Digital Systems - École Normale Supérieurepouzet/cours/mpri/cours3/4BinaryAlgebra.pdf ·...
Transcript of Digital Systems - École Normale Supérieurepouzet/cours/mpri/cours3/4BinaryAlgebra.pdf ·...
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Digital Systems
Why Binary? Physics Information Theory Algebra
Automatic Runs Forever No error Cheap
Algebra Boolean Algebra 2-adic Numbers Binary Data Flow
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Binary Integers
n Bn0 Bn1 Bn2 Bn3 Bn4 [n] {kn} 0 0 0 0 0 0 0 {} 1 1 0 0 0 0 10 {0} 2 0 1 0 0 0 010 {1} 3 1 1 0 0 0 110 {0,1} 4 0 0 1 0 0 0010 {2} 5 1 0 1 0 0 1010 {0,2} 6 0 1 1 0 0 0110 {1,2} 7 1 1 1 0 0 1110 {0,1,2} 8 0 0 0 1 0 00010 {3}
2 2n k kkk k n
n B
= =
n Bn0 Bn1 Bn2 Bn3 Bn4 -1 1 1 1 1 1 -2 0 1 1 1 1 -3 1 0 1 1 1 -4 0 0 1 1 1 -5 1 1 0 1 1 -6 0 1 0 1 1 -7 1 0 0 1 1 -8 0 0 0 1 1 -9 1 1 1 0 1
, : utimately constant: ( )n nk ln l k k l B B s w s > = =
( 1)1
n nn n
= + = +
1n n+ = 1n n+ = 1n n =
? 012345678
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Magic Masks
0
0
: 101010102 : 01010101+
2 22
1(1 0 )1 2
k k
k k
= =+
11 4 160 3
= + + + =
03 1 =
1 2 3 4 5
4 4
1 0 1 0 1 (10)1 1 0 0 1 (1100)1 1
012 1 1 0 (1 0 )
k k k k k kk B B B B B
2 2n n nk kn n k
B
= =
0
0
0
: 101010102 : 010101013 : 11111111
+=
-1/3
-1/5
-1/17
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Infinite Binary Number d 0 { } ( ) (2)d d d d z d=
{ } { : 1}N Nd d= =2. Integer set:
0 0 1d d d= 1. Bit sequence:
( ) NNd z d z= 3. Binary series: (2) 2NNd d= 4. 2-adic Integer:
12
1
Neither unique nor comp = 2utable: nn
>
(1 / 2) 2 NNd d= * Real:
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Examples 0 (0) 00001 1(0) 1000= == =
2
11 4 16
31 1 1 4
12 4 16 3
(2)1(10) { } 2 ( )1 ( )
vv v v z
z v
+ + +
+ + +
= = = = =
= =
011010001000000010{ } {2 : }
hh
=
=
1 2 4 8
1 1 11
2 2 4
(2) 11(1) { } ( )1 ( ) 2
uu u u z
z u
+ + + +
+ + +
= = = = =
= =
10.816421509
2
is transcenden
( )
(ov ral )t e
h =
22
2
( )
is 2-algebr
( )
ai .(
c)
n z h zh z z
z h z+
= =+
Theorem (Adamczewski, Bugeaud 2007) If h(z) is 2-algebraic, then h(1/2) is rational or transcendental.
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Metrics 0 { } ( ) (2)d d d d z d=
0 1
0 1 0
0
(2 1){ } { }
{ } { }
nn
n n
nn
a aa aa a
=
=
Limit
Norm 1
20 0, 1 2 1, 2x x x= + = =
Distance ( , )d a b a b a b= = 0 1 2 nna a
Ultra-metric max( , )a b a b a b+ +
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Digital Function
Continuous: each output bit depends upon finitely many input bits.
R cea onl pr tinuedicates ous constant!: f
Computable: computable primitives composed by finite programs
Causal: current output depends upon past inputs (strict or weak)
Sequential: causal and finite function of finite circuit/program
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8
Serial Opposite
1x x = +
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9
Serial Incr&Decr
1x x = +
1 x x+ = 1x x =
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Binary Addition
1 1 1 0 1 01 0 1 1 0 0
abcs
12t t t t ta b c s c
0 1 0
1 1
1 0
1 0
1 1
12 2 2 2 2 2t t t t t
t t t t ta b c s c 0s a b c
+
0 + =
2-adic
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Parallel Add
0: 2 2 2 2k k k N
k k k Nk N k N k N
N c a b s c
