Parallel addition in non-standard numeration systems

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

kk-block parallel addition versus 11-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 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 $\{0,1,2\}$.Comment: 21 page

Parallel and online arithmetics in imaginary quadratic fields

Nestandardní číselné systémy jsou určené svou bází p é C, p > 1, a svou abecedou cifer A c C. Zabýváme se polygonálními číselnými systémy s abecedou ve tvaru A„= (0, 1, p,..., p" ), kde P = e ~ . Navíc požadujeme, aby báze i abeceda byly v okruhu celých čísel nějakého imaginárního kva-Non-standard numeration systems are given by their base P é C, P > 1, and their alphabet of digits A c C. We focus on the so-called polygonal numeration systems where the alphabet is of the form A„= (0, 1, P,..., P ') where P = e ~ and both the base and the alphabet are in the ring of algebraic integers of some imaginary quadratic field. Feasibility of several arithmetic operations including parallel addition and on-line division and multiplication is discussed. We characterize the complete polygonal numeration systems in imaginary quadratic fields. The Extending Window Method [20] is used to find the algorithms for parallel addition. Then the decision whether the numeration systems satisfy OL property follows along with computation of preprocessing for on-line division using the implementation from [29]

From positional representation of numbers to positional representation of vectors

To represent real m-dimensional vectors, a positional vector system given by a non-singular matrix M ∈ ℤm×m and a digit set Ɗ ⊂ ℤm is used. If m = 1, the system coincides with the well known numeration system used to represent real numbers. We study some properties of the vector systems which are transformable from the case m = 1 to higher dimensions. We focus on an algorithm for parallel addition and on systems allowing an eventually periodic representation of vectors with rational coordinates