1

BinaryArithmetic

(2) (2)(2)(2) (2)

a b a bb ba b a b

+ +

− −

× ×

Integer Ring : + − ×D

[ ]

2

:

( ) 2

ntt

t n

n t B

n

n

∈

= →

=

∈

∑

D

2

binary sequence 2-adic integer

→N BZ

1y x= − − 2a c s r+ = + 2y x= 1 2y x= +

Binary Algebra©Vuillemin:2010-2047

/2n nB Z Z( ) ( )( ) ( )2

a b a b a ba b a b

+ = ∪ − ∩= ⊕ + ∩0 a b a b a b a b= ∩ ⇔ + = ∪ = ⊕

Disjoint Sum Integers modulo 2n

2

Increment[ ]

2

:

( ) 2

ntt

t n

n t B

n

n

∈

= →

=

∈

∑

D

2

binary sequence 2-adic integer

→N BZ

Binary Algebra©Vuillemin:2010-2047

[0] [0] [0] 2 [1][1

[ ]2 ,

] [1] [1] 2 [2]

[ ]2

[0]

n n

a c s ca c s

A a n S s n

S A c

c+ = ++ = +

= =

= +∑ ∑

Parallel Increment

Serial Increment

1 22

1y x

c rx c y r

= ++ =

= ++

3

Bit SerialArithmetic

111

b bb bb b

− = +¬+ = −¬− = ¬−

[ ]

{ } { : 1= }

(2) 2

:nt

nt

t

t n

n t B

t n t B

n

n∈

= →

∈ = ∈

=

∈

∑N

D

2

binary sequence integer subset 2-adic intege

2r

→N

N B

Z

Binary Algebra©Vuillemin:2010-2047

4

HybridFormulas

Binary Algebra©Vuillemin:2010-2047

∩∩

5

0

1

0

2 2 2n n

n nn n

n N n N

cc rc r+

∈ ∈

===∑ ∑

2

2 2 2 2 2 2t t t t t

t t t t tt t t t t

t N t N t N t N t N

a b c s r

a b c s r∈ ∈ ∈ ∈ ∈

+ + = +

+ + = +∑ ∑ ∑ ∑ ∑

Serial Add

a+b+c = s+2r

s = a+b c = 2r

Binary Algebra©Vuillemin:2010-2047

6

d = a-b ' 1' 2

1 2

b b ba b c d rc r

= ¬ =− −+ + = +

= +

Serial Subtract

Binary Algebra©Vuillemin:2010-2047

( )1

a b a ba b a b¬ − =¬ +− = + +¬

ALU

s= f(i)(a,b)

i f(i)01

∩

+

i f(i)0123

⊕

∩

+

−

Number of Instructions 2 4 8 16 32 64 ...

Two’s Complement

7

Parallel Adder

0 2 2 2 2k k k Nk k k N

k N k N k N

c a b s c< < <

+ + = +∑ ∑ ∑

0 0 0 0 1

1 1 1 1 2

22

a b c s ca b c s c+ + = ++ + = +

Binary Algebra©Vuillemin:2010-2047

8

Trade Time for Space

z=4 : 2b/cycle z=2 : 1b/cycle

22

a b c s rc rs a b

+ + = +== +

2

2

( ) ( ) 2

a u v u v

b w x w x

s u w v x

= + =

= + =

= + + + =

z=√2z2=2

Energy is now the DOMINANT design criteria for most soft&hard.Goal: minimal computation at minimal speed!

0.5b/cycle

Binary Algebra©Vuillemin:2010-2047

0 1 0 1 0 0 1 2

0 2

2 2 2 44

a a b b c s s cc cs a b

+ + + + = + +== +

a k b k s kk k ka B z b B z s B z= = =∑ ∑ ∑0 1( 2 )a a k

k ka B B z= +∑ 2 1 2a k a kk ku B z v B z+= =∑ ∑

9

SIMD

Binary Algebra©Vuillemin:2010-2047

8

8

8

8

2

2

kk

k

kk

k

x x

y y<

<

=

=

∑

∑

8

8

2

(mod 256)

kk

k

k k k

s s

s x y<

=

= +

∑

Pack 8 bytes per 64b word

Compute the byte-wise sum!

7 8 642

128(0 1) (2 1)255

m= = −

( ) ( )( ) ( )

c x m y md x m y ms d c

= ∩ ⊕ ∩= ∩¬ + ∩¬= ⊕

MMXGPU 8 operations

Single 64b operation

8 : , 128k kk x y s x y∀ < < ⇒ = +

10

Binary Series

( ) ( )

( ) ( ) ( )

10 0

a b b aa b c a b ca aa

a b c a b a c

⊗ = ⊗⊗ ⊗ = ⊗ ⊗

⊗ =⊗ =

⊗ ⊕ = ⊗ ⊕ ⊗

( ) ( )

2 2 2

00

( )

a b b aa b c a b ca aa aa b a b

⊕ = ⊕⊕ ⊕ = ⊕

=

⊕ =

⊕=

⊕

⊕

⊕

mod( ) ( ) ( 2)a b a z b z⊗ ×

(mod 2)1 1

1n nz a

za= + + +

−

[ ]

(2) 2

:

( )

ntt

t nt

t n

nn t B

n

n z z∈

∈

= →

=

=

∈

∑

∑

D

2

2

binary sequence 2-adic integer

series ) (z

→N BZ

F

Series Ring: ⊕ ⊗D 2 2Polynomial Rings: [z]/ z [z]nnD F F

Convolution “carry-free” Product

11

SSS[ ]

{ } { : 1= }

(2) 2

(

:

)

nt

nt

t

t nt

t n

n t B

t n t B

n

n z z

n

∈

∈

= →

∈ =

∈

∈

=

=

∑

∑

N

D

2

2

binary sequence integer subset 2-adic integer

series

2

( )z

→N

N B

ZF

2 2( ) ( )n m n z zm zN N N P P PA A A D D D

= += == =

Binary Algebra©Vuillemin:2010-2047

Shuffle

( )( )

k

k

n kn k n z zn k n z z

D N∈ ∈

⇑ =

⇓ = ÷

Shift n kn k n kn k n k

D Z∈ ∈⇑− = ⇓⇓− = ⇑

2

2

( )

( ) ( (10) )( )

n n z n n

n n z n z∞

↑ = = ⊗

↓ = = ∩

21(10)

3

n n

n n n∞

↓↑ =−

↑↓ = ∩ = ∩

Sample

( )(( ) / )

n m n z mn n m

m n m z

=↑ + ↑=↓

=↓

2n nd n d d z⇑ = × = ⊗

Homework

64

Let word size be 64b.Let be a word.

Compute the words (l,h) such that:( 64).

Likewise for .

n

n l hn

D∈

↑ = + ⇑↓

12

Binary Rational

n22 = p 2 Write in binar7

y!np = ∑2 3

2 2 222 11 2 1= 0 2 01 2 010 2 ...7 7 7 7

+ = + = + =

222 0101(110)7

∞=

1 2

= ∩ =+ZP D QN

Binary Algebra©Vuillemin:2010-2047

nW = q 2rite in binary :1 2

nnqd

=+ ∑

0

2

(1 2 )1 2

mod

( ) k k

n q dk kk

n nq n

n − ++

= ∈= ∈

= ∈

Z

Z

B

72

22 30101110 27 7

−= +

4 5 62 2 2

22 3 5 60101 2 01011 2 010111 2 ...7 7 7 7

− − −= + = + = + =

32 2 3

3 3 3110 2 (110)7 7 1 2

∞− −= + = =

−

13

HenselDivision

10 0k kd n d n +− ≤ ≤ ⇒ − ≤ ≤

2 0 1 1( )1 2 i i i pn q q q qd − + −=

+

Initial part

Period

mod

( ,2)1 2 (

orde )

rdp

p d=

=

10 / 2k k k kn n n n+> ⇒ ≤ <

1(1 2 ) / 2k k k kn d n n n +< − + ⇒ < ≤

0t t

0 0.. 1

= q 2 q 2 21 2 1 2

(1 2 ) 2

t t k k

t kk

k k

n nqd dn d q n

<

−

= = ++ +

= + +

∑ ∑

- mod2 ( 1 2 )k

kn n d= +

2 0 12 0 1( ) 1 0

1 2 1 2p

p p

q qnq q q qd

−∞−= = = ⇔ − ≤ ≤

+ −

Homework1- Compute sign.2- Compute quotient & remainderfrom

period=0 exact d

initial part & pe

i

r .

vide

iod

⇔

Binary Algebra©Vuillemin:2010-2047

14

RationalSeries

nWrite as a bin22 = p a 7

ry series!np z= ∑01101 1101 011 11= 0 01 010111 111 111 111

01 10101 01010 010(110)111 111

z z z zz z z

z z z z

z zz z z

z z

∞

• = • = • =

• = • =

2

2

[ ]1 [

]

zz z

=+FPF

Binary Algebra©Vuillemin:2010-2047

15

Binary Algebra

11 2

11

n

zn

−

⊕

∈ ⊂ =¬ ∪ ∩ ⊕

+ × −⊗ •

⇓

<

↓ ↑ ⇑

[ ]

(2) 2

:

( )

ntt

t nt

t n

nn t B

n

n z z∈

∈

= →

=

=

∈

∑

∑

D

2

2

binary sequence 2-adic integer

series ) (z

→N BZ

F

⊕ ⊗⊕ ∩+ − ×

DDD

Binary Operations

3 Rings + Many Hybrids !

0 0 ( 0 0)0

x y x y x yx y x y x y x y y x

× = ⇔ ⊗ = ⇔ = ∪ =∩ = ⇔ + = ∪ = ⊕ ⇔ ⊆ ¬

Odd inverses

1 1 21 2

n naa

= + + +−

15/10/2009 Binary Algebra - Vuillemin

(mod 2)1 1

1n nz a

za= + + +

−

2 2 2: nSubrings ZD P A C D

16

Valuation [ ]

(2) 2

:

( )

ntt

t nt

t n

nn t B

n

n z z∈

∈

= →

=

=

∈

∑

∑

D

{ }

2

2

2 2( )

( ) min(0) , (1) 0

0 2 (1 2 ')v d

v d n dv v

d s d

= ∈

=+∞ =

≠ ⇒ = +Valuation

{ }2 2 2 2

2 2 2 2

( ) ( ) ( ) ( )( ) ( ) min ( ), ( )

v x y v x y v x v yv x y v x y v x v y

× = ⊗ = +

− = ⊕ ≥

( 1) 2( 1) 2 1( 1) 2 1

v

v

v

x x xx xx x x

∩ − = −¬ ∩ − = −⊕ − = + −1

22

2

v

v

v

x xx xx x +

∩− =∪− = −⊕− = −

2

2

2 2

(0)(1 2 ) 0

(2 ) 1 ( )

vv xv x v x

= ∞+ =

= +

17

Measures

1 11

n nz aza

= ⊕ ⊕ ⊕⊕

[ ]

(2) 2

:

( )

ntt

t nt

t n

nn t B

n

n z z∈

∈

= →

=

=

∈

∑

∑

D

Distance

and 1 1 21 2

n naa

= + + +−

2 ( )2 [0,1]v dd R−= ∈ ∩

Norm { }ma

0 0

x ,

x x

x y x y x y

x y x yx y

= ⇔ =

× = ⊗ = ×

≤≤+ +

{ }

( , ) ( , )( , ) 0

( , ) (max ( , ), ( , ) , ) ( , )

d x y d y x x y x yd x y x y

d x z d x y d y zd x y d y z

= = − = ⊕

= ⇔ =

≤ +≤

Big-Endian: R Little-Endian: D.

converge!

HomeworkAll triangles isosceles!

converges 0 lim n nnd d

→∞⇔ =∑

Ultra-metric

{ }2 ( ) minv x k x= ∈

18

Endians

2 0 2

1/2 0

Little Endian:

Big Endian

2

:

2 [0,2 ]

nn n

nn n

q q q

q q q −

= ∈

= ∈ ∩

∑∑

Z

R

Integer Operations

11 2

11

n

zn

−

⊕

+ ×

⊂¬ ∪ ∩ ⊕∈

⊗ •

<

⇓

−

÷

↓ ↑ ⇑

LE = Little Endian = from LSB to MSB.BE = Big Endian = from MSB to LSB.EE = Either Endian = one, or the other.AE = Any Endian = any order.

19

Subset: { } : { 1}nb n b= ∈ =N (z-series: ) n

nn

b z b z∈

= ∑N

Digital Number b∈D

2-adic integer: (2) 2nnn

b b∈

= ∑N

1 ( ) !2

Real: 2nn

n

b is not uni ueb q∈

= ∑N

0 1 2Sequence: ( ) b b b b=N

Digital Sign (a : )l ) (nn

b t b t n∈

= ∂ −∑N

Predicate: ( ) nb n b=

Binary Algebra©Vuillemin:2010-2047

20

Digital Function 1. Continuous: each output depends upon finitely many inputs.

2. Computable : presented by a finite program.

3. Causal : present output depends upon past input.

→D D

Binary Algebra©Vuillemin:2010-2047

11 2

11

n

zn

−

⊕

∈ ⊂¬ ∪ ∩ ⊕

+ × −⊗ •

⇓

= <

↓ ↑ ⇑

4. Sequential : causal + finite memory finite circuit.

5. Memoryless : Boolean function

11 2

11

n

zn

−

⊕

= <

×

∈ ⊂¬ ∪ ∩ ⊕

+ −⊗ •

↓ ↑ ⇑ ⇓

21

Binary Algebra

12 2... ... 2

1 2n⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂

+ZF N Z A DZ

NB B

= ∈ ⊂ ¬ ∩ ∪ ⊕

Words:Catenate, shifts, star, residual

Sets:

1 / (1 2 )d+ − × −Integers:

Series: 1 / (1 )zd⊗ ⊕ − ↑ ↓

* −• ⇑ ⇓

Binary Algebra©Vuillemin:2010-2047

22

What am I doing here?

1. Function2. Algorithm3. Specify4. Verify5. Instrument6. Implement7. Optimize8. Synthesize9. Validate10. Sell

Binary Algebra©Vuillemin:2010-2047

Tall Thin Designercuts across all trades