Abstract
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.
Highlights
The concept of parallel addition in a numeration system with a base β and alphabet A was introduced by A
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
We show that whenever we require the alphabet to be finite and the sum of two numbers with finite (β, A)-representations to have again a finite (β, A)representation, we can consider only bases which are algebraic numbers
Summary
The concept of parallel addition in a numeration system with a base β and alphabet A was introduced by A. 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 main result of this paper is generalization of these results to non-integer alphabets, namely A ⊂ Z[β]. Such alphabets might have elements smaller in modulus comparing to integer ones. This paper is organized as follows: in Section 2, we recall the necessary definitions and show that for parallel addition we can consider only bases being algebraic numbers. 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
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