Parallel Addition in Non-standard Numeration Systems

Transcription

1 1 / 23 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 June 6-10, 2011 Numeration 2011 Liege, Belgium

2 Table of contents 2 / 23 Table of contents 1 Preliminaries Numeration Systems Parallel Addition Positive Integer Base 2 Results for Bases being Algebraic Numbers Bases with Strong Representation of Zero Bases with Weak Representation of Zero Parallel Addition for Bases being Algebraic Numbers Base of Golden Mean 3 Concluding Remarks Algorithm Parameters & Properties Symmetric Alphabet & Subtraction Minimizing the Alphabet

3 3 / 23 Table of contents 1 Preliminaries Numeration Systems Parallel Addition Positive Integer Base 2 Results for Bases being Algebraic Numbers Bases with Strong Representation of Zero Bases with Weak Representation of Zero Parallel Addition for Bases being Algebraic Numbers Base of Golden Mean 3 Concluding Remarks Algorithm Parameters & Properties Symmetric Alphabet & Subtraction Minimizing the Alphabet

4 4 / 23 Numeration System with Base β and Alphabet A positional numeration system with base and digits

5 4 / 23 Numeration System with Base β and Alphabet A positional numeration system with base and digits number x M with a number field M = N, Z, Q, R, C

6 4 / 23 Numeration System with Base β and Alphabet A positional numeration system with base and digits number x M with a number field M = N, Z, Q, R, C base β N, Z, Q, R, C, with β > 1

7 4 / 23 Numeration System with Base β and Alphabet A positional numeration system with base and digits number x M with a number field M = N, Z, Q, R, C base β N, Z, Q, R, C, with β > 1 digits from alphabet A Z of consecutive integers:

8 4 / 23 Numeration System with Base β and Alphabet A positional numeration system with base and digits number x M with a number field M = N, Z, Q, R, C base β N, Z, Q, R, C, with β > 1 digits from alphabet A Z of consecutive integers: A = {0, 1,..., c} Z + A = { d,..., 1, 0, 1,..., c} Z

9 4 / 23 Numeration System with Base β and Alphabet A positional numeration system with base and digits number x M with a number field M = N, Z, Q, R, C base β N, Z, Q, R, C, with β > 1 digits from alphabet A Z of consecutive integers: A = {0, 1,..., c} Z + A = { d,..., 1, 0, 1,..., c} Z x M expressed as a β-representation with digits x j A: (x) β = x k x k 1... x 1 x 0 x 1... x l

10 4 / 23 Numeration System with Base β and Alphabet A positional numeration system with base and digits number x M with a number field M = N, Z, Q, R, C base β N, Z, Q, R, C, with β > 1 digits from alphabet A Z of consecutive integers: A = {0, 1,..., c} Z + A = { d,..., 1, 0, 1,..., c} Z x M expressed as a β-representation with digits x j A: (x) β = x k x k 1... x 1 x 0 x 1... x l x Fin A (β) = { j J x j β j J Z, J finite, x j A}

11 4 / 23 Numeration System with Base β and Alphabet A positional numeration system with base and digits number x M with a number field M = N, Z, Q, R, C base β N, Z, Q, R, C, with β > 1 digits from alphabet A Z of consecutive integers: A = {0, 1,..., c} Z + A = { d,..., 1, 0, 1,..., c} Z x M expressed as a β-representation with digits x j A: (x) β = x k x k 1... x 1 x 0 x 1... x l x Fin A (β) = { j J x j β j J Z, J finite, x j A} Note: One aspect to consider is non/redundancy of the systems.

12 Numeration System with Base β and Alphabet A positional numeration system with base and digits number x M with a number field M = N, Z, Q, R, C base β N, Z, Q, R, C, with β > 1 digits from alphabet A Z of consecutive integers: A = {0, 1,..., c} Z + A = { d,..., 1, 0, 1,..., c} Z x M expressed as a β-representation with digits x j A: (x) β = x k x k 1... x 1 x 0 x 1... x l x Fin A (β) = { j J x j β j J Z, J finite, x j A} Note: One aspect to consider is non/redundancy of the systems. Note: For different alphabets A A, the sets Fin A (β) and Fin A (β) can, but do not have to coincide. 4 / 23

13 5 / 23 Numeration System with Base β and Alphabet A Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting rules, or representations of zero:

14 5 / 23 Numeration System with Base β and Alphabet A Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting rules, or representations of zero: basic: given directly by the minimal polynomial of β

15 5 / 23 Numeration System with Base β and Alphabet A Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting rules, or representations of zero: basic: given directly by the minimal polynomial of β derived: from the basic one by various combinations

16 5 / 23 Numeration System with Base β and Alphabet A Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting rules, or representations of zero: basic: given directly by the minimal polynomial of β derived: from the basic one by various combinations therefore, we need the base β to be an algebraic number, in order to operate with integers as digits of the rewriting rules

17 5 / 23 Numeration System with Base β and Alphabet A Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting rules, or representations of zero: basic: given directly by the minimal polynomial of β derived: from the basic one by various combinations therefore, we need the base β to be an algebraic number, in order to operate with integers as digits of the rewriting rules Example Base β = τ, the Golden Mean, with minimal polynomial τ 2 = τ + 1, or τ 2 τ 1 = 0

18 5 / 23 Numeration System with Base β and Alphabet A Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting rules, or representations of zero: basic: given directly by the minimal polynomial of β derived: from the basic one by various combinations therefore, we need the base β to be an algebraic number, in order to operate with integers as digits of the rewriting rules Example Base β = τ, the Golden Mean, with minimal polynomial τ 2 = τ + 1, or τ 2 τ 1 = 0 basic rewriting rule: [1, 1, 1]

19 5 / 23 Numeration System with Base β and Alphabet A Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting rules, or representations of zero: basic: given directly by the minimal polynomial of β derived: from the basic one by various combinations therefore, we need the base β to be an algebraic number, in order to operate with integers as digits of the rewriting rules Example Base β = τ, the Golden Mean, with minimal polynomial τ 2 = τ + 1, or τ 2 τ 1 = 0 basic rewriting rule: [1, 1, 1] derived rewriting rules: infinite number of possibilities, e.g. [1, 0, 3, 0, 1] [1, 0, 0, 0, 7, 0, 0, 0, 1]

20 6 / 23 What is Addition To perform addition of two β-representations x = (x) β = x k... x 1 x 0 x 1... x l and y = (y) β = y k... y 1 y 0 y 1... y l with x j, y j A in fact means

21 6 / 23 What is Addition To perform addition of two β-representations x = (x) β = x k... x 1 x 0 x 1... x l and y = (y) β = y k... y 1 y 0 y 1... y l with x j, y j A in fact means to simply add the digits on each position separately into w j = x j + y j, thus obtaining an interim β-representation x + y = (w) β = w k... w 1 w 0 w 1... w l, with w j A + A;

22 6 / 23 What is Addition To perform addition of two β-representations x = (x) β = x k... x 1 x 0 x 1... x l and y = (y) β = y k... y 1 y 0 y 1... y l with x j, y j A in fact means to simply add the digits on each position separately into w j = x j + y j, thus obtaining an interim β-representation x + y = (w) β = w k... w 1 w 0 w 1... w l, with w j A + A; and then to apply such transformation on the β-representation (w) β that we

23 6 / 23 What is Addition To perform addition of two β-representations x = (x) β = x k... x 1 x 0 x 1... x l and y = (y) β = y k... y 1 y 0 y 1... y l with x j, y j A in fact means to simply add the digits on each position separately into w j = x j + y j, thus obtaining an interim β-representation x + y = (w) β = w k... w 1 w 0 w 1... w l, with w j A + A; and then to apply such transformation on the β-representation (w) β that we reduce the digits w j from alphabet A + A into digits z j within alphabet A,

24 6 / 23 What is Addition To perform addition of two β-representations x = (x) β = x k... x 1 x 0 x 1... x l and y = (y) β = y k... y 1 y 0 y 1... y l with x j, y j A in fact means to simply add the digits on each position separately into w j = x j + y j, thus obtaining an interim β-representation x + y = (w) β = w k... w 1 w 0 w 1... w l, with w j A + A; and then to apply such transformation on the β-representation (w) β that we reduce the digits w j from alphabet A + A into digits z j within alphabet A, while preserving the value of the resulting β-representation (z j ) β, i.e. x + y = (z) β = z k... z 1 z 0 z 1... z l

25 6 / 23 What is Addition To perform addition of two β-representations x = (x) β = x k... x 1 x 0 x 1... x l and y = (y) β = y k... y 1 y 0 y 1... y l with x j, y j A in fact means to simply add the digits on each position separately into w j = x j + y j, thus obtaining an interim β-representation x + y = (w) β = w k... w 1 w 0 w 1... w l, with w j A + A; and then to apply such transformation on the β-representation (w) β that we reduce the digits w j from alphabet A + A into digits z j within alphabet A, while preserving the value of the resulting β-representation (z j ) β, i.e. x + y = (z) β = z k... z 1 z 0 z 1... z l So, the task is to reduce digits from A + A back into A, while preserving the value of the β-representation.

26 7 / 23 What is Addition When constructing algorithms for performing (parallel) reduction from A + A into A, the main trick is the ability to reduce the digits just on the border of the original digit set A:

27 7 / 23 What is Addition When constructing algorithms for performing (parallel) reduction from A + A into A, the main trick is the ability to reduce the digits just on the border of the original digit set A: Example A = {0, 1,..., c}

28 7 / 23 What is Addition When constructing algorithms for performing (parallel) reduction from A + A into A, the main trick is the ability to reduce the digits just on the border of the original digit set A: Example A = {0, 1,..., c} A + A = {0, 1,..., c, c + 1, c + 2,..., 2c}

29 7 / 23 What is Addition When constructing algorithms for performing (parallel) reduction from A + A into A, the main trick is the ability to reduce the digits just on the border of the original digit set A: Example Example A = {0, 1,..., c} A + A = {0, 1,..., c, c + 1, c + 2,..., 2c} A = { d,..., 1, 0, 1,..., c}

30 7 / 23 What is Addition When constructing algorithms for performing (parallel) reduction from A + A into A, the main trick is the ability to reduce the digits just on the border of the original digit set A: Example Example A = {0, 1,..., c} A + A = {0, 1,..., c, c + 1, c + 2,..., 2c} A = { d,..., 1, 0, 1,..., c} A + A = { 2d,..., d 2, d 1, d,..., 1, 0, 1,..., c, c + 1, c + 2,..., 2c}

31 8 / 23 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.

32 8 / 23 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.

33 8 / 23 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.

34 8 / 23 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.

35 8 / 23 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!

36 9 / 23 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: we perform addition of two elements x, y Fin A (β)

37 9 / 23 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: 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

38 9 / 23 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: 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 w j = x j + y j... w j+t... w j+1 w j w j 1... w j r }{{}... w j A + A

39 9 / 23 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: 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 w j = x j + y j... w j+t... w j+1 w j w j 1... w j r }{{}... w j A + A z j = φ(w j+t... w j r )... z j+t... z j+1 z j z j 1... z j r... z j A

40 9 / 23 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: 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 w j = x j + y j... w j+t... w j+1 w j w j 1... w j r }{{}... w j A + A z j = φ(w j+t... w j r )... z j+t... z j+1 z j z j 1... z j r... z j A At the end of the process, we obtain a finite sum x + y = z j β j, with z j A; i.e. z = (z) β is back again in the original set Fin A (β).

41 9 / 23 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: 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 w j = x j + y j... w j+t... w j+1 w j w j 1... w j r }{{}... w j A + A z j = φ(w j+t... w j r )... z j+t... z j+1 z j z j 1... z j r... z j A At the end of the process, we obtain a finite sum x + y = z j β j, with z j A; i.e. z = (z) β is back again in the original set Fin A (β). Note: Application of the p-local function ϕ(w) = z generally enlarges the set of indices j Z for which z j 0, compared to the set of indices for which w j 0, by r positions to the left and by t positions to the right.

42 10 / 23 Parallel Addition for Bases β = b N For base β a positive integer β = b 2, there are known results:

43 10 / 23 Parallel Addition for Bases β = b N For base β a positive integer β = b 2, there are known results: by A.Avizienis (1961)

44 10 / 23 Parallel Addition for Bases β = b N For base β a positive integer β = b 2, there are known results: by A.Avizienis (1961) by C.Y.Chow & J.E.Robertson (1978)

45 10 / 23 Parallel Addition for Bases β = b N For base β a positive integer β = b 2, there are known results: by A.Avizienis (1961) by C.Y.Chow & J.E.Robertson (1978) by B.Parhami (1990)

46 10 / 23 Parallel Addition for Bases β = b N For base β a positive integer β = b 2, there are known results: by A.Avizienis (1961) by C.Y.Chow & J.E.Robertson (1978) by B.Parhami (1990) base 2-local algorithm 3-local algorithm 3-local algorithm b N of Avizienis of Chow & Robertson of Parhami 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}

47 Parallel Addition for Bases β = b N For base β a positive integer β = b 2, there are known results: by A.Avizienis (1961) by C.Y.Chow & J.E.Robertson (1978) by B.Parhami (1990) base 2-local algorithm 3-local algorithm 3-local algorithm b N of Avizienis of Chow & Robertson of Parhami 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} * Apart { from alphabet } A = {0,..., b}, we can also use shifted alphabets, { thereby also a symmetric alphabet A = b 2,..., b for b even, and an almost symmetric alphabet A = b 1 },..., b+1 for b odd / 23

48 11 / 23 Table of contents 1 Preliminaries Numeration Systems Parallel Addition Positive Integer Base 2 Results for Bases being Algebraic Numbers Bases with Strong Representation of Zero Bases with Weak Representation of Zero Parallel Addition for Bases being Algebraic Numbers Base of Golden Mean 3 Concluding Remarks Algorithm Parameters & Properties Symmetric Alphabet & Subtraction Minimizing the Alphabet

49 12 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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

50 12 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 1 β 1 + b 0 + b 1 β b h β h = S(β)

51 12 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 1 β 1 + b 0 + b 1 β b h β h = S(β) with b j Z and b 0 =: B > 2M := 2 j 0 b j.

52 12 / 23 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 1 β 1 + b 0 + b 1 β b h β h = S(β) with b j Z and b 0 =: B > 2M := 2 j 0 b j. S is then called a strong polynomial for β.

53 12 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 1 β 1 + b 0 + b 1 β b h β h = S(β) with b j 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)

54 12 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 1 β 1 + b 0 + b 1 β b h β h = S(β) with b j 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,

55 12 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 1 β 1 + b 0 + b 1 β b h β h = S(β) with b j 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.

56 13 / 23 Parallel Addition for Bases β C with Strong Representation of Zero Example: irrational base β = = :

57 13 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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

58 13 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 a = 18; memory k = 1, anticipation h = 1 p = 3

59 13 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 a = 18; memory k = 1, anticipation h = 1 p = 3 we obtain a 3-local function of addition working on alphabet A = { 18,..., 0,..., +18} as follows:

60 13 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 a = 18; memory k = 1, anticipation h = 1 p = 3 we obtain a 3-local function of addition working on alphabet A = { 18,..., 0,..., +18} as follows: x Fin A (β) x j A y Fin A (β) y j A

61 13 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 a = 18; memory k = 1, anticipation h = 1 p = 3 we obtain a 3-local function of addition working on alphabet A = { 18,..., 0,..., +18} as follows: x Fin A (β) x j A y Fin A (β) y j A w j = x j + y j w j A + A

62 13 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 a = 18; memory k = 1, anticipation h = 1 p = 3 we obtain a 3-local function of addition working on alphabet A = { 18,..., 0,..., +18} as follows: x Fin A (β) x j A y Fin A (β) y j A w j = x j + y j w j A + A

63 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 a = 18; memory k = 1, anticipation h = 1 p = 3 we obtain a 3-local function of addition working on alphabet A = { 18,..., 0,..., +18} as follows: x Fin A (β) x j A y Fin A (β) y j A w j = x j + y j w j A + A z j = φ(w j+1 w j w j 1 ) z 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. 13 / 23

64 14 / 23 Bases β with Strong Representation of Zero Theorem Let β be an algebraic number of degree d, and let β > 1.

65 14 / 23 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

66 14 / 23 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,

67 14 / 23 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.

68 14 / 23 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

69 15 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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

70 15 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 1 β 1 + b 0 + b 1 β b h β h = W (β)

71 15 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 1 β 1 + b 0 + b 1 β b h β h = W (β) with b j Z and b 0 =: B > M := j 0 b j.

72 15 / 23 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 1 β 1 + b 0 + b 1 β b h β h = W (β) with b j Z and b 0 =: B > M := j 0 b j. W is then called a weak polynomial for β.

73 15 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 1 β 1 + b 0 + b 1 β b h β h = W (β) with b j 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.

74 15 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 1 β 1 + b 0 + b 1 β b h β h = W (β) with b j 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,

75 15 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 1 β 1 + b 0 + b 1 β b h β h = W (β) with b j 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

76 16 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks Parallel Addition for Bases β C with Weak Representation of Zero Example: irrational base β = = τ, the Golden Mean:

77 16 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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

78 16 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 a = 3; memory = anticipation = = 6 p = 13

79 16 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 a = 3; memory = anticipation = = 6 p = local function on alphabet A = { 3,..., 0,..., +3}:

80 16 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 a = 3; memory = anticipation = = 6 p = local function on alphabet A = { 3,..., 0,..., +3}: x Fin A (β) x j A y Fin A (β) y j A w j = x j + y j w j A + A

81 16 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 a = 3; memory = anticipation = = 6 p = local function on alphabet A = { 3,..., 0,..., +3}: x Fin A (β) x j A y Fin A (β) y j A w j = x j + y j w j A + A

82 16 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 a = 3; memory = anticipation = = 6 p = local function on alphabet A = { 3,..., 0,..., +3}: x Fin A (β) x j A y Fin A (β) y j A w j = x j + y j w j A + A

83 16 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks 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 a = 3; memory = anticipation = = 6 p = local function on alphabet A = { 3,..., 0,..., +3}: x Fin A (β) x j A y Fin A (β) y j A w j = x j + y j w j A + A z j = φ(w j+6... w j 6 ) z j A

84 17 / 23 Preliminaries Results for Bases being Algebraic Numbers Concluding Remarks Parallel Addition 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.

85 17 / 23 Parallel Addition 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.

86 17 / 23 Parallel Addition 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.

87 17 / 23 Parallel Addition 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.

88 18 / 23 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

89 18 / 23 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 memory = anticipation = 4.

90 18 / 23 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 memory = anticipation = 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 memory = anticipation = 6.

91 18 / 23 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 memory = anticipation = 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 memory = anticipation = 6. Algorithm III: The weak polynomial τ τ 2 = 0 enables to construct an algorithm, specific for base τ, for parallel addition by means of a 21-local function in alphabet A = { 1, 0, 1}

92 18 / 23 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 memory = anticipation = 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 memory = anticipation = 6. Algorithm III: The weak polynomial τ τ 2 = 0 enables to construct an algorithm, specific for base τ, for parallel addition by means of a 21-local function in alphabet A = { 1, 0, 1}...and A = { 1, 0, 1} is the minimal alphabet allowing parallel addition by means of a p-local function for base τ

93 19 / 23 Table of contents 1 Preliminaries Numeration Systems Parallel Addition Positive Integer Base 2 Results for Bases being Algebraic Numbers Bases with Strong Representation of Zero Bases with Weak Representation of Zero Parallel Addition for Bases being Algebraic Numbers Base of Golden Mean 3 Concluding Remarks Algorithm Parameters & Properties Symmetric Alphabet & Subtraction Minimizing the Alphabet

94 20 / 23 Algorithm Parameters & Properties One aspect we distinguish on the parallel algorithms is their neighbor sensitivity, in the sense that we call the algorithm

95 20 / 23 Algorithm Parameters & Properties One aspect we distinguish on the parallel algorithms is their neighbor sensitivity, in the sense that we call the algorithm neighbor-free when the prescription what to do on the respective position w j depends only on the value w j itself, and not on its neighbors

96 20 / 23 Algorithm Parameters & Properties One aspect we distinguish on the parallel algorithms is their neighbor sensitivity, in the sense that we call the algorithm neighbor-free when the prescription what to do on the respective position w j depends only on the value w j itself, and not on its neighbors neighbor-sensitive when the prescription for position w j is dependent on the values of some of its neighbors (e.g. w j 1, or w j+1, or w j 2, or a combination...)

97 20 / 23 Algorithm Parameters & Properties One aspect we distinguish on the parallel algorithms is their neighbor sensitivity, in the sense that we call the algorithm Example neighbor-free when the prescription what to do on the respective position w j depends only on the value w j itself, and not on its neighbors neighbor-sensitive when the prescription for position w j is dependent on the values of some of its neighbors (e.g. w j 1, or w j+1, or w j 2, or a combination...) Avizienis algorithm: neighbor-free

98 20 / 23 Algorithm Parameters & Properties One aspect we distinguish on the parallel algorithms is their neighbor sensitivity, in the sense that we call the algorithm Example neighbor-free when the prescription what to do on the respective position w j depends only on the value w j itself, and not on its neighbors neighbor-sensitive when the prescription for position w j is dependent on the values of some of its neighbors (e.g. w j 1, or w j+1, or w j 2, or a combination...) Avizienis algorithm: neighbor-free Chow-Robertson algorithm: neighbor-sensitive

99 20 / 23 Algorithm Parameters & Properties One aspect we distinguish on the parallel algorithms is their neighbor sensitivity, in the sense that we call the algorithm Example neighbor-free when the prescription what to do on the respective position w j depends only on the value w j itself, and not on its neighbors neighbor-sensitive when the prescription for position w j is dependent on the values of some of its neighbors (e.g. w j 1, or w j+1, or w j 2, or a combination...) Avizienis algorithm: neighbor-free Chow-Robertson algorithm: neighbor-sensitive algorithms based on strong or weak representation of zero: neighbor-free

100 Algorithm Parameters & Properties One aspect we distinguish on the parallel algorithms is their neighbor sensitivity, in the sense that we call the algorithm Example neighbor-free when the prescription what to do on the respective position w j depends only on the value w j itself, and not on its neighbors neighbor-sensitive when the prescription for position w j is dependent on the values of some of its neighbors (e.g. w j 1, or w j+1, or w j 2, or a combination...) Avizienis algorithm: neighbor-free Chow-Robertson algorithm: neighbor-sensitive algorithms based on strong or weak representation of zero: neighbor-free Algorithm III for base the Golden Mean: (very) neighbor-sensitive 20 / 23

101 21 / 23 Algorithm Parameters & Properties The p-locality has two main contributors :

102 21 / 23 Algorithm Parameters & Properties The p-locality has two main contributors : neighbor-sensitivity increases the parameter p... depending on how many neighbors we must consider when deciding what to do on position w j

103 21 / 23 Algorithm Parameters & Properties The p-locality has two main contributors : neighbor-sensitivity increases the parameter p... depending on how many neighbors we must consider when deciding what to do on position w j width of the rewriting rule we are using within the algorithm

104 21 / 23 Algorithm Parameters & Properties The p-locality has two main contributors : neighbor-sensitivity increases the parameter p... depending on how many neighbors we must consider when deciding what to do on position w j width of the rewriting rule we are using within the algorithm We can make choices between the size of the alphabet and the p-locality:

105 21 / 23 Algorithm Parameters & Properties The p-locality has two main contributors : neighbor-sensitivity increases the parameter p... depending on how many neighbors we must consider when deciding what to do on position w j width of the rewriting rule we are using within the algorithm We can make choices between the size of the alphabet and the p-locality: bigger cardinality #A of alphabet A smaller p-locality = narrower sliding window

106 21 / 23 Algorithm Parameters & Properties The p-locality has two main contributors : neighbor-sensitivity increases the parameter p... depending on how many neighbors we must consider when deciding what to do on position w j width of the rewriting rule we are using within the algorithm We can make choices between the size of the alphabet and the p-locality: bigger cardinality #A of alphabet A smaller p-locality = narrower sliding window smaller cardinality #A of alphabet A bigger p-locality = wider sliding window

107 22 / 23 Symmetric Alphabet & Subtraction For the parallelism as such, there is no need for a symmetric alphabet, however,

108 22 / 23 Symmetric Alphabet & Subtraction For the parallelism as such, there is no need for a symmetric alphabet, however, having a symmetric alphabet means that the same algorithm as for (parallel) addition can be used for (parallel) subtraction,

109 22 / 23 Symmetric Alphabet & Subtraction For the parallelism as such, there is no need for a symmetric alphabet, however, having a symmetric alphabet means that the same algorithm as for (parallel) addition can be used for (parallel) subtraction, since (parallel) subtraction is a reduction of digits from A A back into A

110 22 / 23 Symmetric Alphabet & Subtraction For the parallelism as such, there is no need for a symmetric alphabet, however, having a symmetric alphabet means that the same algorithm as for (parallel) addition can be used for (parallel) subtraction, since (parallel) subtraction is a reduction of digits from A A back into A... symmetric alphabet means A = A, so A A equals A + A, so subtraction algorithm equals addition algorithm...

111 22 / 23 Symmetric Alphabet & Subtraction For the parallelism as such, there is no need for a symmetric alphabet, however, having a symmetric alphabet means that the same algorithm as for (parallel) addition can be used for (parallel) subtraction, since (parallel) subtraction is a reduction of digits from A A back into A... symmetric alphabet means A = A, so A A equals A + A, so subtraction algorithm equals addition algorithm... Note: Requiring symmetry of the alphabet A may incur an increased cardinality #A.

112 22 / 23 Symmetric Alphabet & Subtraction For the parallelism as such, there is no need for a symmetric alphabet, however, having a symmetric alphabet means that the same algorithm as for (parallel) addition can be used for (parallel) subtraction, since (parallel) subtraction is a reduction of digits from A A back into A... symmetric alphabet means A = A, so A A equals A + A, so subtraction algorithm equals addition algorithm... Note: Requiring symmetry of the alphabet A may incur an increased cardinality #A. Note: Symmetric alphabet is a simplification, but not necessity for (parallel) subtraction.

113 Symmetric Alphabet & Subtraction For the parallelism as such, there is no need for a symmetric alphabet, however, having a symmetric alphabet means that the same algorithm as for (parallel) addition can be used for (parallel) subtraction, since (parallel) subtraction is a reduction of digits from A A back into A... symmetric alphabet means A = A, so A A equals A + A, so subtraction algorithm equals addition algorithm... Note: Requiring symmetry of the alphabet A may incur an increased cardinality #A. Note: Symmetric alphabet is a simplification, but not necessity for (parallel) subtraction. Note: Also for subtraction, the crucial trick for parallelism is the ability to reduce the digits on the border of the original digit set A. 22 / 23

114 23 / 23 Minimizing the Alphabet We now have results - algorithms for parallel addition - for a broad class of algebraic numbers.

115 23 / 23 Minimizing the Alphabet We now have results - algorithms for parallel addition - for a broad class of algebraic numbers. These algorithms are quite general, but therefore not yet optimized in any way:

116 23 / 23 Minimizing the Alphabet We now have results - algorithms for parallel addition - for a broad class of algebraic numbers. These algorithms are quite general, but therefore not yet optimized in any way: neither in terms of the cardinality #A of the alphabet,

117 23 / 23 Minimizing the Alphabet We now have results - algorithms for parallel addition - for a broad class of algebraic numbers. These algorithms are quite general, but therefore not yet optimized in any way: neither in terms of the cardinality #A of the alphabet, nor in terms of the p-locality.

118 23 / 23 Minimizing the Alphabet We now have results - algorithms for parallel addition - for a broad class of algebraic numbers. These algorithms are quite general, but therefore not yet optimized in any way: neither in terms of the cardinality #A of the alphabet, nor in terms of the p-locality. For specific bases β, the minimal alphabets for parallelism are known:

119 23 / 23 Minimizing the Alphabet We now have results - algorithms for parallel addition - for a broad class of algebraic numbers. These algorithms are quite general, but therefore not yet optimized in any way: neither in terms of the cardinality #A of the alphabet, nor in terms of the p-locality. For specific bases β, the minimal alphabets for parallelism are known: Positive integer bases: minimal alphabet for parallelism in base β = b N has cardinality #A = b + 1

120 23 / 23 Minimizing the Alphabet We now have results - algorithms for parallel addition - for a broad class of algebraic numbers. These algorithms are quite general, but therefore not yet optimized in any way: neither in terms of the cardinality #A of the alphabet, nor in terms of the p-locality. For specific bases β, the minimal alphabets for parallelism are known: Positive integer bases: minimal alphabet for parallelism in base β = b N has cardinality #A = b + 1 Base of Golden Mean: minimal alphabet for parallelism is A = { 1, 0, 1}

121 23 / 23 Minimizing the Alphabet We now have results - algorithms for parallel addition - for a broad class of algebraic numbers. These algorithms are quite general, but therefore not yet optimized in any way: neither in terms of the cardinality #A of the alphabet, nor in terms of the p-locality. For specific bases β, the minimal alphabets for parallelism are known: Positive integer bases: minimal alphabet for parallelism in base β = b N has cardinality #A = b + 1 Base of Golden Mean: minimal alphabet for parallelism is A = { 1, 0, 1} Next task: what are the minimal alphabets for parallelism for bases β being algebraic number of higher orders, of course starting with quadratic first... the clue is neighbor sensitivity