Parallel Addition in Non-standard Numeration Systems

Transcription

1 Parallel Addition in Non-standard Numeration Systems Christiane Frougny*, Edita Pelantová**, Milena Svobodová** * LIAFA, UMR 7089 CNRS, Université Paris 7 & Université Paris 8, Paris, France ** Dept. of Mathematics, FNSPE, Czech Technical University, Prague, Czech Republic March 19-20, 2011 AAMP VIII - Words, Numeration, and Automata Prague, Czech Republic

2 Abstract We consider numeration systems where digits are integers and the base is an algebraic number β such that β > 1 and the modulus of all the conjugates of β differs from 1. For this broad class of bases β, we can find an alphabet of signed-digits on which addition is realizable by a parallel algorithm in constant time. An important property of this algorithm is that it is a p-local function. We also discuss the question of cardinality of the used alphabet, and we are able to modify our algorithm in order to work in a smaller alphabet. For the particular case of the base β being the Golden Mean, we further refine the construction to obtain a parallel algorithm (again a p-local function) on the alphabet { 1, 0, 1}, which is the minimum alphabet allowing parallel addition in this base.

3 Non-standard Numeration Systems We work with positional numeration systems, given by base β C, β > 1, and a finite set of integer digits called alphabet A Z Let us consider finite β-representations of numbers in this alphabet A: Fin A (β) = {x = j I x j β j I Z, I finite, x j A} C Note: By calling these numeration systems non-standard we refer to the broad choice of the base β: the base does not have to be positive integer, in fact it does not have to be integer at all, neither it has to be real. Note: Non/redundancy of these numeration systems is dependant on the choice of the alphabet A. Note: For different alphabets A A, the sets Fin A (β) and Fin A (β) can, but do not have to coincide.

4 Parallel Addition as a Local Function We are looking for algorithms allowing parallel addition in these numeration systems; more specifically, addition by means of a local function. Definition Let A, B be alphabets, and let A Z, B Z be sets of words on these alphabets. Let r, t be non-negative integers, p = r + t + 1 N, and let φ : A p B. A function ϕ : A Z B Z acting on u A Z by ϕ(u) = v B Z, where v j = φ(u j+t... u j... u j r ), is called a p-local function, with memory r and anticipation t. Parallel addition by means of a p-local function necessarily requires a redundant numeration system!

5 Parallel Addition as a Local Function Algorithm of such p-local function is also called a sliding block code; as illustrated by the following scheme, wherein p = r + t + 1, with memory r and anticipation t: we perform addition of two elements x, y Fin A (β) x Fin A (β)... x j+t... x j+1 x j x j 1... x j r... x j A y Fin A (β)... y j+t... y j+1 y j y j 1... y j r... y j A u j = x j + y j... u j+t... u j+1 u j u j 1... u j r }{{}... u j A + A v j = φ(u j+t... u j r )... v j+t... v j+1 v j v j 1... v j r... v j A At the end of the process, we obtain the sum x + y = v j β j, with v = (v j ) A Z ; i.e. back again in the original set Fin A (β). Note: Application of the p-local function ϕ(u) = v generally enlarges the set of indices j Z for which v j 0, compared to the set of indices for which u j 0, by r positions to the left and by t positions to the right.

6 Parallel Addition as p-local Function for Bases β C being Algebraic Numbers Developing an algorithm of parallel addition as a p-local function, for a given numeration system with base β, comprises not only describing the steps of the algorithm, but also finding a suitable alphabet in which parallelization is possible, and also determining the parameter p itself. We develop two algorithms that perform parallel addition as p-local function, for quite a large class of bases β C being algebraic numbers. They work on symmetric alphabets A = { a,..., 0,..., a}, therefore, they perform not only parallel addition, but also parallel subtraction. These algorithms, being quite general, are not yet focused on minimizing the size of alphabet A.

7 Parallel Addition for Bases β C with Strong Representation of Zero Definition (Strong Representation of Zero Property) β C, β > 1 has the strong representation of zero property if 0 = b k β k + b k 1 β k b 1 β 1 + b 0 + b 1 β b h β h = S(β) with b i Z and b 0 =: B > 2M := 2 j 0 b j. S is then called a strong polynomial for β. Theorem Let β C satisfy the strong representation of zero property. Denote A = { a,..., a} for a = B M. B 1 2(B 2M) Then, addition in Fin A (β) can be realized as a p-local function, with p = }{{} k memory + }{{} h anticipation +1.

8 Parallel Addition for Bases β C with Strong Representation of Zero Example: irrational base β = = : β is root of the strong polynomial S(β) = 2β β 1 B = 17, M = 5 a = 18 memory k = 1, anticipation h = 1 p = 3 We obtain a 3-local function on alphabet A = { 18,..., 0,..., +18}. See the action of the algorithm using this 3-local function on an example: x Fin A (β) x j A y Fin A (β) y j A u j = x j + y j u j A + A v j = φ(u j+1 u j u j 1 ) v j A... we are in fact subtracting a convenient multiple of the strong polynomial on each position separately, and then summing up the result of all these subtractions.

9 Bases β with Strong Representation of Zero Theorem Let β be an algebraic number of degree d, and let β > 1. If d is odd, or if d is even and the minimal polynomial of β is not reciprocal, then β has the strong representation of zero property. We have a constructive proof of this Theorem, providing e.g. the following results: base β minimal polynomial of β strong polynomial for β β = 2 β 2 = 0 S(X ) = X β = β 2 β 1 = 0 S(X ) = X 8 + 7X 4 1 β = 1 + ı β 2 + 2β + 2 = 0 S(X ) = X 4 + 4

10 Parallel Addition for Bases β C with Weak Representation of Zero Definition (Weak Representation of Zero Property) β C, β > 1 has the weak representation of zero property if 0 = b k β k + b k 1 β k b 1 β 1 + b 0 + b 1 β b h β h = W (β) with b i Z and b 0 =: B > M := j 0 b j. W is then called a weak polynomial for β. Theorem Let β C satisfy the weak representation of zero property. Denote A = { a,..., a} for a = B M. Then, addition in Fin A (β) can be realized as a p-local function, with p = a B M }{{ k + a B M } h +1. }{{} memory anticipation

11 Parallel Addition for Bases β C with Weak Representation of Zero Example: irrational base β = = τ, the Golden Mean: τ is root of the weak polynomial W (τ) = τ τ 2 B = 3, M = 2, k = h = 2 a = 3 memory = anticipation = = 6 p = 13 We obtain a 13-local function on alphabet A = { 3,..., 0,..., +3}. x Fin A (β) x j A y Fin A (β) y j A u j = x j + y j u j A + A v j = φ(u j+6... u j 6 ) v j A

12 Algorithms for Base β = = τ, the Golden Mean minimal polynomial: τ 2 + τ + 1 = 0 weak polynomial: τ τ 2 = 0 strong polynomial: τ τ 4 = 0 Algorithm I: Using the strong polynomial τ τ 4 = 0, we obtain a 9-local function for parallel addition in alphabet A = { 5,..., 0,..., 5}, with both memory and anticipation equal to 4. Algorithm II: Using the weak polynomial τ τ 2 = 0, we obtain a 13-local function for parallel addition in alphabet A = { 3,..., 0,..., 3}, with both memory and anticipation equal to 6. Algorithm III: The weak polynomial τ τ 2 = 0 enables to construct another algorithm, specific for base τ, for parallel addition by means of a 21-local function in alphabet A = { 1, 0, 1}, with both memory and anticipation equal to 10.

13 Algorithms for Base β = = τ, the Golden Mean The alphabet A = { 1, 0, 1} is the minimal alphabet allowing parallel addition by means of a p-local function for base τ; it is proved that alphabet {0, 1} would not be sufficient. All the three algorithms I, II, III as described for base τ, with just minor modifications, work also for the Fibonacci numeration system.

14 Parallel Addition as p-local Function for Bases β = b N For base β being a positive integer β = b 2, there were already known results, by A.Avizienis (1961) and by C.Y.Chow & J.E.Robertson (1978); and we now enrich these results still a bit further see the following summary table of algorithms for parallel addition by means of p-local function, in numeration systems with positive integer base β = b: base 2-local algorithm 3-local algorithm 3-local algorithm b N of Avizienis of Chow & Robertson new* b = 2 not working A = { 1, 0, 1} A = {0, 1, 2} b = 3 A = { 2,..., 2} not working A = {0, 1, 2, 3} b = 4 A = { 3,..., 3} A = { 2,..., 2} A = {0,..., 4} b = 5 A = { 3,..., 3} not working A = {0,..., 5} b = 6 A = { 4,..., 4} A = { 3,..., 3} A = {0,..., 6}.. } {. }. b = 2m A = { b 2 1,..., b A = b 2,..., b A = {0,..., b} 2 = { m { 1,..., m + } 1} = { m,..., m} = {0,..., 2m} b = 2m + 1 A = b+1 2,..., b+1 not working A = {0,..., b} 2 = { m 1,..., m + 1} = {0,..., 2m + 1} * In the new algorithm, we can choose not only alphabet A = {0,..., b}, but also a symmetric alphabet A = { b 2,..., } b 2 for b even, and an almost symmetric alphabet A = { b 1 2,..., } b+1 2 for b odd.