Parallel addition in non-standard numeration systems

TL;DR

This paper develops parallel addition algorithms for numeration systems with algebraic number bases satisfying certain dominance conditions, including the Golden Mean, enabling efficient computation with signed-digit alphabets.

Contribution

It introduces a generalized parallel addition algorithm for algebraic bases with a dominance condition, and refines it for the Golden Mean with a minimal alphabet.

Findings

Parallel addition is achievable in constant time for these bases.

The dominance condition is characterized by the absence of conjugates of modulus 1.

For the Golden Mean, the minimal alphabet is , 1, .

Abstract

We consider numeration systems where digits are integers and the base is an algebraic number $\beta$ such that $|\beta|>1$ and $\beta$ satisfies a polynomial where one coefficient is dominant in a certain sense. For this class of bases $\beta$, we can find an alphabet of signed-digits on which addition is realizable by a parallel algorithm in constant time. This algorithm is a kind of generalization of the one of Avizienis. We also discuss the question of cardinality of the used alphabet, and we are able to modify our algorithm in order to work with a smaller alphabet. We then prove that $\beta$ satisfies this dominance condition if and only if it has no conjugate of modulus 1. When the base $\beta$ is the Golden Mean, we further refine the construction to obtain a parallel algorithm on the alphabet $\{-1,0,1\}$. This alphabet cannot be reduced any more.