-K)$. The main result is that the set of values of $\delta^\ell$ is a well-ordered subset of $[0,\infty)$, with order type $\omega^\omega$. The results obtained here are analogous to the results for integer complexity obtained in [1] and [3]. We also prove similar well-ordering results for restricted forms of addition chain length, such as star chain length and Hansen chain length.