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

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 [14]: 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 block 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 + · · · + X + 1. If in a base being a d-bonacci number 1-block parallel addition is possible on an alphabet A, then #A d + 1; on the other hand, there exists a k ∈ 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}.

On periodic representations in non-Pisot bases

“…• For every x = k∈Z x k β −k with x k = 0 for k < L for some L and x k ∈ B, it holds that x = k∈Z z k β −k , where z k = Φ(x k−t · · · x k x k+1 · · · x k+r ) ∈ A. Theorem 22 ([14], [12]). Let β ∈ C, |β| > 1.…”

Minimal Non-Integer Alphabets Allowing Parallel Addition

“…We remark that 1-block parallel addition is the same as parallel addition. C. Frougny, P. Heller, E. Pelantová and M. Svobodová [3] showed that for a given base β, there exists an integer alphabet A such that (β, A) allows parallel addition if and only if β is an algebraic number with no conjugates of modulus 1. Moreover, it was shown that the concept of k-block parallel addition does not enlarge the class of basis allowing parallel addition in case of integer alphabets.…”

“…It is know that the alphabet A must be redundant [2], otherwise parallel addition is not possible. Necessary conditions on the base and alphabet were further studied by C. Frougny, P. Heller, E. Pelantová, and M. Svobodová [3][4][5] under assumption that the alphabet A consists of consecutive integers containing 0. It was shown that there exists an integer alphabet allowing parallel addition if and only if the base is an algebraic number with no conjugates of modulus 1.…”

“…The key result from [3] is generalized to A ⊂ Z[β] in Section 4. Namely, there is an alphabet in Z[β] allowing so-called k-block parallel addition if and only if β is an algebraic number with no conjugates of modulus one.…”

Periodic representations in algebraic bases

“…Given two (β, A)-representations, one can study their behaviour under elementary arithmetic operations. In [4,5], the authors proved that if β has no conjugates on the unit circle, then there exists A ⊂ Z such that (β, A)-representations allow a parallel addition algorithm defined as follows.…”