Minimal Non-Integer Alphabets Allowing Parallel Addition

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.

Construction of Algorithms for Parallel Addition

“…Z[ω] -see the discussion in conclusion of [14]. Nevertheless, the statements in Theorem 5.1 (β must be expanding), Theorem 5.7 (convergence of Phase 1), and Theorems 5.10 and 5.12 (control of convergence of Phase 2) can be proven also for Z[ω], see [12].…”

“…For the case of alphabets A ⊂ Z of consecutive integers, the condition on base β to allow parallel addition was proved in [3], and later in [14] it was generalized to alphabets A ⊂ Z[β]: Theorem 2.3. Let β be a complex number such that |β| > 1.…”

“…The problem of the minimal alphabet size for parallel addition was again studied first for the case of alphabets of consecutive integers, e.g. in [5], and then extended in [14] to the more general case of A ⊂ Z[β]. Those results provide not just the lower bound on the alphabet size, but also the form of digits that have to be contained in the alphabet, based on congruence classes modulo β and β − 1.…”

“…We see from Theorem 2.4 that the requirements put on the alphabet A are in line with the necessary conditions on (β, A) for parallel addition. Proof: By Corollary 3.6 in [14], if the numeration system (β, A) allows parallel addition without anticipation (i.e., t = 0 in Definition 2.1), then β is expanding. Parallel addition produced by the EWM is indeed without anticipation, since there is no carry to the right when using the rewriting rule x − β.…”

“…Although the alphabet A is most often selected as a subset of integers, the problem is relevant also in a more general setting, namely considering A ⊂ Z [β]. Some of the results and conditions on the alphabet A to enable parallel addition, obtained earlier for integer alphabets, were generalized in [14] also to A ⊂ Z [β]. Especially the lower bound on minimal cardinality of A in the case of A[β] = Z [β].…”