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Minimal Digit Sets for Parallel Addition
in Non-standard Numeration Systems
Christiane Frougny*, Edita Pelantová**,
Milena Svobodová**
* LIAFA, CNRS UMR 7089 & Université Paris 7 & Université Paris 8, Paris, France
** Dept. of Mathematics, FNSPE, Czech Technical University, Prague, Czech Republic
May 21-25, 2012
CANT 2012
Marseille, France
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Table of contents
Table of contents
1
Preliminaries
Positional Numeration Systems
Parallel Addition as a Local Function
Rewriting Rules & Representations of Zero
2
Existence of Parallel Addition
Positive Integer Base
Bases being Algebraic Numbers
Minimizing the Alphabet for Parallel Addition
3
Lower Bounds on the Cardinality of Alphabet for Parallelism
Lower Bounds for Bases being Algebraic Numbers
4
Addition versus Conversion & Subtraction
Addition versus Conversion & Subtraction
5
Minimal Alphabets for Parallel Addition in Selected Numeration Systems
Integer Bases
Quadratic Pisot Unit Bases
Rational Bases
Complex Base
6
Summary of Minimal Alphabets for Parallel Addition
Summary of Minimal Alphabets for Parallel Addition
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Table of contents
1
Preliminaries
Positional Numeration Systems
Parallel Addition as a Local Function
Rewriting Rules & Representations of Zero
2
Existence of Parallel Addition
Positive Integer Base
Bases being Algebraic Numbers
Minimizing the Alphabet for Parallel Addition
3
Lower Bounds on the Cardinality of Alphabet for Parallelism
Lower Bounds for Bases being Algebraic Numbers
4
Addition versus Conversion & Subtraction
Addition versus Conversion & Subtraction
5
Minimal Alphabets for Parallel Addition in Selected Numeration Systems
Integer Bases
Quadratic Pisot Unit Bases
Rational Bases
Complex Base
6
Summary of Minimal Alphabets for Parallel Addition
Summary of Minimal Alphabets for Parallel Addition
3 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Numeration System with Base β and Alphabet A
positional numeration system with base β and alphabet A
4 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Numeration System with Base β and Alphabet A
positional numeration system with base β and alphabet A
base β ∈ N, Z, Q, R, C, with |β| > 1
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Numeration System with Base β and Alphabet A
positional numeration system with base β and alphabet A
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
4 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Numeration System with Base β and Alphabet A
positional numeration system with base β and alphabet A
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
V number x expressed as a β-representation with digits xj ∈ A:
(x)β = xk xk−1 . . . x1 x0 • x−1 . . . x−l
x ∈ FinA (β) = {
X
xj β j | J ⊂ Z , J finite, xj ∈ A}
j∈J
4 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition as a Local Function
We search for algorithms doing addition in parallel;
5 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition as a Local Function
We search for algorithms doing addition in parallel; i.e., by means of a p-local
function,
5 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition as a Local Function
We search for algorithms doing addition in parallel; i.e., by means of a p-local
function, or via a sliding block code of length p; as illustrated by the following scheme:
5 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition as a Local Function
We search for algorithms doing addition in parallel; i.e., by means of a p-local
function, or via a sliding block code of length p; as illustrated by the following scheme:
V we perform addition of two elements x, y ∈ FinA (β)
5 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition as a Local Function
We search for algorithms doing addition in parallel; i.e., by means of a p-local
function, or via a sliding block code of length p; as illustrated by the following scheme:
V we perform addition of two elements x, y ∈ FinA (β)
x ∈ FinA (β)
y ∈ FinA (β)
. . . xj+t . . . xj+1 xj xj−1 . . . xj−r . . .
. . . yj+t . . . yj+1 yj yj−1 . . . yj−r . . .
xj ∈ A
yj ∈ A
5 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition as a Local Function
We search for algorithms doing addition in parallel; i.e., by means of a p-local
function, or via a sliding block code of length p; as illustrated by the following scheme:
V we perform addition of two elements x, y ∈ FinA (β)
x ∈ FinA (β)
y ∈ FinA (β)
wj = xj + yj
. . . xj+t . . . xj+1 xj xj−1 . . . xj−r . . .
. . . yj+t . . . yj+1 yj yj−1 . . . yj−r . . .
. . . wj+t . . . wj+1 wj wj−1 . . . wj−r . . .
|
{z
}
xj ∈ A
yj ∈ A
wj ∈ A + A
5 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition as a Local Function
We search for algorithms doing addition in parallel; i.e., by means of a p-local
function, or via a sliding block code of length p; as illustrated by the following scheme:
V we perform addition of two elements x, y ∈ FinA (β)
x ∈ FinA (β)
y ∈ FinA (β)
wj = xj + yj
zj = φ(wj+t . . . wj−r )
. . . xj+t . . . xj+1 xj xj−1 . . . xj−r . . .
. . . yj+t . . . yj+1 yj yj−1 . . . yj−r . . .
. . . wj+t . . . wj+1 wj wj−1 . . . wj−r . . .
|
{z
}
. . . zj+t . . . zj+1 zj zj−1 . . . zj−r . . .
xj ∈ A
yj ∈ A
wj ∈ A + A
zj ∈ A
5 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition as a Local Function
We search for algorithms doing addition in parallel; i.e., by means of a p-local
function, or via a sliding block code of length p; as illustrated by the following scheme:
V we perform addition of two elements x, y ∈ FinA (β)
x ∈ FinA (β)
y ∈ FinA (β)
wj = xj + yj
. . . xj+t . . . xj+1 xj xj−1 . . . xj−r . . .
xj ∈ A
. . . yj+t . . . yj+1 yj yj−1 . . . yj−r . . .
yj ∈ A
. . . wj+t . . . wj+1 wj wj−1 . . . wj−r . . .
wj ∈ A + A
|
{z
}
zj = φ(wj+t . . . wj−r )
. . . zj+t . . . zj+1 zj zj−1 . . . zj−r . . .
zj ∈ A
P
j
Finally, we obtain a finite sum x + y = z =
zj β , with zj ∈ A; i.e. z = (z)β is back
again in the original set FinA (β).
5 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition as a Local Function
We search for algorithms doing addition in parallel; i.e., by means of a p-local
function, or via a sliding block code of length p; as illustrated by the following scheme:
V we perform addition of two elements x, y ∈ FinA (β)
x ∈ FinA (β)
y ∈ FinA (β)
wj = xj + yj
. . . xj+t . . . xj+1 xj xj−1 . . . xj−r . . .
xj ∈ A
. . . yj+t . . . yj+1 yj yj−1 . . . yj−r . . .
yj ∈ A
. . . wj+t . . . wj+1 wj wj−1 . . . wj−r . . .
wj ∈ A + A
|
{z
}
zj = φ(wj+t . . . wj−r )
. . . zj+t . . . zj+1 zj zj−1 . . . zj−r . . .
zj ∈ A
P
j
Finally, we obtain a finite sum x + y = z =
zj β , with zj ∈ A; i.e. z = (z)β is back
again in the original set FinA (β).
W So, the task is to reduce digits from A + A back into A, while preserving the value
of the β-representation.
5 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition as a Local Function
We search for algorithms doing addition in parallel; i.e., by means of a p-local
function, or via a sliding block code of length p; as illustrated by the following scheme:
V we perform addition of two elements x, y ∈ FinA (β)
x ∈ FinA (β)
y ∈ FinA (β)
wj = xj + yj
. . . xj+t . . . xj+1 xj xj−1 . . . xj−r . . .
xj ∈ A
. . . yj+t . . . yj+1 yj yj−1 . . . yj−r . . .
yj ∈ A
. . . wj+t . . . wj+1 wj wj−1 . . . wj−r . . .
wj ∈ A + A
|
{z
}
zj = φ(wj+t . . . wj−r )
. . . zj+t . . . zj+1 zj zj−1 . . . zj−r . . .
zj ∈ A
P
j
Finally, we obtain a finite sum x + y = z =
zj β , with zj ∈ A; i.e. z = (z)β is back
again in the original set FinA (β).
W So, the task is to reduce digits from A + A back into A, while preserving the value
of the β-representation.
The p-local function enlarges the set of non-zero digits zj 6= 0 in the
representation of the sum by r positions to the left (so-called memory) and by t
positions to the right (so-called anticipation).
5 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition as a Local Function
We search for algorithms doing addition in parallel; i.e., by means of a p-local
function, or via a sliding block code of length p; as illustrated by the following scheme:
V we perform addition of two elements x, y ∈ FinA (β)
x ∈ FinA (β)
y ∈ FinA (β)
wj = xj + yj
. . . xj+t . . . xj+1 xj xj−1 . . . xj−r . . .
xj ∈ A
. . . yj+t . . . yj+1 yj yj−1 . . . yj−r . . .
yj ∈ A
. . . wj+t . . . wj+1 wj wj−1 . . . wj−r . . .
wj ∈ A + A
|
{z
}
zj = φ(wj+t . . . wj−r )
. . . zj+t . . . zj+1 zj zj−1 . . . zj−r . . .
zj ∈ A
P
j
Finally, we obtain a finite sum x + y = z =
zj β , with zj ∈ A; i.e. z = (z)β is back
again in the original set FinA (β).
W So, the task is to reduce digits from A + A back into A, while preserving the value
of the β-representation.
The p-local function enlarges the set of non-zero digits zj 6= 0 in the
representation of the sum by r positions to the left (so-called memory) and by t
positions to the right (so-called anticipation).
In the algorithm for parallel addition, we must not forget the necessary condition
φ(0p ) = 0, i.e. only a finite number of non-zero digits in the image of
φ : (A + A)p 7→ A !!
5 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition as a Local Function
We search for algorithms doing addition in parallel; i.e., by means of a p-local
function, or via a sliding block code of length p; as illustrated by the following scheme:
V we perform addition of two elements x, y ∈ FinA (β)
x ∈ FinA (β)
y ∈ FinA (β)
wj = xj + yj
. . . xj+t . . . xj+1 xj xj−1 . . . xj−r . . .
xj ∈ A
. . . yj+t . . . yj+1 yj yj−1 . . . yj−r . . .
yj ∈ A
. . . wj+t . . . wj+1 wj wj−1 . . . wj−r . . .
wj ∈ A + A
|
{z
}
zj = φ(wj+t . . . wj−r )
. . . zj+t . . . zj+1 zj zj−1 . . . zj−r . . .
zj ∈ A
P
j
Finally, we obtain a finite sum x + y = z =
zj β , with zj ∈ A; i.e. z = (z)β is back
again in the original set FinA (β).
W So, the task is to reduce digits from A + A back into A, while preserving the value
of the β-representation.
The p-local function enlarges the set of non-zero digits zj 6= 0 in the
representation of the sum by r positions to the left (so-called memory) and by t
positions to the right (so-called anticipation).
In the algorithm for parallel addition, we must not forget the necessary condition
φ(0p ) = 0, i.e. only a finite number of non-zero digits in the image of
φ : (A + A)p 7→ A !!
Parallel addition by means of a p-local function necessarily requires redundancy
in the numeration system!
5 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Rewriting Rules & Representations of Zero
Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting
rules, or representations of zero
6 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Rewriting Rules & Representations of Zero
Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting
rules, or representations of zero
V we need the base β to be an algebraic number, in order to operate with integer
rewriting rules:
6 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Rewriting Rules & Representations of Zero
Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting
rules, or representations of zero
V we need the base β to be an algebraic number, in order to operate with integer
rewriting rules:
basic: given directly by the minimal polynomial of β
6 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Rewriting Rules & Representations of Zero
Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting
rules, or representations of zero
V we need the base β to be an algebraic number, in order to operate with integer
rewriting rules:
basic: given directly by the minimal polynomial of β
derived: from the basic one by various combinations
6 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Rewriting Rules & Representations of Zero
Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting
rules, or representations of zero
V we need the base β to be an algebraic number, in order to operate with integer
rewriting rules:
basic: given directly by the minimal polynomial of β
derived: from the basic one by various combinations
Example (Base β = 10)
Base β = 10, the standard decimal numeration system, with minimal polynomial
β = 10, or β − 10 = 0
6 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Rewriting Rules & Representations of Zero
Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting
rules, or representations of zero
V we need the base β to be an algebraic number, in order to operate with integer
rewriting rules:
basic: given directly by the minimal polynomial of β
derived: from the basic one by various combinations
Example (Base β = 10)
Base β = 10, the standard decimal numeration system, with minimal polynomial
β = 10, or β − 10 = 0
basic rewriting rule: [1, −10]
6 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Rewriting Rules & Representations of Zero
Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting
rules, or representations of zero
V we need the base β to be an algebraic number, in order to operate with integer
rewriting rules:
basic: given directly by the minimal polynomial of β
derived: from the basic one by various combinations
Example (Base β = 10)
Base β = 10, the standard decimal numeration system, with minimal polynomial
β = 10, or β − 10 = 0
basic rewriting rule: [1, −10]
derived rewriting rules: e.g. [1, 0, −100], [1, 0, 0, −1000], etc.
6 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Rewriting Rules & Representations of Zero
Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting
rules, or representations of zero
V we need the base β to be an algebraic number, in order to operate with integer
rewriting rules:
basic: given directly by the minimal polynomial of β
derived: from the basic one by various combinations
Example (Base β = 10)
Base β = 10, the standard decimal numeration system, with minimal polynomial
β = 10, or β − 10 = 0
basic rewriting rule: [1, −10]
derived rewriting rules: e.g. [1, 0, −100], [1, 0, 0, −1000], etc.
Example (Base β = τ the Golden Mean)
Base β = τ , the Golden Mean, with minimal polynomial τ 2 = τ + 1, or
τ2 − τ − 1 = 0
6 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Rewriting Rules & Representations of Zero
Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting
rules, or representations of zero
V we need the base β to be an algebraic number, in order to operate with integer
rewriting rules:
basic: given directly by the minimal polynomial of β
derived: from the basic one by various combinations
Example (Base β = 10)
Base β = 10, the standard decimal numeration system, with minimal polynomial
β = 10, or β − 10 = 0
basic rewriting rule: [1, −10]
derived rewriting rules: e.g. [1, 0, −100], [1, 0, 0, −1000], etc.
Example (Base β = τ the Golden Mean)
Base β = τ , the Golden Mean, with minimal polynomial τ 2 = τ + 1, or
τ2 − τ − 1 = 0
basic rewriting rule: [1, −1, −1]
6 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Rewriting Rules & Representations of Zero
Tools to develop algorithms for (parallel) addition (subtraction) = so-called rewriting
rules, or representations of zero
V we need the base β to be an algebraic number, in order to operate with integer
rewriting rules:
basic: given directly by the minimal polynomial of β
derived: from the basic one by various combinations
Example (Base β = 10)
Base β = 10, the standard decimal numeration system, with minimal polynomial
β = 10, or β − 10 = 0
basic rewriting rule: [1, −10]
derived rewriting rules: e.g. [1, 0, −100], [1, 0, 0, −1000], etc.
Example (Base β = τ the Golden Mean)
Base β = τ , the Golden Mean, with minimal polynomial τ 2 = τ + 1, or
τ2 − τ − 1 = 0
basic rewriting rule: [1, −1, −1]
derived rewriting rules: again, infinite number of possibilities, e.g. [1, 0, −3, 0, 1]
or [1, 0, 0, 0, −7, 0, 0, 0, 1]
6 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Table of contents
1
Preliminaries
Positional Numeration Systems
Parallel Addition as a Local Function
Rewriting Rules & Representations of Zero
2
Existence of Parallel Addition
Positive Integer Base
Bases being Algebraic Numbers
Minimizing the Alphabet for Parallel Addition
3
Lower Bounds on the Cardinality of Alphabet for Parallelism
Lower Bounds for Bases being Algebraic Numbers
4
Addition versus Conversion & Subtraction
Addition versus Conversion & Subtraction
5
Minimal Alphabets for Parallel Addition in Selected Numeration Systems
Integer Bases
Quadratic Pisot Unit Bases
Rational Bases
Complex Base
6
Summary of Minimal Alphabets for Parallel Addition
Summary of Minimal Alphabets for Parallel Addition
7 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
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)
8 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
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
b∈N
b
b
b
b
b
=
=
=
=
=
2
3
4
5
6
2-local algorithm
Avizienis
A
A
A
A
not working
= {−2, . . . , 2}
= {−3, . . . , 3}
= {−3, . . . , 3}
= {−4, . . . , 4}
.
.
.
b = 2m
b = 2m + 1
3-local algorithm
Chow & Robertson
3-local algorithm
Parhami
A = {−1, 0, 1}
not working
A = {−2, . . . , 2}
not working
A = {−3, . . . , 3}
A = {0, 1, 2}
A = {0, 1, 2, 3}
A = {0, . . . , 4}
A = {0, . . . , 5}
A = {0, . . . , 6}
.
.
.
.
.
.
.
.
.
n
− b+2
, . . . , b+2
2
2
o
A=
= {−m
n − 1, . . . , m + 1}
o
, . . . , b+1
A = − b+1
2
2
= {−m − 1, . . . , m + 1}
n
− b2 , . . . , b2
A=
= {−m, . . . , m}
not working
o
A = {0, . . . , b}
= {0, . . . , 2m}
A = {0, . . . , b}
= {0, . . . , 2m + 1}
8 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
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
b∈N
b
b
b
b
b
=
=
=
=
=
2
3
4
5
6
2-local algorithm
Avizienis
A
A
A
A
not working
= {−2, . . . , 2}
= {−3, . . . , 3}
= {−3, . . . , 3}
= {−4, . . . , 4}
.
.
.
b = 2m
b = 2m + 1
3-local algorithm
Chow & Robertson
3-local algorithm
Parhami
A = {−1, 0, 1}
not working
A = {−2, . . . , 2}
not working
A = {−3, . . . , 3}
A = {0, 1, 2}
A = {0, 1, 2, 3}
A = {0, . . . , 4}
A = {0, . . . , 5}
A = {0, . . . , 6}
.
.
.
.
.
.
.
.
.
n
− b+2
, . . . , b+2
2
2
o
A=
= {−m
n − 1, . . . , m + 1}
o
, . . . , b+1
A = − b+1
2
2
= {−m − 1, . . . , m + 1}
n
− b2 , . . . , b2
A=
= {−m, . . . , m}
not working
o
A = {0, . . . , b}
= {0, . . . , 2m}
A = {0, . . . , b}
= {0, . . . , 2m + 1}
V for positive integer bases β = b, size #A = b + 1 is the minimal cardinality for
alphabet allowing parallel addition
8 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition for Bases being Algebraic Numbers
Given a base β, we try to find a suitable alphabet in which parallelism is possible:
9 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition for Bases being Algebraic Numbers
Given a base β, we try to find a suitable alphabet in which parallelism is possible:
Theorem (Parallel Addition for Bases being Algebraic Numbers)
Let the base β be an algebraic number such that |β| > 1 and all its conjugates in
modulus differ from 1.
Then there exists an alphabet A ⊂ Z such that addition on FinA (β) can be
performed in parallel.
9 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition for Bases being Algebraic Numbers
Given a base β, we try to find a suitable alphabet in which parallelism is possible:
Theorem (Parallel Addition for Bases being Algebraic Numbers)
Let the base β be an algebraic number such that |β| > 1 and all its conjugates in
modulus differ from 1.
Then there exists an alphabet A ⊂ Z such that addition on FinA (β) can be
performed in parallel.
The proof is constructive,
9 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition for Bases being Algebraic Numbers
Given a base β, we try to find a suitable alphabet in which parallelism is possible:
Theorem (Parallel Addition for Bases being Algebraic Numbers)
Let the base β be an algebraic number such that |β| > 1 and all its conjugates in
modulus differ from 1.
Then there exists an alphabet A ⊂ Z such that addition on FinA (β) can be
performed in parallel.
The proof is constructive,
the obtained alphabet has the form of a symmetric set of contiguous integers
A = {−a, . . . , 0 . . . , a};
9 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition for Bases being Algebraic Numbers
Given a base β, we try to find a suitable alphabet in which parallelism is possible:
Theorem (Parallel Addition for Bases being Algebraic Numbers)
Let the base β be an algebraic number such that |β| > 1 and all its conjugates in
modulus differ from 1.
Then there exists an alphabet A ⊂ Z such that addition on FinA (β) can be
performed in parallel.
The proof is constructive,
the obtained alphabet has the form of a symmetric set of contiguous integers
A = {−a, . . . , 0 . . . , a};
in general, the parameter a is rather big here; i.e. these algorithms do not (yet)
focus on minimizing the size of alphabet A.
9 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition for Bases being Algebraic Numbers
Given a base β, we try to find a suitable alphabet in which parallelism is possible:
Theorem (Parallel Addition for Bases being Algebraic Numbers)
Let the base β be an algebraic number such that |β| > 1 and all its conjugates in
modulus differ from 1.
Then there exists an alphabet A ⊂ Z such that addition on FinA (β) can be
performed in parallel.
The proof is constructive,
the obtained alphabet has the form of a symmetric set of contiguous integers
A = {−a, . . . , 0 . . . , a};
in general, the parameter a is rather big here; i.e. these algorithms do not (yet)
focus on minimizing the size of alphabet A.
Regarding the (non)-symmetry of the alphabet A:
9 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition for Bases being Algebraic Numbers
Given a base β, we try to find a suitable alphabet in which parallelism is possible:
Theorem (Parallel Addition for Bases being Algebraic Numbers)
Let the base β be an algebraic number such that |β| > 1 and all its conjugates in
modulus differ from 1.
Then there exists an alphabet A ⊂ Z such that addition on FinA (β) can be
performed in parallel.
The proof is constructive,
the obtained alphabet has the form of a symmetric set of contiguous integers
A = {−a, . . . , 0 . . . , a};
in general, the parameter a is rather big here; i.e. these algorithms do not (yet)
focus on minimizing the size of alphabet A.
Regarding the (non)-symmetry of the alphabet A:
having a symmetric alphabet means that the same algorithm as for (parallel)
addition can be used for (parallel) subtraction;
9 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition for Bases being Algebraic Numbers
Given a base β, we try to find a suitable alphabet in which parallelism is possible:
Theorem (Parallel Addition for Bases being Algebraic Numbers)
Let the base β be an algebraic number such that |β| > 1 and all its conjugates in
modulus differ from 1.
Then there exists an alphabet A ⊂ Z such that addition on FinA (β) can be
performed in parallel.
The proof is constructive,
the obtained alphabet has the form of a symmetric set of contiguous integers
A = {−a, . . . , 0 . . . , a};
in general, the parameter a is rather big here; i.e. these algorithms do not (yet)
focus on minimizing the size of alphabet A.
Regarding the (non)-symmetry of the alphabet A:
having a symmetric alphabet means that the same algorithm as for (parallel)
addition can be used for (parallel) subtraction;
symmetric alphabet is a simplification, but not necessity for parallelism;
9 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Parallel Addition for Bases being Algebraic Numbers
Given a base β, we try to find a suitable alphabet in which parallelism is possible:
Theorem (Parallel Addition for Bases being Algebraic Numbers)
Let the base β be an algebraic number such that |β| > 1 and all its conjugates in
modulus differ from 1.
Then there exists an alphabet A ⊂ Z such that addition on FinA (β) can be
performed in parallel.
The proof is constructive,
the obtained alphabet has the form of a symmetric set of contiguous integers
A = {−a, . . . , 0 . . . , a};
in general, the parameter a is rather big here; i.e. these algorithms do not (yet)
focus on minimizing the size of alphabet A.
Regarding the (non)-symmetry of the alphabet A:
having a symmetric alphabet means that the same algorithm as for (parallel)
addition can be used for (parallel) subtraction;
symmetric alphabet is a simplification, but not necessity for parallelism;
requiring symmetry of the alphabet A may increase the cardinality #A.
9 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Examples of Parallel Addition
Example (Base with Strong Representation of Zero)
√
.
base β = −17−4 265 = −8.3197: root of (strong) polynomial S(β) = 2β+17+3β −1
V 3-local function of addition on alphabet A = {−18, . . . , 0, . . . , +18}:
10 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Examples of Parallel Addition
Example (Base with Strong Representation of Zero)
√
.
base β = −17−4 265 = −8.3197: root of (strong) polynomial S(β) = 2β+17+3β −1
V 3-local function of addition on alphabet A = {−18, . . . , 0, . . . , +18}:
x ∈ FinA (β)
y ∈ FinA (β)
wj = xj + yj
3
−7
3
−7
17
13
30
−1
−9
−10
9
−3
6
xj ∈ A
yj ∈ A
wj ∈ A + A
10 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Examples of Parallel Addition
Example (Base with Strong Representation of Zero)
√
.
base β = −17−4 265 = −8.3197: root of (strong) polynomial S(β) = 2β+17+3β −1
V 3-local function of addition on alphabet A = {−18, . . . , 0, . . . , +18}:
x ∈ FinA (β)
y ∈ FinA (β)
wj = xj + yj
3
−7
3
−7
−4
zj = φ(wj+1 wj wj−1 )
3
−11
17
13
30
−34
2
−2
−1
−9
−10
−6
17
1
9
−3
6
3
9
xj ∈ A
yj ∈ A
wj ∈ A + A
zj ∈ A
10 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Examples of Parallel Addition
Example (Base with Strong Representation of Zero)
√
.
base β = −17−4 265 = −8.3197: root of (strong) polynomial S(β) = 2β+17+3β −1
V 3-local function of addition on alphabet A = {−18, . . . , 0, . . . , +18}:
x ∈ FinA (β)
y ∈ FinA (β)
wj = xj + yj
3
−7
3
−7
−4
zj = φ(wj+1 wj wj−1 )
3
−11
Example (Base β =
√
1+ 5
2
17
13
30
−34
2
−2
−1
−9
−10
−6
17
1
9
−3
6
3
9
xj ∈ A
yj ∈ A
wj ∈ A + A
zj ∈ A
= τ , the Golden Mean)
base β = τ , the Golden Mean, with minimal polynomial −τ 2 + τ + 1 = 0
V does not seem to lead to any parallel algorithm. . .
10 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Examples of Parallel Addition
Example (Base with Strong Representation of Zero)
√
.
base β = −17−4 265 = −8.3197: root of (strong) polynomial S(β) = 2β+17+3β −1
V 3-local function of addition on alphabet A = {−18, . . . , 0, . . . , +18}:
x ∈ FinA (β)
y ∈ FinA (β)
wj = xj + yj
3
−7
3
−7
−4
zj = φ(wj+1 wj wj−1 )
3
−11
Example (Base β =
√
1+ 5
2
17
13
30
−34
2
−2
−1
−9
−10
−6
17
1
9
−3
6
3
9
xj ∈ A
yj ∈ A
wj ∈ A + A
zj ∈ A
= τ , the Golden Mean)
base β = τ , the Golden Mean, with minimal polynomial −τ 2 + τ + 1 = 0
V does not seem to lead to any parallel algorithm. . .
strong polynomial: −τ 4 + 7 − τ −4 = 0
V
#A = 11
V 9-local function for parallel addition in A = {−5, . . . , 0, . . . , 5}
10 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Examples of Parallel Addition
Example (Base with Strong Representation of Zero)
√
.
base β = −17−4 265 = −8.3197: root of (strong) polynomial S(β) = 2β+17+3β −1
V 3-local function of addition on alphabet A = {−18, . . . , 0, . . . , +18}:
x ∈ FinA (β)
y ∈ FinA (β)
wj = xj + yj
3
−7
3
−7
−4
zj = φ(wj+1 wj wj−1 )
3
−11
Example (Base β =
√
1+ 5
2
17
13
30
−34
2
−2
−1
−9
−10
−6
17
1
9
−3
6
3
9
xj ∈ A
yj ∈ A
wj ∈ A + A
zj ∈ A
= τ , the Golden Mean)
base β = τ , the Golden Mean, with minimal polynomial −τ 2 + τ + 1 = 0
V does not seem to lead to any parallel algorithm. . .
strong polynomial: −τ 4 + 7 − τ −4 = 0
V
#A = 11
V 9-local function for parallel addition in A = {−5, . . . , 0, . . . , 5}
weak polynomial: −τ 2 + 3 − τ −2 = 0
V
#A = 3
V 21-local function for parallel addition in A = {−1, 0, 1}
V A = {−1, 0, 1} is the minimal alphabet allowing parallelism for base τ
10 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Minimizing the Alphabet for Parallel Addition
We can make choices between the size of the alphabet and the p-locality:
bigger cardinality #A ↔ smaller p-locality = narrower sliding window
smaller cardinality #A ↔ bigger p-locality = wider sliding window
11 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Minimizing the Alphabet for Parallel Addition
We can make choices between the size of the alphabet and the p-locality:
bigger cardinality #A ↔ smaller p-locality = narrower sliding window
smaller cardinality #A ↔ bigger p-locality = wider sliding window
The up-to-now results - algorithms for parallel addition for a broad class of
algebraic numbers - are quite general, and not yet optimized in any way:
neither in terms of the cardinality #A of the alphabet,
nor in terms of the p-locality.
11 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Minimizing the Alphabet for Parallel Addition
We can make choices between the size of the alphabet and the p-locality:
bigger cardinality #A ↔ smaller p-locality = narrower sliding window
smaller cardinality #A ↔ bigger p-locality = wider sliding window
The up-to-now results - algorithms for parallel addition for a broad class of
algebraic numbers - are quite general, and not yet optimized in any way:
neither in terms of the cardinality #A of the alphabet,
nor in terms of the p-locality.
The minimal cardinality #A of alphabets for parallelism are known for:
Positive integer bases: for β = b ∈ N, minimal #A = b + 1
Base of Golden Mean: minimal #A = 3
11 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Minimizing the Alphabet for Parallel Addition
We can make choices between the size of the alphabet and the p-locality:
bigger cardinality #A ↔ smaller p-locality = narrower sliding window
smaller cardinality #A ↔ bigger p-locality = wider sliding window
The up-to-now results - algorithms for parallel addition for a broad class of
algebraic numbers - are quite general, and not yet optimized in any way:
neither in terms of the cardinality #A of the alphabet,
nor in terms of the p-locality.
The minimal cardinality #A of alphabets for parallelism are known for:
Positive integer bases: for β = b ∈ N, minimal #A = b + 1
Base of Golden Mean: minimal #A = 3
Next task: What are the minimal alphabets for parallelism?
for bases β being algebraic integers of higher orders (e.g. quadratic Pisot units)
for bases being algebraic numbers, but not algebraic integers
11 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Table of contents
1
Preliminaries
Positional Numeration Systems
Parallel Addition as a Local Function
Rewriting Rules & Representations of Zero
2
Existence of Parallel Addition
Positive Integer Base
Bases being Algebraic Numbers
Minimizing the Alphabet for Parallel Addition
3
Lower Bounds on the Cardinality of Alphabet for Parallelism
Lower Bounds for Bases being Algebraic Numbers
4
Addition versus Conversion & Subtraction
Addition versus Conversion & Subtraction
5
Minimal Alphabets for Parallel Addition in Selected Numeration Systems
Integer Bases
Quadratic Pisot Unit Bases
Rational Bases
Complex Base
6
Summary of Minimal Alphabets for Parallel Addition
Summary of Minimal Alphabets for Parallel Addition
12 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Lower Bounds for Bases being Algebraic Numbers
Theorem (Lower Bound on #A for Bases being Positive Real Algebraic Numbers)
Let base β > be a positive real algebraic number, and let A be a finite set of
contiguous integers containing 0 and 1.
13 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Lower Bounds for Bases being Algebraic Numbers
Theorem (Lower Bound on #A for Bases being Positive Real Algebraic Numbers)
Let base β > be a positive real algebraic number, and let A be a finite set of
contiguous integers containing 0 and 1.
If addition in FinA (β) can be performed in parallel, then #A ≥ dβe.
13 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Lower Bounds for Bases being Algebraic Numbers
Theorem (Lower Bound on #A for Bases being Positive Real Algebraic Numbers)
Let base β > be a positive real algebraic number, and let A be a finite set of
contiguous integers containing 0 and 1.
If addition in FinA (β) can be performed in parallel, then #A ≥ dβe.
Theorem (Lower Bound on #A for Bases being Algebraic Integers)
Let base β, with |β| > 1, be an algebraic integer of degree d with minimal polynomial
f (X ). Let A be an alphabet of contiguous integers containing 0 and 1.
13 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Lower Bounds for Bases being Algebraic Numbers
Theorem (Lower Bound on #A for Bases being Positive Real Algebraic Numbers)
Let base β > be a positive real algebraic number, and let A be a finite set of
contiguous integers containing 0 and 1.
If addition in FinA (β) can be performed in parallel, then #A ≥ dβe.
Theorem (Lower Bound on #A for Bases being Algebraic Integers)
Let base β, with |β| > 1, be an algebraic integer of degree d with minimal polynomial
f (X ). Let A be an alphabet of contiguous integers containing 0 and 1.
1
If addition in FinA (β) is computable in parallel, then #A ≥ |f (1)|.
13 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Lower Bounds for Bases being Algebraic Numbers
Theorem (Lower Bound on #A for Bases being Positive Real Algebraic Numbers)
Let base β > be a positive real algebraic number, and let A be a finite set of
contiguous integers containing 0 and 1.
If addition in FinA (β) can be performed in parallel, then #A ≥ dβe.
Theorem (Lower Bound on #A for Bases being Algebraic Integers)
Let base β, with |β| > 1, be an algebraic integer of degree d with minimal polynomial
f (X ). Let A be an alphabet of contiguous integers containing 0 and 1.
1
If addition in FinA (β) is computable in parallel, then #A ≥ |f (1)|.
2
If, moreover, β is a positive real number, β > 1, then #A ≥ |f (1)| + 2.
13 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Lower Bounds for Bases being Algebraic Numbers
Theorem (Lower Bound on #A for Bases being Positive Real Algebraic Numbers)
Let base β > be a positive real algebraic number, and let A be a finite set of
contiguous integers containing 0 and 1.
If addition in FinA (β) can be performed in parallel, then #A ≥ dβe.
Theorem (Lower Bound on #A for Bases being Algebraic Integers)
Let base β, with |β| > 1, be an algebraic integer of degree d with minimal polynomial
f (X ). Let A be an alphabet of contiguous integers containing 0 and 1.
1
If addition in FinA (β) is computable in parallel, then #A ≥ |f (1)|.
2
If, moreover, β is a positive real number, β > 1, then #A ≥ |f (1)| + 2.
The assumption ’β is a positive real / algebraic integer / algebraic number’ in these
theorems can be replaced by:
13 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Lower Bounds for Bases being Algebraic Numbers
Theorem (Lower Bound on #A for Bases being Positive Real Algebraic Numbers)
Let base β > be a positive real algebraic number, and let A be a finite set of
contiguous integers containing 0 and 1.
If addition in FinA (β) can be performed in parallel, then #A ≥ dβe.
Theorem (Lower Bound on #A for Bases being Algebraic Integers)
Let base β, with |β| > 1, be an algebraic integer of degree d with minimal polynomial
f (X ). Let A be an alphabet of contiguous integers containing 0 and 1.
1
If addition in FinA (β) is computable in parallel, then #A ≥ |f (1)|.
2
If, moreover, β is a positive real number, β > 1, then #A ≥ |f (1)| + 2.
The assumption ’β is a positive real / algebraic integer / algebraic number’ in these
theorems can be replaced by:
’β or
1
β
is a positive real / algebraic integer / algebraic number’; or
13 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Lower Bounds for Bases being Algebraic Numbers
Theorem (Lower Bound on #A for Bases being Positive Real Algebraic Numbers)
Let base β > be a positive real algebraic number, and let A be a finite set of
contiguous integers containing 0 and 1.
If addition in FinA (β) can be performed in parallel, then #A ≥ dβe.
Theorem (Lower Bound on #A for Bases being Algebraic Integers)
Let base β, with |β| > 1, be an algebraic integer of degree d with minimal polynomial
f (X ). Let A be an alphabet of contiguous integers containing 0 and 1.
1
If addition in FinA (β) is computable in parallel, then #A ≥ |f (1)|.
2
If, moreover, β is a positive real number, β > 1, then #A ≥ |f (1)| + 2.
The assumption ’β is a positive real / algebraic integer / algebraic number’ in these
theorems can be replaced by:
’β or
1
β
is a positive real / algebraic integer / algebraic number’; or
β or one of its conjugates is a positive real / algebraic integer / algebraic
number’.
13 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Table of contents
1
Preliminaries
Positional Numeration Systems
Parallel Addition as a Local Function
Rewriting Rules & Representations of Zero
2
Existence of Parallel Addition
Positive Integer Base
Bases being Algebraic Numbers
Minimizing the Alphabet for Parallel Addition
3
Lower Bounds on the Cardinality of Alphabet for Parallelism
Lower Bounds for Bases being Algebraic Numbers
4
Addition versus Conversion & Subtraction
Addition versus Conversion & Subtraction
5
Minimal Alphabets for Parallel Addition in Selected Numeration Systems
Integer Bases
Quadratic Pisot Unit Bases
Rational Bases
Complex Base
6
Summary of Minimal Alphabets for Parallel Addition
Summary of Minimal Alphabets for Parallel Addition
14 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Addition in FinA (β) is a digit set conversion from A + A into A; but,
15 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Addition in FinA (β) is a digit set conversion from A + A into A; but, if addition of 2
numbers in parallel ⇒ addition of 3 (4, . . .) summands in parallel. . . therefore:
15 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Addition in FinA (β) is a digit set conversion from A + A into A; but, if addition of 2
numbers in parallel ⇒ addition of 3 (4, . . .) summands in parallel. . . therefore:
Proposition (Parallel Addition versus Conversion)
Let A = {m, . . . , 0, 1, . . . , M} be an alphabet of contiguous integers (containing 0 and
1), and let β be the base of the numeration system.
15 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Addition in FinA (β) is a digit set conversion from A + A into A; but, if addition of 2
numbers in parallel ⇒ addition of 3 (4, . . .) summands in parallel. . . therefore:
Proposition (Parallel Addition versus Conversion)
Let A = {m, . . . , 0, 1, . . . , M} be an alphabet of contiguous integers (containing 0 and
1), and let β be the base of the numeration system.
1
If m = 0, then addition in FinA (β) can be performed in parallel if, and only if,
conversion from A ∪ {M + 1} into A (GDE - ’greatest digit elimination’) can be
performed in parallel.
15 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Addition in FinA (β) is a digit set conversion from A + A into A; but, if addition of 2
numbers in parallel ⇒ addition of 3 (4, . . .) summands in parallel. . . therefore:
Proposition (Parallel Addition versus Conversion)
Let A = {m, . . . , 0, 1, . . . , M} be an alphabet of contiguous integers (containing 0 and
1), and let β be the base of the numeration system.
1
If m = 0, then addition in FinA (β) can be performed in parallel if, and only if,
conversion from A ∪ {M + 1} into A (GDE - ’greatest digit elimination’) can be
performed in parallel.
2
If {−1, 0, 1} ⊂ A, then addition in FinA (β) can be performed in parallel if, and
only if, conversion from A ∪ {M + 1} into A (GDE) and conversion from
{m − 1} ∪ A into A (SDE - ’smallest digit elimination’) can be performed in
parallel.
15 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Addition in FinA (β) is a digit set conversion from A + A into A; but, if addition of 2
numbers in parallel ⇒ addition of 3 (4, . . .) summands in parallel. . . therefore:
Proposition (Parallel Addition versus Conversion)
Let A = {m, . . . , 0, 1, . . . , M} be an alphabet of contiguous integers (containing 0 and
1), and let β be the base of the numeration system.
1
If m = 0, then addition in FinA (β) can be performed in parallel if, and only if,
conversion from A ∪ {M + 1} into A (GDE - ’greatest digit elimination’) can be
performed in parallel.
2
If {−1, 0, 1} ⊂ A, then addition in FinA (β) can be performed in parallel if, and
only if, conversion from A ∪ {M + 1} into A (GDE) and conversion from
{m − 1} ∪ A into A (SDE - ’smallest digit elimination’) can be performed in
parallel.
1
A ⊂ [0, +∞):
15 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Addition in FinA (β) is a digit set conversion from A + A into A; but, if addition of 2
numbers in parallel ⇒ addition of 3 (4, . . .) summands in parallel. . . therefore:
Proposition (Parallel Addition versus Conversion)
Let A = {m, . . . , 0, 1, . . . , M} be an alphabet of contiguous integers (containing 0 and
1), and let β be the base of the numeration system.
1
If m = 0, then addition in FinA (β) can be performed in parallel if, and only if,
conversion from A ∪ {M + 1} into A (GDE - ’greatest digit elimination’) can be
performed in parallel.
2
If {−1, 0, 1} ⊂ A, then addition in FinA (β) can be performed in parallel if, and
only if, conversion from A ∪ {M + 1} into A (GDE) and conversion from
{m − 1} ∪ A into A (SDE - ’smallest digit elimination’) can be performed in
parallel.
1
A ⊂ [0, +∞): the set FinA (β) ⊂ [0, +∞) is not closed under subtraction;
15 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Addition in FinA (β) is a digit set conversion from A + A into A; but, if addition of 2
numbers in parallel ⇒ addition of 3 (4, . . .) summands in parallel. . . therefore:
Proposition (Parallel Addition versus Conversion)
Let A = {m, . . . , 0, 1, . . . , M} be an alphabet of contiguous integers (containing 0 and
1), and let β be the base of the numeration system.
1
If m = 0, then addition in FinA (β) can be performed in parallel if, and only if,
conversion from A ∪ {M + 1} into A (GDE - ’greatest digit elimination’) can be
performed in parallel.
2
If {−1, 0, 1} ⊂ A, then addition in FinA (β) can be performed in parallel if, and
only if, conversion from A ∪ {M + 1} into A (GDE) and conversion from
{m − 1} ∪ A into A (SDE - ’smallest digit elimination’) can be performed in
parallel.
1
A ⊂ [0, +∞): the set FinA (β) ⊂ [0, +∞) is not closed under subtraction; even
if FinA (β) is closed under subtraction of y − x for y ≥ x, it is not possible to
find any parallel algorithm for subtraction.
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Addition in FinA (β) is a digit set conversion from A + A into A; but, if addition of 2
numbers in parallel ⇒ addition of 3 (4, . . .) summands in parallel. . . therefore:
Proposition (Parallel Addition versus Conversion)
Let A = {m, . . . , 0, 1, . . . , M} be an alphabet of contiguous integers (containing 0 and
1), and let β be the base of the numeration system.
1
If m = 0, then addition in FinA (β) can be performed in parallel if, and only if,
conversion from A ∪ {M + 1} into A (GDE - ’greatest digit elimination’) can be
performed in parallel.
2
If {−1, 0, 1} ⊂ A, then addition in FinA (β) can be performed in parallel if, and
only if, conversion from A ∪ {M + 1} into A (GDE) and conversion from
{m − 1} ∪ A into A (SDE - ’smallest digit elimination’) can be performed in
parallel.
1
A ⊂ [0, +∞): the set FinA (β) ⊂ [0, +∞) is not closed under subtraction; even
if FinA (β) is closed under subtraction of y − x for y ≥ x, it is not possible to
find any parallel algorithm for subtraction.
2
{−1, 0, 1} ⊂ A ⊂ Z:
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Addition in FinA (β) is a digit set conversion from A + A into A; but, if addition of 2
numbers in parallel ⇒ addition of 3 (4, . . .) summands in parallel. . . therefore:
Proposition (Parallel Addition versus Conversion)
Let A = {m, . . . , 0, 1, . . . , M} be an alphabet of contiguous integers (containing 0 and
1), and let β be the base of the numeration system.
1
If m = 0, then addition in FinA (β) can be performed in parallel if, and only if,
conversion from A ∪ {M + 1} into A (GDE - ’greatest digit elimination’) can be
performed in parallel.
2
If {−1, 0, 1} ⊂ A, then addition in FinA (β) can be performed in parallel if, and
only if, conversion from A ∪ {M + 1} into A (GDE) and conversion from
{m − 1} ∪ A into A (SDE - ’smallest digit elimination’) can be performed in
parallel.
1
A ⊂ [0, +∞): the set FinA (β) ⊂ [0, +∞) is not closed under subtraction; even
if FinA (β) is closed under subtraction of y − x for y ≥ x, it is not possible to
find any parallel algorithm for subtraction.
2
{−1, 0, 1} ⊂ A ⊂ Z: here, subtraction of two numbers from FinA (β) can be
viewed as addition of a fixed number of summands, and thus it is not needed to
study specifically the parallelism for subtraction of (β, A)-representations.
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Definition (Letters Fixed by p-local Function)
For a base β and alphabets A and B, let the conversion in base β from A to B be
computable in parallel by a p-local function ϕ.
If ϕ(. . . hhh • hhh . . .) = . . . hhh • hhh . . . for a letter h ∈ A, we say that h is fixed by ϕ.
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Definition (Letters Fixed by p-local Function)
For a base β and alphabets A and B, let the conversion in base β from A to B be
computable in parallel by a p-local function ϕ.
If ϕ(. . . hhh • hhh . . .) = . . . hhh • hhh . . . for a letter h ∈ A, we say that h is fixed by ϕ.
Proposition (Shifting Alphabets with Parallel Addition Algorithm)
For a base β an algebraic number, and for K , d ∈ Z, 0 ≤ d ≤ K − 1, denote
A−d = {−d, . . . , 0, . . . , K − 1 − d}.
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Definition (Letters Fixed by p-local Function)
For a base β and alphabets A and B, let the conversion in base β from A to B be
computable in parallel by a p-local function ϕ.
If ϕ(. . . hhh • hhh . . .) = . . . hhh • hhh . . . for a letter h ∈ A, we say that h is fixed by ϕ.
Proposition (Shifting Alphabets with Parallel Addition Algorithm)
For a base β an algebraic number, and for K , d ∈ Z, 0 ≤ d ≤ K − 1, denote
A−d = {−d, . . . , 0, . . . , K − 1 − d}.
Let a p-local function ϕ realize parallel conversion from A0 ∪ {K } to A0 in base β.
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Addition versus Conversion & Subtraction
Definition (Letters Fixed by p-local Function)
For a base β and alphabets A and B, let the conversion in base β from A to B be
computable in parallel by a p-local function ϕ.
If ϕ(. . . hhh • hhh . . .) = . . . hhh • hhh . . . for a letter h ∈ A, we say that h is fixed by ϕ.
Proposition (Shifting Alphabets with Parallel Addition Algorithm)
For a base β an algebraic number, and for K , d ∈ Z, 0 ≤ d ≤ K − 1, denote
A−d = {−d, . . . , 0, . . . , K − 1 − d}.
Let a p-local function ϕ realize parallel conversion from A0 ∪ {K } to A0 in base β.
If both letters d and K − 1 − d are fixed by ϕ, then addition is performable in parallel
on A−d as well.
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Table of contents
1
Preliminaries
Positional Numeration Systems
Parallel Addition as a Local Function
Rewriting Rules & Representations of Zero
2
Existence of Parallel Addition
Positive Integer Base
Bases being Algebraic Numbers
Minimizing the Alphabet for Parallel Addition
3
Lower Bounds on the Cardinality of Alphabet for Parallelism
Lower Bounds for Bases being Algebraic Numbers
4
Addition versus Conversion & Subtraction
Addition versus Conversion & Subtraction
5
Minimal Alphabets for Parallel Addition in Selected Numeration Systems
Integer Bases
Quadratic Pisot Unit Bases
Rational Bases
Complex Base
6
Summary of Minimal Alphabets for Parallel Addition
Summary of Minimal Alphabets for Parallel Addition
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Positive Integer Base β = b
Recall the known result for positive integer bases β = b, b ∈ Z+ , b ≥ 2.
Parallel addition uses the ’rewriting rule’ [1, −b] derived directly from the
minimal polynomial β − b = 0.
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Positive Integer Base β = b
Recall the known result for positive integer bases β = b, b ∈ Z+ , b ≥ 2.
Parallel addition uses the ’rewriting rule’ [1, −b] derived directly from the
minimal polynomial β − b = 0.
Algorithm GDE(β = b): Base β = b, b ∈ Z+ , b ≥ 2, parallel conversion (greatest digit elimination) from
{0, . . . , b, b + 1} to {0, . . . , b} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , b, b + 1}, with z =
P
Output: a finite sequence of digits {0, . . . , b}, with z =
zj β j .
P
zj β j .
for each j in parallel do
1. if zj ≥ b
then qj := 1
else
qj := 0
2. zj := zj − bqj + qj−1
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Positive Integer Base β = b
Recall the known result for positive integer bases β = b, b ∈ Z+ , b ≥ 2.
Parallel addition uses the ’rewriting rule’ [1, −b] derived directly from the
minimal polynomial β − b = 0.
Algorithm GDE(β = b): Base β = b, b ∈ Z+ , b ≥ 2, parallel conversion (greatest digit elimination) from
{0, . . . , b, b + 1} to {0, . . . , b} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , b, b + 1}, with z =
P
Output: a finite sequence of digits {0, . . . , b}, with z =
zj β j .
P
zj β j .
for each j in parallel do
1. if zj ≥ b
then qj := 1
else
qj := 0
2. zj := zj − bqj + qj−1
Letters fixed by this local function are all h ∈ {0, . . . , b − 1}, but not h = b; still:
The algorithm can be modified into parallel addition / conversion for alphabets
A−d = {−d, . . . , 0, . . . , b − d} with any d ∈ {0, . . . , b}; all with
Cardinality #A−d = b + 1, which is minimal for parallelism in this base.
18 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Positive Integer Base β = b
Recall the known result for positive integer bases β = b, b ∈ Z+ , b ≥ 2.
Parallel addition uses the ’rewriting rule’ [1, −b] derived directly from the
minimal polynomial β − b = 0.
Algorithm GDE(β = b): Base β = b, b ∈ Z+ , b ≥ 2, parallel conversion (greatest digit elimination) from
{0, . . . , b, b + 1} to {0, . . . , b} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , b, b + 1}, with z =
P
Output: a finite sequence of digits {0, . . . , b}, with z =
zj β j .
P
zj β j .
for each j in parallel do
1. if zj ≥ b
then qj := 1
else
qj := 0
2. zj := zj − bqj + qj−1
Letters fixed by this local function are all h ∈ {0, . . . , b − 1}, but not h = b; still:
The algorithm can be modified into parallel addition / conversion for alphabets
A−d = {−d, . . . , 0, . . . , b − d} with any d ∈ {0, . . . , b}; all with
Cardinality #A−d = b + 1, which is minimal for parallelism in this base.
In the latter, we provide a set of new results for several selected classes of bases β. . .
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Negative Integer Base β = −b
Considered for b ∈ Z+ , b ≥ 2.
Parallel addition uses the ’rewriting rule’ [−1, −b] derived directly from the
minimal polynomial −β − b = 0.
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Negative Integer Base β = −b
Considered for b ∈ Z+ , b ≥ 2.
Parallel addition uses the ’rewriting rule’ [−1, −b] derived directly from the
minimal polynomial −β − b = 0.
Algorithm GDE(β = −b): Base β = −b, b ∈ Z+ , b ≥ 2, parallel conversion (greatest digit elimination) from
{0, . . . , b, b + 1} to {0, . . . , b} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , b, b + 1}, with z =
P
Output: a finite sequence of digits {0, . . . , b}, with z =
zj β j .
for eachj in parallel do
zj = b + 1
zj = b and zj−1 = 0
1. case
if
zj = 0 and zj−1 ≥ b
else
P
zj β j .
then qj := 1
then qj := −1
qj := 0
2. zj := zj − bqj − qj−1
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Negative Integer Base β = −b
Considered for b ∈ Z+ , b ≥ 2.
Parallel addition uses the ’rewriting rule’ [−1, −b] derived directly from the
minimal polynomial −β − b = 0.
Algorithm GDE(β = −b): Base β = −b, b ∈ Z+ , b ≥ 2, parallel conversion (greatest digit elimination) from
{0, . . . , b, b + 1} to {0, . . . , b} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , b, b + 1}, with z =
P
Output: a finite sequence of digits {0, . . . , b}, with z =
zj β j .
for eachj in parallel do
zj = b + 1
zj = b and zj−1 = 0
1. case
zj = 0 and zj−1 ≥ b
if
else
P
zj β j .
then qj := 1
then qj := −1
qj := 0
2. zj := zj − bqj − qj−1
Letters fixed by this local function are all h ∈ {0, . . . , b}; therefore:
The algorithm can be modified into parallel addition / conversion for alphabets
A−d = {−d, . . . , 0, . . . , b − d} with any d ∈ {0, . . . , b}; all with
Cardinality #A−d = b + 1, minimal for parallelism in this base.
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Quadratic Pisot Unit Base β 2 = aβ − 1
Considered for a ∈ Z+ , a ≥ 3, when β is a Pisot unit.
Parallel addition uses the ’rewriting rule’ [1, −a, 1] derived directly from the
minimal polynomial β 2 − aβ + 1 = 0.
20 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Quadratic Pisot Unit Base β 2 = aβ − 1
Considered for a ∈ Z+ , a ≥ 3, when β is a Pisot unit.
Parallel addition uses the ’rewriting rule’ [1, −a, 1] derived directly from the
minimal polynomial β 2 − aβ + 1 = 0.
Algorithm GDE(β 2 = aβ − 1): Base β satisfying β 2 = aβ − 1, with a ∈ Z+ , a ≥ 3, parallel conversion
(greatest digit elimination) from {0, . . . , a} to {0, . . . , a − 1} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , a}, with z =
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , a − 1}, with z =
for each j








1. case







in parallel do
zj = a
zj
zj
zj
zj
zj
=
=
=
=
=
a−1
a−2
a−2
a−2
a−2
and
and
and
and
and
zj β j .
zj+1 ≥ a − 1 or zj−1 ≥ a − 1
zj+1 = a and zj−1 = a
zj+1 = a and zj−1 = a − 1 and zj−2 ≥ a − 1
zj−1 = a and zj+1 = a − 1 and zj+2 ≥ a − 1
zj±1 = a − 1 and zj±2 ≥ a − 1
else








then qj := 1







qj := 0
2. zj := zj − aqj + qj+1 + qj−1
20 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Quadratic Pisot Unit Base β 2 = aβ − 1
Considered for a ∈ Z+ , a ≥ 3, when β is a Pisot unit.
Parallel addition uses the ’rewriting rule’ [1, −a, 1] derived directly from the
minimal polynomial β 2 − aβ + 1 = 0.
Algorithm GDE(β 2 = aβ − 1): Base β satisfying β 2 = aβ − 1, with a ∈ Z+ , a ≥ 3, parallel conversion
(greatest digit elimination) from {0, . . . , a} to {0, . . . , a − 1} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , a}, with z =
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , a − 1}, with z =
for each j








1. case







in parallel do
zj = a
zj
zj
zj
zj
zj
=
=
=
=
=
a−1
a−2
a−2
a−2
a−2
and
and
and
and
and
zj β j .
zj+1 ≥ a − 1 or zj−1 ≥ a − 1
zj+1 = a and zj−1 = a
zj+1 = a and zj−1 = a − 1 and zj−2 ≥ a − 1
zj−1 = a and zj+1 = a − 1 and zj+2 ≥ a − 1
zj±1 = a − 1 and zj±2 ≥ a − 1
else








then qj := 1







qj := 0
2. zj := zj − aqj + qj+1 + qj−1
Letters fixed by this local function are all h ∈ {0, . . . , a − 2}, but not h = a − 1;
still:
The algorithm can be modified into parallel addition / conversion for alphabets
A−d = {−d, . . . , 0, . . . , a − 1 − d} with any d ∈ {0, . . . , a − 1}; all with
Cardinality #A−d = a, which is minimal for parallelism in this base.
20 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Quadratic Pisot Unit Base β 2 = aβ + 1
Considered for a ∈ Z+ , a ≥ 1, when β is a Pisot unit; wherein
the cases a ≥ 2 are covered by the algorithm below, and
a = 1 is a special case, solved separately.
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Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Quadratic Pisot Unit Base β 2 = aβ + 1
Considered for a ∈ Z+ , a ≥ 1, when β is a Pisot unit; wherein
the cases a ≥ 2 are covered by the algorithm below, and
a = 1 is a special case, solved separately.
Parallel addition uses the ’rewriting rule’ [1, −a, −1] derived directly from the
minimal polynomial β 2 − aβ − 1 = 0.
21 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Quadratic Pisot Unit Base β 2 = aβ + 1
Considered for a ∈ Z+ , a ≥ 1, when β is a Pisot unit; wherein
the cases a ≥ 2 are covered by the algorithm below, and
a = 1 is a special case, solved separately.
Parallel addition uses the ’rewriting rule’ [1, −a, −1] derived directly from the
minimal polynomial β 2 − aβ − 1 = 0.
Algorithm GDE(β 2 = aβ + 1): Base β satisfying β 2 = aβ + 1, with a ∈ Z+ , a ≥ 2, parallel conversion (greatest
digit elimination) from {0, . . . , a + 2} to {0, . . . , a + 1} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , a + 2}, with z =
for eachj in parallel do
 zj = a + 2
z = a + 1 and (zj+1 = 0 or zj−1 ≥ a + 1)
 j
zj = a and zj+1 = 0 and zj−1 ≥ a + 1
if
zj = 0 and zj+1 ≥ a + 1 and zj−1 ≤ a
1. case
else


zj β j .
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , a + 1}, with z =
then qj := 1

then qj := −1
qj := 0
2. zj := zj − aqj − qj+1 + qj−1
21 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Quadratic Pisot Unit Base β 2 = aβ + 1
Considered for a ∈ Z+ , a ≥ 1, when β is a Pisot unit; wherein
the cases a ≥ 2 are covered by the algorithm below, and
a = 1 is a special case, solved separately.
Parallel addition uses the ’rewriting rule’ [1, −a, −1] derived directly from the
minimal polynomial β 2 − aβ − 1 = 0.
Algorithm GDE(β 2 = aβ + 1): Base β satisfying β 2 = aβ + 1, with a ∈ Z+ , a ≥ 2, parallel conversion (greatest
digit elimination) from {0, . . . , a + 2} to {0, . . . , a + 1} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , a + 2}, with z =
for eachj in parallel do
 zj = a + 2
z = a + 1 and (zj+1 = 0 or zj−1 ≥ a + 1)
 j
zj = a and zj+1 = 0 and zj−1 ≥ a + 1
if
zj = 0 and zj+1 ≥ a + 1 and zj−1 ≤ a
1. case
else


zj β j .
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , a + 1}, with z =
then qj := 1

then qj := −1
qj := 0
2. zj := zj − aqj − qj+1 + qj−1
Letters fixed by this local function are all h ∈ {0, . . . , a}, but not h = a + 1; still:
The algorithm can be modified into parallel addition / conversion for alphabets
A−d = {−d, . . . , 0, . . . , a + 1 − d} with any d ∈ {0, . . . , a + 1}; all with
Cardinality #A−d = a + 2, which is minimal for parallelism in this base.
21 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Positive Rational Base β =
a
b
Considered for a > b ≥ 1, a, b ∈ Z+ , and a, b co-primes.
Parallel addition uses the ’rewriting rule’ [b, −a] derived directly from the
minimal polynomial bβ − a = 0.
22 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Positive Rational Base β =
a
b
Considered for a > b ≥ 1, a, b ∈ Z+ , and a, b co-primes.
Parallel addition uses the ’rewriting rule’ [b, −a] derived directly from the
minimal polynomial bβ − a = 0.
Algorithm GDE(β = a/b): Base β = a/b, with a > b ≥ 1, a, b co-primes, parallel conversion (greatest digit
elimination) from {0, . . . , a + b} to {0, . . . , a + b − 1} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , a + b}, with z =
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , a + b − 1}, with z =
zj β j .
for each j in parallel do
1. if a ≤ zj ≤ a + b
then qj := 1
else
qj := 0
2. zj := zj − aqj + bqj−1
22 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Positive Rational Base β =
a
b
Considered for a > b ≥ 1, a, b ∈ Z+ , and a, b co-primes.
Parallel addition uses the ’rewriting rule’ [b, −a] derived directly from the
minimal polynomial bβ − a = 0.
Algorithm GDE(β = a/b): Base β = a/b, with a > b ≥ 1, a, b co-primes, parallel conversion (greatest digit
elimination) from {0, . . . , a + b} to {0, . . . , a + b − 1} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , a + b}, with z =
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , a + b − 1}, with z =
zj β j .
for each j in parallel do
1. if a ≤ zj ≤ a + b
then qj := 1
else
qj := 0
2. zj := zj − aqj + bqj−1
Letters fixed be this local function are only h ∈ {0, . . . , a − 1}; therefore:
22 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Positive Rational Base β =
a
b
Considered for a > b ≥ 1, a, b ∈ Z+ , and a, b co-primes.
Parallel addition uses the ’rewriting rule’ [b, −a] derived directly from the
minimal polynomial bβ − a = 0.
Algorithm GDE(β = a/b): Base β = a/b, with a > b ≥ 1, a, b co-primes, parallel conversion (greatest digit
elimination) from {0, . . . , a + b} to {0, . . . , a + b − 1} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , a + b}, with z =
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , a + b − 1}, with z =
zj β j .
for each j in parallel do
1. if a ≤ zj ≤ a + b
then qj := 1
else
qj := 0
2. zj := zj − aqj + bqj−1
Letters fixed be this local function are only h ∈ {0, . . . , a − 1}; therefore:
The algorithm can be modified into parallel addition / conversion for alphabets
A−d = {−d, . . . , 0, . . . , a + b − 1 − d} with only a limited number of the shifts
d, namely d ∈ {0} ∪ {b, . . . , a − 1} ∪ {a + b − 1}; again, all with
Cardinality #A−d = a + b, minimal for parallelism in this base.
22 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Positive Rational Base β =
a
b
Considered for a > b ≥ 1, a, b ∈ Z+ , and a, b co-primes.
Parallel addition uses the ’rewriting rule’ [b, −a] derived directly from the
minimal polynomial bβ − a = 0.
Algorithm GDE(β = a/b): Base β = a/b, with a > b ≥ 1, a, b co-primes, parallel conversion (greatest digit
elimination) from {0, . . . , a + b} to {0, . . . , a + b − 1} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , a + b}, with z =
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , a + b − 1}, with z =
zj β j .
for each j in parallel do
1. if a ≤ zj ≤ a + b
then qj := 1
else
qj := 0
2. zj := zj − aqj + bqj−1
Letters fixed be this local function are only h ∈ {0, . . . , a − 1}; therefore:
The algorithm can be modified into parallel addition / conversion for alphabets
A−d = {−d, . . . , 0, . . . , a + b − 1 − d} with only a limited number of the shifts
d, namely d ∈ {0} ∪ {b, . . . , a − 1} ∪ {a + b − 1}; again, all with
Cardinality #A−d = a + b, minimal for parallelism in this base.
In other words, if the alphabet A−d for parallel addition / conversion contains
{−1, 1}, then it must contain the entire set {−b, . . . , 0, . . . , b}.
22 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Positive Rational Base β =
a
b
Considered for a > b ≥ 1, a, b ∈ Z+ , and a, b co-primes.
Parallel addition uses the ’rewriting rule’ [b, −a] derived directly from the
minimal polynomial bβ − a = 0.
Algorithm GDE(β = a/b): Base β = a/b, with a > b ≥ 1, a, b co-primes, parallel conversion (greatest digit
elimination) from {0, . . . , a + b} to {0, . . . , a + b − 1} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , a + b}, with z =
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , a + b − 1}, with z =
zj β j .
for each j in parallel do
1. if a ≤ zj ≤ a + b
then qj := 1
else
qj := 0
2. zj := zj − aqj + bqj−1
Letters fixed be this local function are only h ∈ {0, . . . , a − 1}; therefore:
The algorithm can be modified into parallel addition / conversion for alphabets
A−d = {−d, . . . , 0, . . . , a + b − 1 − d} with only a limited number of the shifts
d, namely d ∈ {0} ∪ {b, . . . , a − 1} ∪ {a + b − 1}; again, all with
Cardinality #A−d = a + b, minimal for parallelism in this base.
In other words, if the alphabet A−d for parallel addition / conversion contains
{−1, 1}, then it must contain the entire set {−b, . . . , 0, . . . , b}.
For b = 1, we obtain exactly the algorithm given for positive integer base.
22 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Negative Rational Base β = − ba
Considered for a > b ≥ 1, a, b ∈ Z+ , and a, b co-primes.
Parallel addition uses the ’rewriting rule’ [−b, −a] derived directly from the
minimal polynomial −bβ − a = 0.
23 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Negative Rational Base β = − ba
Considered for a > b ≥ 1, a, b ∈ Z+ , and a, b co-primes.
Parallel addition uses the ’rewriting rule’ [−b, −a] derived directly from the
minimal polynomial −bβ − a = 0.
Algorithm GDE(β = −a/b): Base β = −a/b, with a > b ≥ 1, a, b co-primes, parallel conversion (greatest digit
elimination) from {0, . . . , a + b} to {0, . . . , a + b − 1} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , a + b}, with z =
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , a + b − 1}, with z =
zj β j .
for eachj in parallel do
zj = a + b
then qj := 1
a ≤ zj ≤ a + b − 1 and 0 ≤ zj−1 ≤ b − 1
if
0 ≤ zj ≤ b − 1 and a ≤ zj−1 ≤ a + b
then qj := −1
else
qj := 0
1. case
2. zj := zj − aqj − bqj−1
23 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Negative Rational Base β = − ba
Considered for a > b ≥ 1, a, b ∈ Z+ , and a, b co-primes.
Parallel addition uses the ’rewriting rule’ [−b, −a] derived directly from the
minimal polynomial −bβ − a = 0.
Algorithm GDE(β = −a/b): Base β = −a/b, with a > b ≥ 1, a, b co-primes, parallel conversion (greatest digit
elimination) from {0, . . . , a + b} to {0, . . . , a + b − 1} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , a + b}, with z =
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , a + b − 1}, with z =
zj β j .
for eachj in parallel do
zj = a + b
then qj := 1
a ≤ zj ≤ a + b − 1 and 0 ≤ zj−1 ≤ b − 1
if
0 ≤ zj ≤ b − 1 and a ≤ zj−1 ≤ a + b
then qj := −1
else
qj := 0
1. case
2. zj := zj − aqj − bqj−1
Letters fixed by this local function are all h ∈ {0, . . . , a + b − 1}; therefore:
The algorithm can be modified into parallel addition / conversion for alphabets
A−d = {−d, . . . , 0, . . . , a + b − 1 − d} with any d ∈ {0, . . . , a + b − 1}; all with
Cardinality #A−d = a + b, which is minimal for parallelism in this base.
23 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Negative Rational Base β = − ba
Considered for a > b ≥ 1, a, b ∈ Z+ , and a, b co-primes.
Parallel addition uses the ’rewriting rule’ [−b, −a] derived directly from the
minimal polynomial −bβ − a = 0.
Algorithm GDE(β = −a/b): Base β = −a/b, with a > b ≥ 1, a, b co-primes, parallel conversion (greatest digit
elimination) from {0, . . . , a + b} to {0, . . . , a + b − 1} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , a + b}, with z =
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , a + b − 1}, with z =
zj β j .
for eachj in parallel do
zj = a + b
then qj := 1
a ≤ zj ≤ a + b − 1 and 0 ≤ zj−1 ≤ b − 1
if
0 ≤ zj ≤ b − 1 and a ≤ zj−1 ≤ a + b
then qj := −1
else
qj := 0
1. case
2. zj := zj − aqj − bqj−1
Letters fixed by this local function are all h ∈ {0, . . . , a + b − 1}; therefore:
The algorithm can be modified into parallel addition / conversion for alphabets
A−d = {−d, . . . , 0, . . . , a + b − 1 − d} with any d ∈ {0, . . . , a + b − 1}; all with
Cardinality #A−d = a + b, which is minimal for parallelism in this base.
For b = 1, we obtain exactly the algorithm given for negative integer base.
23 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Complex Base β = −1 + ı
The minimal polynomial of this algebraic integer is β 2 + 2β + 2 = 0; however,
For constructing the algorithm for parallel addition / conversion, we use another
’rewriting rule’, namely [−1, 0, 0, 0, −4] derived from the equality β 4 + 4 = 0.
24 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Complex Base β = −1 + ı
The minimal polynomial of this algebraic integer is β 2 + 2β + 2 = 0; however,
For constructing the algorithm for parallel addition / conversion, we use another
’rewriting rule’, namely [−1, 0, 0, 0, −4] derived from the equality β 4 + 4 = 0.
Algorithm GDE(β = −1 + ı): Base β = −1 + ı, parallel conversion (greatest digit elimination) from {0, . . . , 5}
to {0, . . . , 4} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , 5}, with z =
for eachj in parallel do
zj = 5
zj = 4 and zj−4 = 0
1. case
if
zj = 0 and zj−4 ≥ 4
else
zj β j .
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , 4}, with z =
then qj := 1
then qj := −1
qj := 0
2. zj := zj − 4qj − qj−4
24 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Complex Base β = −1 + ı
The minimal polynomial of this algebraic integer is β 2 + 2β + 2 = 0; however,
For constructing the algorithm for parallel addition / conversion, we use another
’rewriting rule’, namely [−1, 0, 0, 0, −4] derived from the equality β 4 + 4 = 0.
Algorithm GDE(β = −1 + ı): Base β = −1 + ı, parallel conversion (greatest digit elimination) from {0, . . . , 5}
to {0, . . . , 4} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , 5}, with z =
for eachj in parallel do
zj = 5
zj = 4 and zj−4 = 0
1. case
zj = 0 and zj−4 ≥ 4
if
else
zj β j .
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , 4}, with z =
then qj := 1
then qj := −1
qj := 0
2. zj := zj − 4qj − qj−4
Letters fixed by this local function are all h ∈ {0, . . . , 4}; therefore:
The algorithm can be modified into parallel addition / conversion for alphabets
A−d = {−d, . . . , 0, . . . , 4 − d} with any d ∈ {0, . . . , 4}; all with
Cardinality #A−d = 5, minimal for parallelism in this base.
24 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Complex Base β = −1 + ı
The minimal polynomial of this algebraic integer is β 2 + 2β + 2 = 0; however,
For constructing the algorithm for parallel addition / conversion, we use another
’rewriting rule’, namely [−1, 0, 0, 0, −4] derived from the equality β 4 + 4 = 0.
Algorithm GDE(β = −1 + ı): Base β = −1 + ı, parallel conversion (greatest digit elimination) from {0, . . . , 5}
to {0, . . . , 4} = A.
Input: a finite sequence of digits (zj ) of {0, . . . , 5}, with z =
for eachj in parallel do
zj = 5
zj = 4 and zj−4 = 0
1. case
zj = 0 and zj−4 ≥ 4
if
else
zj β j .
P
zj β j .
P
Output: a finite sequence of digits (zj ) of {0, . . . , 4}, with z =
then qj := 1
then qj := −1
qj := 0
2. zj := zj − 4qj − qj−4
Letters fixed by this local function are all h ∈ {0, . . . , 4}; therefore:
The algorithm can be modified into parallel addition / conversion for alphabets
A−d = {−d, . . . , 0, . . . , 4 − d} with any d ∈ {0, . . . , 4}; all with
Cardinality #A−d = 5, minimal for parallelism in this base.
Attempts to build an algorithm for parallel addition in base β = −1 + ı using an
alphabet of the same size #A = 5, but A * Z; e.g. A = {0, 1, −1, ı, −ı}, with
the rewriting rule [1, 1 − ı]. . . so far unsuccessfully
24 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Table of contents
1
Preliminaries
Positional Numeration Systems
Parallel Addition as a Local Function
Rewriting Rules & Representations of Zero
2
Existence of Parallel Addition
Positive Integer Base
Bases being Algebraic Numbers
Minimizing the Alphabet for Parallel Addition
3
Lower Bounds on the Cardinality of Alphabet for Parallelism
Lower Bounds for Bases being Algebraic Numbers
4
Addition versus Conversion & Subtraction
Addition versus Conversion & Subtraction
5
Minimal Alphabets for Parallel Addition in Selected Numeration Systems
Integer Bases
Quadratic Pisot Unit Bases
Rational Bases
Complex Base
6
Summary of Minimal Alphabets for Parallel Addition
Summary of Minimal Alphabets for Parallel Addition
25 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Minimal Alphabets A for Parallel Addition
Minimal alphabets A−d
(of contiguous integers containing 0)
for parallel addition:
26 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Minimal Alphabets A for Parallel Addition
Minimal alphabets A−d
(of contiguous integers containing 0)
for parallel addition:
Base β
Canonical
alphabet C
Minimal alphabet A−d
for parallel addition
Cardinality
#A−d
β =b
b ∈ Z+ , b ≥ 2
{0, . . . , b − 1}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
b+1
26 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Minimal Alphabets A for Parallel Addition
Minimal alphabets A−d
(of contiguous integers containing 0)
for parallel addition:
Base β
Canonical
alphabet C
Minimal alphabet A−d
for parallel addition
Cardinality
#A−d
β =b
b ∈ Z+ , b ≥ 2
β = −b
b ∈ Z+ , b ≥ 2
{0, . . . , b − 1}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
b+1
{0, . . . , b − 1}
b+1
26 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Minimal Alphabets A for Parallel Addition
Minimal alphabets A−d
(of contiguous integers containing 0)
for parallel addition:
Base β
Canonical
alphabet C
Minimal alphabet A−d
for parallel addition
Cardinality
#A−d
β =b
b ∈ Z+ , b ≥ 2
β = −b
b ∈ Z+ , b ≥ 2
{0, . . . , b − 1}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
b+1
β 2 = aβ − 1
a ∈ Z+ , a ≥ 3
{0, . . . , a − 1}
A−d = {−d, . . . , 0, . . . , a − 1 − d}
d ∈ {0, . . . , a − 1}
a
{0, . . . , b − 1}
b+1
26 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Minimal Alphabets A for Parallel Addition
Minimal alphabets A−d
(of contiguous integers containing 0)
for parallel addition:
Base β
Canonical
alphabet C
Minimal alphabet A−d
for parallel addition
Cardinality
#A−d
β =b
b ∈ Z+ , b ≥ 2
β = −b
b ∈ Z+ , b ≥ 2
{0, . . . , b − 1}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
b+1
β 2 = aβ − 1
a ∈ Z+ , a ≥ 3
β 2 = aβ + 1
a ∈ Z+ , a ≥ 1
{0, . . . , a − 1}
A−d = {−d, . . . , 0, . . . , a − 1 − d}
d ∈ {0, . . . , a − 1}
A−d = {−d, . . . , 0, . . . , a + 1 − d}
d ∈ {0, . . . , a + 1}
a
{0, . . . , b − 1}
{0, . . . , a}
b+1
a+2
26 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Minimal Alphabets A for Parallel Addition
Minimal alphabets A−d
(of contiguous integers containing 0)
for parallel addition:
Base β
Canonical
alphabet C
Minimal alphabet A−d
for parallel addition
Cardinality
#A−d
β =b
b ∈ Z+ , b ≥ 2
β = −b
b ∈ Z+ , b ≥ 2
{0, . . . , b − 1}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
b+1
β 2 = aβ − 1
a ∈ Z+ , a ≥ 3
β 2 = aβ + 1
a ∈ Z+ , a ≥ 1
{0, . . . , a − 1}
A−d = {−d, . . . , 0, . . . , a − 1 − d}
d ∈ {0, . . . , a − 1}
A−d = {−d, . . . , 0, . . . , a + 1 − d}
d ∈ {0, . . . , a + 1}
a
β = ba
a, b ∈ Z+ , a ⊥ b, a > b
{0, . . . , a − 1}
{0, . . . , b − 1}
{0, . . . , a}
A−d = {−d, . . . , 0, . . . , a + b − 1 − d}
d ∈ {0} ∪ {b, . . . , a − 1} ∪ {a + b − 1}
b+1
a+2
a+b
26 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Minimal Alphabets A for Parallel Addition
Minimal alphabets A−d
(of contiguous integers containing 0)
for parallel addition:
Base β
Canonical
alphabet C
Minimal alphabet A−d
for parallel addition
Cardinality
#A−d
β =b
b ∈ Z+ , b ≥ 2
β = −b
b ∈ Z+ , b ≥ 2
{0, . . . , b − 1}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
b+1
β 2 = aβ − 1
a ∈ Z+ , a ≥ 3
β 2 = aβ + 1
a ∈ Z+ , a ≥ 1
{0, . . . , a − 1}
A−d = {−d, . . . , 0, . . . , a − 1 − d}
d ∈ {0, . . . , a − 1}
A−d = {−d, . . . , 0, . . . , a + 1 − d}
d ∈ {0, . . . , a + 1}
a
β = ba
a, b ∈ Z+ , a ⊥ b, a > b
β = − ba
a, b ∈ Z+ , a ⊥ b, a > b
{0, . . . , a − 1}
{0, . . . , b − 1}
{0, . . . , a}
{0, . . . , a − 1}
A−d = {−d, . . . , 0, . . . , a + b − 1 − d}
d ∈ {0} ∪ {b, . . . , a − 1} ∪ {a + b − 1}
A−d = {−d, . . . , 0, . . . , a + b − 1 − d}
d ∈ {0, . . . , a + b − 1}
b+1
a+2
a+b
a+b
26 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Minimal Alphabets A for Parallel Addition
Minimal alphabets A−d
(of contiguous integers containing 0)
for parallel addition:
Base β
Canonical
alphabet C
Minimal alphabet A−d
for parallel addition
Cardinality
#A−d
β =b
b ∈ Z+ , b ≥ 2
β = −b
b ∈ Z+ , b ≥ 2
{0, . . . , b − 1}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
b+1
β 2 = aβ − 1
a ∈ Z+ , a ≥ 3
β 2 = aβ + 1
a ∈ Z+ , a ≥ 1
{0, . . . , a − 1}
A−d = {−d, . . . , 0, . . . , a − 1 − d}
d ∈ {0, . . . , a − 1}
A−d = {−d, . . . , 0, . . . , a + 1 − d}
d ∈ {0, . . . , a + 1}
a
β = ba
a, b ∈ Z+ , a ⊥ b, a > b
β = − ba
a, b ∈ Z+ , a ⊥ b, a > b
{0, . . . , a − 1}
β = −1 + ı
{0, . . . , b − 1}
{0, . . . , a}
{0, . . . , a − 1}
{0, 1}
A−d = {−d, . . . , 0, . . . , a + b − 1 − d}
d ∈ {0} ∪ {b, . . . , a − 1} ∪ {a + b − 1}
A−d = {−d, . . . , 0, . . . , a + b − 1 − d}
d ∈ {0, . . . , a + b − 1}
A−d = {−d, . . . , 0, . . . , 4 − d}
d ∈ {0, . . . , 4}
b+1
a+2
a+b
a+b
5
26 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Minimal Alphabets A for Parallel Addition
Minimal alphabets A−d
(of contiguous integers containing 0)
for parallel addition:
Base β
Canonical
alphabet C
Minimal alphabet A−d
for parallel addition
Cardinality
#A−d
β =b
b ∈ Z+ , b ≥ 2
β = −b
b ∈ Z+ , b ≥ 2
{0, . . . , b − 1}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
A−d = {−d, . . . , 0, . . . , b − d}
d ∈ {0, . . . , b}
b+1
β 2 = aβ − 1
a ∈ Z+ , a ≥ 3
β 2 = aβ + 1
a ∈ Z+ , a ≥ 1
{0, . . . , a − 1}
A−d = {−d, . . . , 0, . . . , a − 1 − d}
d ∈ {0, . . . , a − 1}
A−d = {−d, . . . , 0, . . . , a + 1 − d}
d ∈ {0, . . . , a + 1}
a
β = ba
a, b ∈ Z+ , a ⊥ b, a > b
β = − ba
a, b ∈ Z+ , a ⊥ b, a > b
{0, . . . , a − 1}
β = −1 + ı
{0, . . . , b − 1}
{0, . . . , a}
{0, . . . , a − 1}
{0, 1}
A−d = {−d, . . . , 0, . . . , a + b − 1 − d}
d ∈ {0} ∪ {b, . . . , a − 1} ∪ {a + b − 1}
A−d = {−d, . . . , 0, . . . , a + b − 1 − d}
d ∈ {0, . . . , a + b − 1}
A−d = {−d, . . . , 0, . . . , 4 − d}
d ∈ {0, . . . , 4}
b+1
a+2
a+b
a+b
5
Notice the important difference between the positive and the negative rational bases:
positive β = ba : shifted alphabet A−d allows parallel addition for only a limited
subset of the shifts d ∈ {0} ∪ {b, . . . , a − 1} ∪ {a + b − 1} ⊂ {0, . . . , a + b − 1};
negative β = − ba : we can use the shifted alphabet A−d for parallel addition
with any d ∈ {0, . . . , a + b − 1}.
26 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
lower bound dβe ≤ #A: for β a positive real algebraic number
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
lower bound dβe ≤ #A: for β a positive real algebraic number
lower bound |f (1)| ≤ #A: for β an algebraic integer
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
lower bound dβe ≤ #A: for β a positive real algebraic number
lower bound |f (1)| ≤ #A: for β an algebraic integer
lower bound |f (1)| + 2 ≤ #A: for β a positive real algebraic integer
(minimality for alphabets of contiguous integers containing 0; f (X ) being the minimal polynomial of β)
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
lower bound dβe ≤ #A: for β a positive real algebraic number
lower bound |f (1)| ≤ #A: for β an algebraic integer
lower bound |f (1)| + 2 ≤ #A: for β a positive real algebraic integer
(minimality for alphabets of contiguous integers containing 0; f (X ) being the minimal polynomial of β)
Base
β
Cardinality
#C
Cardinality
#A
Lower bound
dβe
β =b
b
b+1
b
Lower bound
|f (1)|
Lower bound
|f (1)| + 2
b+1
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
lower bound dβe ≤ #A: for β a positive real algebraic number
lower bound |f (1)| ≤ #A: for β an algebraic integer
lower bound |f (1)| + 2 ≤ #A: for β a positive real algebraic integer
(minimality for alphabets of contiguous integers containing 0; f (X ) being the minimal polynomial of β)
Base
β
Cardinality
#C
Cardinality
#A
Lower bound
dβe
Lower bound
|f (1)|
Lower bound
|f (1)| + 2
β =b
β = −b
b
b
b+1
b+1
b
not applicable
b+1
b+1
not applicable
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
lower bound dβe ≤ #A: for β a positive real algebraic number
lower bound |f (1)| ≤ #A: for β an algebraic integer
lower bound |f (1)| + 2 ≤ #A: for β a positive real algebraic integer
(minimality for alphabets of contiguous integers containing 0; f (X ) being the minimal polynomial of β)
Base
β
Cardinality
#C
Cardinality
#A
Lower bound
dβe
Lower bound
|f (1)|
Lower bound
|f (1)| + 2
β =b
β = −b
b
b
b+1
b+1
b
not applicable
b+1
b+1
not applicable
β 2 = aβ − 1
a
a
a
a
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
lower bound dβe ≤ #A: for β a positive real algebraic number
lower bound |f (1)| ≤ #A: for β an algebraic integer
lower bound |f (1)| + 2 ≤ #A: for β a positive real algebraic integer
(minimality for alphabets of contiguous integers containing 0; f (X ) being the minimal polynomial of β)
Base
β
Cardinality
#C
Cardinality
#A
Lower bound
dβe
Lower bound
|f (1)|
Lower bound
|f (1)| + 2
β =b
β = −b
b
b
b+1
b+1
b
not applicable
b+1
b+1
not applicable
β 2 = aβ − 1
β 2 = aβ + 1
a
a+1
a
a+2
a
a+1
a
a+2
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
lower bound dβe ≤ #A: for β a positive real algebraic number
lower bound |f (1)| ≤ #A: for β an algebraic integer
lower bound |f (1)| + 2 ≤ #A: for β a positive real algebraic integer
(minimality for alphabets of contiguous integers containing 0; f (X ) being the minimal polynomial of β)
Base
β
Cardinality
#C
Cardinality
#A
Lower bound
dβe
Lower bound
|f (1)|
Lower bound
|f (1)| + 2
β =b
β = −b
b
b
b+1
b+1
b
not applicable
b+1
b+1
not applicable
β 2 = aβ − 1
β 2 = aβ + 1
a
a+1
a
a+2
a
a+1
β = ba
a
a+b
d ba e
a
a+2
not applicable
not applicable
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
lower bound dβe ≤ #A: for β a positive real algebraic number
lower bound |f (1)| ≤ #A: for β an algebraic integer
lower bound |f (1)| + 2 ≤ #A: for β a positive real algebraic integer
(minimality for alphabets of contiguous integers containing 0; f (X ) being the minimal polynomial of β)
Base
β
Cardinality
#C
Cardinality
#A
Lower bound
dβe
Lower bound
|f (1)|
Lower bound
|f (1)| + 2
β =b
β = −b
b
b
b+1
b+1
b
not applicable
b+1
b+1
not applicable
β 2 = aβ − 1
β 2 = aβ + 1
a
a+1
a
a+2
a
a+1
β = ba
β = − ba
a
a
a+b
a+b
d ba e
not applicable
a
a+2
not applicable
not applicable
not applicable
not applicable
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
lower bound dβe ≤ #A: for β a positive real algebraic number
lower bound |f (1)| ≤ #A: for β an algebraic integer
lower bound |f (1)| + 2 ≤ #A: for β a positive real algebraic integer
(minimality for alphabets of contiguous integers containing 0; f (X ) being the minimal polynomial of β)
Base
β
Cardinality
#C
Cardinality
#A
Lower bound
dβe
Lower bound
|f (1)|
Lower bound
|f (1)| + 2
β =b
β = −b
b
b
b+1
b+1
b
not applicable
b+1
b+1
not applicable
β 2 = aβ − 1
β 2 = aβ + 1
a
a+1
a
a+2
a
a+1
β = ba
β = − ba
a
a
a+b
a+b
d ba e
not applicable
not applicable
not applicable
not applicable
not applicable
β = −1 + ı
2
5
not applicable
5
not applicable
a
a+2
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
lower bound dβe ≤ #A: for β a positive real algebraic number
lower bound |f (1)| ≤ #A: for β an algebraic integer
lower bound |f (1)| + 2 ≤ #A: for β a positive real algebraic integer
(minimality for alphabets of contiguous integers containing 0; f (X ) being the minimal polynomial of β)
Base
β
Cardinality
#C
Cardinality
#A
Lower bound
dβe
Lower bound
|f (1)|
Lower bound
|f (1)| + 2
β =b
β = −b
b
b
b+1
b+1
b
not applicable
b+1
b+1
not applicable
β 2 = aβ − 1
β 2 = aβ + 1
a
a+1
a
a+2
a
a+1
β = ba
β = − ba
a
a
a+b
a+b
d ba e
not applicable
not applicable
not applicable
not applicable
not applicable
β = −1 + ı
2
5
not applicable
5
not applicable
a
a+2
For the integer bases, quadratic Pisot unit bases, and the base −1 + ı, the
cardinality #A achieved in our parallel algorithms is minimal. . . reaching the
lower bound given by one of our general theorems.
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
Summary: Achieving Lower Bounds on Cardinality #A
Applicability of our general theorems setting lower bound on cardinality #A of
alphabet for parallel addition is limited as follows, depending on the base β:
lower bound dβe ≤ #A: for β a positive real algebraic number
lower bound |f (1)| ≤ #A: for β an algebraic integer
lower bound |f (1)| + 2 ≤ #A: for β a positive real algebraic integer
(minimality for alphabets of contiguous integers containing 0; f (X ) being the minimal polynomial of β)
Base
β
Cardinality
#C
Cardinality
#A
Lower bound
dβe
Lower bound
|f (1)|
Lower bound
|f (1)| + 2
β =b
β = −b
b
b
b+1
b+1
b
not applicable
b+1
b+1
not applicable
β 2 = aβ − 1
β 2 = aβ + 1
a
a+1
a
a+2
a
a+1
β = ba
β = − ba
a
a
a+b
a+b
d ba e
not applicable
not applicable
not applicable
not applicable
not applicable
β = −1 + ı
2
5
not applicable
5
not applicable
a
a+2
For the integer bases, quadratic Pisot unit bases, and the base −1 + ı, the
cardinality #A achieved in our parallel algorithms is minimal. . . reaching the
lower bound given by one of our general theorems.
For the rational bases, the minimality of the cardinality #A = a + b from our
algorithms had to be proved by other means.
27 / 28
Preliminaries Existence of Parallel Addition Lower Bounds on the Cardinality of Alphabet for Parallelism Addition versus Conversio
The End
Thanks for your attention.
. . . any questions. . .?
28 / 28