$k$-block parallel addition versus $1$-block parallel addition in non-standard numeration systems

TL;DR

This paper investigates how $k$-block parallel addition affects the relationship between base and alphabet size in non-standard numeration systems, providing bounds, algorithms, and specific results for Parry, quadratic Pisot, and $d$-bonacci bases.

Contribution

It establishes lower bounds on alphabet sizes for block parallel addition, offers explicit algorithms for quadratic Pisot bases, and analyzes the $d$-bonacci base's properties.

Findings

Lower bounds on alphabet size for block parallel addition

Explicit algorithms for quadratic Pisot bases

Minimum alphabet size for $d$-bonacci base addition

Abstract

Parallel addition in integer base is used for speeding up multiplication and division algorithms. $k$-block parallel addition has been introduced by Kornerup in 1999: instead of manipulating single digits, one works with blocks of fixed length $k$. The aim of this paper is to investigate how such notion influences the relationship between the base and the cardinality of the alphabet allowing parallel addition. In this paper, we mainly focus on a certain class of real bases --- the so-called Parry numbers. We give lower bounds on the cardinality of alphabets of non-negative integer digits allowing block parallel addition. By considering quadratic Pisot bases, we are able to show that these bounds cannot be improved in general and we give explicit parallel algorithms for addition in these cases. We also consider the $d$-bonacci base, which satisfies the equation $X^d = X^{d-1} + X^{d-2} + \cdots + X + 1$. If in a base being a $d$-bonacci number $1$-block parallel addition is possible on the alphabet $\mathcal{A}$, then $\#\mathcal{A} \geq d+1$; on the other hand, there exists a $k\in\mathbb{N}$ such that $k$-block parallel addition in this base is possible on the alphabet $\{0,1,2\}$, which cannot be reduced. In particular, addition in the Tribonacci base is $14$-block parallel on alphabet $\{0,1,2\}$.