Parallel addition in non-standard numeration systems

Minimal digit sets for parallel addition in non-standard numeration systems

We study parallel algorithms for addition of numbers having finite representation in a positional numeration system defined by a base $β$ in $\mathbb{C}$ and a finite digit set $\mathcal{A}$ of contiguous integers containing $0$. For a fixed base $β$, we focus on the question of the size of the alphabet allowing to perform addition in constant time independently of the length of representation of the summands. We produce lower bounds on the size of such alphabet $\mathcal{A}$. For several types of well studied bases (negative integer, complex numbers $ -1 + \imath$, $2 \imath$, and $\imath \sq…

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

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 allow…

Arithmetics in numeration systems with negative quadratic base

We consider positional numeration system with negative base $-β$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $β$ is a quadratic Pisot number. We study a class of roots $β>1$ of polynomials $x^2-mx-n$, $m\geq n\geq 1$, and show that in this case the set ${\rm Fin}(-β)$ of finite $(-β)$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $β=τ=\frac12(1+\sqrt5)$, the golden ratio. For such $β$, we determine the exact bound on the number of fractional digits appearing in arithmetic…

二进制补码计数系统的斐波那契类比

A Fibonacci analogue of the two's complement numeration system

本文提出斐波那契计数系统的补码表示法，证明其加法可由确定性有限状态转换器实现，并基于Berstel加法器给出构造性证明，同时刻画了该表示法与整数集之间的递增双射关系。

Minimal non-integer alphabets allowing parallel addition

Parallel addition, i.e., addition with limited carry propagation, has been so far studied for complex bases and integer alphabets. We focus on alphabets consisting of integer combinations of powers of the base. We give necessary conditions on the alphabet allowing parallel addition. Under certain assumptions, we prove the same lower bound on the size of the generalized alphabet that is known for alphabets consisting of consecutive integers. We also extend the characterization of bases allowing parallel addition to numeration systems with non-integer alphabets.

使用对乘法封闭的数字集的高斯整数与爱森斯坦整数的最优表示

Optimal Representations of Gaussian and Eisenstein Integers using digit sets closed under multiplication

本文研究高斯整数和爱森斯坦整数在特定基与数字集下的最优表示，给出最优表示个数的上界并刻画达到上界的整数，推广了有符号二进制记数系统的相关结果。