EE5393, Circuits, Computation, and
Biology
Computing with Probabilities
1,1,0,0,0,0,1,0
1,1,0,1,0,1,1,1
a
= 6/8
c
= 3/8
b
= 4/8
AND
A
B
C
Positional Encodings
75710 = 7·102 + 5·101 + 7·100
•
A positional representation scheme is compact:
2
n
distinct numbers can be represented with
n
bits.
•
Operating on this representation is complex.
Human
Computer
10101112 = 26 + 24 + 22 + 21 + 20
Multiplication
•
HA: Half adder, 2 basic
gates (AND and XOR)
•
FA: Full adder, 5 basic
gates (AND, OR, and
XOR)
In total 30 gates!
a
x
b
=
c
HA
a
1
HA
b
1
a
0
b
1
FA
a
0
b
2
a
2
b
0
a
1
b
0
a
1
FA
b
2
a
2
b
1
HA
FA
a
0
b
0
c
0
c
1
c
2
c
3
c
4
c
5
a
2
b
2
a
2
a
1
a
0
b
2
b
1
b
0
c
2
c
1
c
0
c
5
c
4
c
3
a
b
c
Representing a Value by a Sequence of
Random Bits
A real value
x
in
[0, 1]
is represented by a sequence of
random bits, each of which has
probability
x
of being
one and
probability
of
1 −
x
of being zero.
0,1,0,1,1,0,0
x
= 3/7
1,1,0,0,0,0,1,0
1,1,0,1,0,1,1,1
a
= 6/8
b
= 4/8
c
= 3/8
Assume two input bit streams are independent
6/8
·
4/8 = 3/8
Multiplication
a
x
b
=
c
AND
A
B
C
Arithmetic Operations
Multiplication
(Scaled) Addition
b
a
B
P
A
P
C
P
c
=
=
=
)
(
)
(
)
(
)
)
1
(
(
)]
(
1
[
)
(
)
(
)
(
b
s
a
s
B
P
S
P
A
P
S
P
C
P
c

+
=

+
=
=
AND
A
B
C
A
B
C
MUX
S
Serial versus Parallel
0,1,0,1,1,0, 0
Stochastic Bit Streams
x
= 3/7
Probabilistic Bundles
x
= 3/7
0
1
0
1
1
0
0
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 Spring '14
 Riedel,Marc
 Addition, Probability, output signals