A binary version of the Mahler-Popken complexity function

Abstract

The (Mahler-Popken) complexity $\| n \|$ of a natural number $n$ is the smallest number of ones that can be used via combinations of multiplication and addition to express $n$, with parentheses arranged in such a way so as to form legal nestings. We generalize $\| \cdot \|$ by defining $\| n \|_{m}$ as the smallest number of possibly repeated selections from $\{ 1, 2, \ldots, m \}$ (counting repetitions), for fixed $m \in \mathbb{N}$, that can be used to express $n$ with the same operational and bracket symbols as before. There is a close relationship, as we explore, between $\|\cdot\|_{2}$ and lengths of shortest addition chains for a given natural number. This illustrates how remarkable it is that $(\| n \|_{2} : n \in \mathbb{N} )$ is not currently included in the On-Line Encyclopedia of Integer Sequences and has, apparently, not been studied previously. This, in turn, motivates our exploration of the complexity function $\| \cdot\|_{2}$, in which we prove explicit upper and lower bounds for $\|\cdot\|_{2}$ and describe some problems and further areas of research concerning $\|\cdot\|_{2}$.