MINIMAL NON-INTEGER ALPHABETS ALLOWING PARALLEL ADDITION

. Parallel addition, i


n-integer alp
abets.

Introduction

The concept of parallel addition in a numeration system with a base β and alphabet A was introduced by A. Avizienis [1].The crucial difference from standard addition is that carry propagation is limited and hence an output digit depends only on bounded number of input digits.Therefore, the whole operation can run in constan time in parallel.

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. Lower bounds on the size of the al habet were given.

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 is useful for instance in online multiplication and division [6].Parallel addition algorithms that use noninteger alphabets are discussed in [7].The paper [8] discusses consequences of parallel addition for eventually periodic represent tions in Q(β).

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.In Section 3, we prove that if (β, A) allows parallel addition and β is a conjugate of β, then there is an alphabet A such that (β , A ) allows parallel addition.If A[β] = Z[β], we show that A must contain all representatives modul β and β −1.If β is an algebraic integer, a consequence is the same lower bound on the size of A ⊂ Z[β] as for integer alphabets.The assumption A[β] = Z[β] or existence of parallel addition without anticipation implies that β is expanding, i.e., all its conjugates are greater than one i modulus.

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 mo

lus one.


Pre
iminaries

The concept of positional numeration systems with integer bases and digits is very old and can be easily generalized:
Definition 2.1. If β ∈ C is such that |β| > 1 and
A ⊂ C is a finite set containing 0, then the pair (β, A) is called a numeration system with a base β and digit set A, usually called an alphabet.

Numbers in a numeration system (β, A) are represented in the following way: let x be a complex number of zeros.

The set of all numbers which have a (β, A)representation with only finitely many non-zero digits is ∈ A    .
The set of all numbers with a finite (β, A)representation with only non-negative p Jan Legerský


Acta Polytechnica


PREPRINT is denoted by
A[β] :=    n j=0 x j β j : n ∈ N, x j ∈ A    .
We remark that the definition of A[β] is analogous to the one of Z[β], i.e. the smallest ring containing Z and β, which is equivalent to the set of all sums of powers of β with integer coefficients.Now 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)represen

tion (which
is the case of parallel addition), then we can consider only bases which are algebraic numbers.

Lemma 2.2.Let β be a complex number such that |β| > 1 and A Z[β] be a finite alphabet with 0 ∈ A and 1 ∈ Fin A (β).If N ⊂ Fin A (β), then β is an algebraic number.

Proof.Since A ⊂ Z[β], all digits can be expressed as finite integer combinations of powers of β.Let d be the maximal exponent of β occurring i these expressions and C be the maximal absolute value of the integer coefficients of all digits in A.

Hence, for every N ∈ N, there exist m d j=0 α ij β j with α ij ∈ Z and |α ij | ≤ C, such that N = n i=−m a i β i = n i=−m d j=0 α ij β i+j .
Suppose for contradiction that β is transcendental.Therefore, the corresponding integer coefficients of powers of β on the left hand side and on the right hand side must be equal, particularly Proof.The closedness of Fin A (β) under addition and
1 ∈ Fin A (β) implies N ⊂ Fin A (β). If A[β] is closed under addition and 1 ∈ A[β] ⊂ Fin A (β), then N ⊂ A[β] ⊂ Fin A (β). In both cases, Lemma 2.2 applies.
The concept of parallelism for operations on representations is formalized by the following definition.Definition 2.4.Let A and B be alphabets.A function ϕ : B Z → A Z is said to be p-local if there exist r, t ∈ N satisfying p = r + t + 1 and a function φ : B p → A such that, for any w = (w j ) j∈Z ∈ B and its image z = ϕ(w) = (z j ) j∈Z ∈ A Z , we have z j = φ(w j+t , • • • , w j−r ) for every j ∈ Z.The parameter t, resp.r, is called anticipation, resp.memory.

In other words, every digit can by determined from only limited number of neighboring input digits.Since a (β, A + A)-representation of sum of two numbers can be easily obtained by digit-wise addition, the crucial part of parallel addition is conversion from the alphabet A + A to A. Definition 2.5.Let β be a base and let A and B be alphabets containing 0. A function ϕ : B Z → A Z such that (1.) for any w = (w j ) j∈Z ∈ B Z with finitely many non-zero digits, z = ϕ(w) = (z j ) j∈Z ∈ A Z has only finite number of non-zero digits, and

) j∈Z w j β j = j∈Z z j β j , is called a digit set conversion in the base β from B to A. Such a conversion ϕ is said to be computable in parallel if ϕ is a p-local fun

ion for some p ∈ N. Parallel addition in a numeration system (β, A) is a
igit set conversion in the base β from A + A to A, which is computable in parallel.


Necessary conditions on alphabets allowing parallel addition in A ⊂ Z[β]

Through this section, we assume that the base β is an algebraic number and the alphabet A is a finite subset of Z[β] such that {0} A. The finiteness of the alphabet is a natural assumption for a practic braic number β, if α, γ, δ are elements of
Z[β], then γ is congruent to δ modulo α in Z[β], denoted by γ ≡ α δ, if there exists ε ∈ Z[β] such that γ − δ = αε.
In this section, we recall the known results on h that A ⊂ Z[β]. If there exists a p-local parallel addition in (β, A) defined by a function φ : (A+A) p → A, then φ(b, . . . , b) ≡ β−1 b for any b ∈ A + A.
The same paper explain that, when considering only integer alphabets A ⊂ Z from the perspective of parallel addition algorithms, all the numbers β, 1/β, and their algebraic conjugates behave analogously: parallel a thms exist either for all, or for none of them.This statement can be extended to non-integer alphabets as well -the following lemma summarizes PREPRINT vol.

no. / Minimal non-integer alphabets allowing parallel addition that if we have a parallel addition algorithm for a base β, then we easily obtain such an algorithm also for conjugates of β by field isomorphism.Regarding the base 1/β, we can use the equa lgorithm, and thus in fact drop the requirement on the base to be greater than 1 in modulus.Proof.Let φ : A p → A be a mapping which defines ϕ with p = r + t + 1.We define a mapping φ : (A ) p → A by
φ (w j+t , . . . , w j−r ) = σ φ σ −1 (w j+t ), . . . , t set conversion ϕ : (A + A ) → A by ϕ (w ) = (z j ) j∈Z , where w = (w j ) j∈Z and z j = φ (w j e is only finitely many non-zeros in (z j ) 0 .
The value of the number represented by w is also preserved:
j∈Z w j β j = j∈Z σ(w j )σ(β) j = σ   j∈Z w j β j   = σ . . , w j−r ))β j = j∈Z z j β j ,
where
w j = σ −1 (w j ) f 3.3.Let (β, A) be a numeration system such that A ⊂ Z[β] and 1 < β ∈ R. Let λ = min A and Λ = max A. If there exists a p-local parallel addition in (β, A), with p = r + t + 1, defined by a mapping φ : (A + A) p → A, then:
(1.) φ(b, . . . , b) = λ for all b ∈ A + A such that b > λ ∧ (b ≥ 0 ∨ t = 0), (2.) φ(b, . . . , b) = Λ for all b ∈ A + A such that b < Λ ∧ (b ≤ 0 ∨ t = 0), (3.
) If Λ = 0, then φ(Λ, . . ., Λ) = Λ,
β j = β n W + we get −b 1 β r 1 β − 1 = −b ∞ j=r+1 1 β j ≥ −λ ∞ j=r+t+1 1 β j = −λ 1 β r+t 1 β − 1 . Thus, we have λ mption b > λ. If b ≥ 0, then λ ≥ bβ t ≥ b since β > 1, which is following statement.


Jan Legerský

Acta Polytechnica P hat A ⊂ Z[β], β is an algebraic number with Let φ be a mapping which defines the parallel addition.Since {0} A, we know that Λ > 0 or λ < 0. Assume that Λ > 0, the latter one is analogous.By Theorem 3.1 and Lemma 3.3, Λ ≡ β−1 φ(Λ, . . ., Λ) ∈ A and λ = φ(Λ, . . ., Λ) = Λ.Hence, φ(Λ, . . ., Λ) is a di

longs to the same
ongruence class as Λ.

If λ ≡ β−1 Λ, the claim follows, with c = φ(Λ, . . ., Λ).

The case that λ ≡ β−1 Λ is divided into two subcases.If λ = 0, then we have λ ≡ β−1 φ(λ, . . ., λ) ∈ A and φ(λ, . . ., λ) e.Suppose, for contradiction, that there is no nonzero element of the alphabet A congruent to 0 modulo β − 1.Let k be the number of congruence classes occurring in A and let R be a subset of A such that there is exactly one representative of each of those k congruence classes.For d ∈ Λ+R, the value φ(d, . . ., d) ∈ A is not congruent to 0, as φ(d, . . ., d) = λ = 0 by Lemma 3.3 and the congruence class con aining zero has only one element in A, by the previous assu ption.Therefore, the values f j = φ(d j , . . ., d j ) ∈ A for k distinct digits
d j = Λ + e j ∈ Λ + R belong to only k − 1 congruence classes modulo β − 1. Hence, there exist two distinct elements d 1 , d 2 ∈ Λ + R such that f 1 ≡ β−1 f 2 . Du to f j = φ(d j , . . . , d j ) ≡ β−1 d j = Λ + e j for each j ,
we obtain also e 1 ≡ β−1 e 2 which contradicts the construction of the set R.


A[β] closed under addition

In order to express the properties of the alphabet A allowing parallel addition in terms of representatives modulo β and β − 1, we restrict ourselves to alphabets such that A[β] is closed under addition, or even slightly stronger condition that A
[β] = Z[β].
The following theorem summarizes some consequences of these assumptions.Theorem 3.5.Let (β, A) be a numeration system such that A ⊂ Z[β] and 1 ∈ A[β].The following statements hold:  (3.):By Lemma 2.2, β is an algebraic number since A[β] ⊂ Fin A (β).The proof that β must be expanding is based on the paper of S. Akiyama and T. Zäimi [9].Let β d under addition, then N[β] ⊂ A[β]. (2Q(β) → Q(β ) be the field isomorphism su

that σ(β) = β . Since N ⊂
[β], for all n ∈ N there exist a 0 , . . . , a N ∈ A such that N i=0 a i β i = n = σ(n) = N i=0 σ(a i )(β ) i .
Denoting m := max{|σ(a)| : a ∈ A}, we have
n = |n| ≤ N i=0 |σ(a i )| • |β | i ≤ ∞ i=0 |σ(a i )| • |β | i ≤ m ∞ i=0 |β | i .
As n is arbitrarily large, the sum on the right side diverges, which implies that |β | ≥ 1.Thus, all conjugates of β are at least one in modulus.

If the degree of β is one, the statement is obvious.Therefore, we may assume that deg β ≥ 2.

Suppose for contradiction that |β | = 1 for an alge o an algebraic conjugate of β.Moreover, 1 γ ≥ 1 and |γ| ≥ 1, which implies that |γ| = 1.We may choose γ = β, which contradicts |β| > 1.Thus all con ugates of β are greater than one in modulus, i.e., β is an expanding algebraic number.

Let us remark that the assumption on A[β] to be closed under addition is satisfied by a wide class of numeration systems.Namely, if a numeration system (β, A) allows p-local parallel addition such that p = r+ 1, i.e., there is no anticipatio .)For β expanding, Lemma 8 in [10] provides a so called weak representation of zero property such that the absolute term is dominant, and hence parallel addition in the base β without anticipation is obtained for some integer alphabet A int according Theo em 4.3. in [5].

In what follows, we assume A[β] = Z[β], although the weaker assumption, A[β] being closed under addition, would be sufficient.The reason is that subtraction is also required in applications using parallel addition, such as on-line multiplication and division.Hence the assumption is justified by (2.) of Theorem 3.5.Let us me r instance, the numeration system (2, {0, 1, 2}) allows parallel addition, but (2, {0, 1, 2}) is obviously not closed under subtraction.
Proof. Let x = N i=0 x i β i be an element of Z[β]. Since x 0 ∈ Z ⊂ A[β], we have x ≡ x 0 = n i=0 a i β i ≡ β a 0 , where a i ∈ A.
Hence, for any element x ∈ Z[β], there is a digit a 0 ∈ A such that x ≡ β a 0 .

In order to prove that there is an element of A congruent to x modulo β − 1, we use the binomial theorem:
x = N i=0 x i β i = N i=0 x i (β − 1 1) i = N i=0 x j (β − 1) j , for some x j ∈ Z. Hence x ≡ β−1 x 0 = n i=0 a i β i ,
for some a i ∈ A. We prove by induction with respect to n that x 0 ≡ β−1 a for some a ∈ A. If n = 0, then x 0 = a 0 ∈ A. For n + 1, we have
x 0 = n+1 i=0 a i β i = a 0 + (β − 1) n i=0 a i+1 β i + n i=0 a i+1 β i ≡ β−1 a 0 + a ≡ β−1 a ∈ A ,
where we use the induction assumption n i=0 a i+1 β i ≡ β−1 a ∈ A and the statement of Theorem 3.1, i.e, for each digit b ∈ A + A there is a digit a ∈ A suc s an algebraic integer (in this whole subsection), since it enables us to count he number of congruence classes, and hence to provide an explicit lower bound on the size of alphabet allo of an algebraic integer be the degree of β.It is well known that
Z[β] = { d−1 i=0 x i β i : x i ∈ Z} if
and only if β is an algebraic integer.Hence, there is an obv i β i ∈ Z[β]
. Moreover, the additive group Z d can be equipped with a multiplication such that π is a ring isomorphism.In order to do that, we recall the concept of compani

s the sam
as the minimal polynomial of β.Since minimal polynomials have no multiple roots, S is diagonalizable over C, i.e., S = P −1 DP where D is a diagonal matrix with the conjugates of β on the diagonal, and P is a non-singular complex matrix.The matrix S α is also diagonalized by P :
S α = d−1 i=0 a ay.Since α = Finally, we put together the fact that the alphabet A for parallel addition in base β contains all representatives modulo β and modulo β 1, the derived formula for the number of congruence classes, and also specific restrictions on alphabets for parallel addition in a base with some positive real conjugate.We remark that the obtained bound is basically the same as the one for integer alphabets in [4].


Necessary and sufficient

condition on bases for parallel addition P. Kornerup [2] proposed a more general concept of parallel addition called k-block digit in the new numeration system with base being the k-th power of the original one.

Definition 4.1.For a positive integer k, the numeration system (β, A) allows k-block parallel addition if there exists parallel addition in (β k , A (k) ) , where
A (k) = {a k−1 β k−1 + • • • + a 1 β + a 0 : a i ∈ A}.
We remark that 1-block parallel addition is the same as parallel addition.C. Frougny, P. Heller, E. Pelantová and M. Svobodová [3] showed hat 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.We prove an extension of these statements also to alphabets being subsets of Z[β] in Theorem 4.2.Although the k-block concept does not enlarge the class of bases for parallel addition, it might decrease the minimal size of the alphabet.Theorem 4.2.Let β be a complex number such that |β| > 1.There exists an alphabet A ⊂ Z[β] with 0 ∈ A and 1 ∈ Fin A (β) which allows k-block parallel addition in (β, A) for some k ∈ N, if and only if β is an algebraic number with no conjugate of modulus 1.If this is the case, then there also exists an alphabet in Z allowing 1-block para lel addition in base β.

Proof.If the base β is an algebraic number with no conjugates of modulus

fact that j
+1 − j i > r + s, we have


Conclusion

We have shown that the necessary conditions on β and A allowing parallel addition that were known for alphabets consisting of consecutive integers can be largely extended to alphabets A being subsets of Z[β].

During our investigation, we also considered even more general case: i.e., A does not contain represen atives of all congruence classes.See [14] for further elaboration.

We conjecture the converse of Corollary 3.6, that is: let (β, A) be a numeration system such that 1 ∈ A[β] and A ⊂ Z[β].Let (β, A) allow parallel addition by p-local functi