EE5393, Circuits, Computation, and
Biology
Computing with Probabilities
a = 6/8
A
B
c = 3/8
1,1,0,1,0,1,1,1
1,1,0,0,0,0,1,0
C
1,1,0,0,1,0,1,0
b = 4/8
AND
Sequential Constructs
What about complex functions such as tanh, exp, and abs?
Sequential Constructs
Sequential Constructs
1
Original tanh function
FSM approximation
Y
0.5
0
-0.5
-1
-1
y
-0.5
e
e
N
x
2
N
x
2
e
e
0
X

N
x
2

N
x
2
0.5
1
Sequential Constructs
1
Original linear gain function
FSM approximation
Y
0.5
0
when 0  PX 
-0.5
when
-1
-1 when
PK
,
1  PK
PY  0 ;
PK
1
 PX 
,
1  PK
1  PK
1-0.5
 PX  1,
1  PK
0P
X Y
PY 
 1.
1  PK
P
 PX  K ;
1  PK
1  PK
0.5
1
Sequential Constructs
1
Original Function
8-State FSM Approximation
16-State FSM Approximation
32-State FSM Approximation
64-State FSM Approximation
0.9
0.8
0.7
Y
0.6
0.5
0.4
0.3
ye
0.2
0.1
0
-1
-0.8
-0.6
-0.4
-0.2
2 G| x|
0
X
0.2
0.4
0.6
0.8
1
Sequential Constructs
1
Original Function
8-State FSM Approximation
16-State FSM Approximation
32-State FSM Approximation
64-State FSM Approximation
0.9
0.8
0.7
Y
0.6
0.5
0.4
0.3
y | x |
0.2
0.1
0
-1
-0.8
-0.6
-0.4
-0.2
0
X
0.2
0.4
0.6
0.8
1
Sensing Applications
Median Filter-Based Image Noise Reduction
Sensing Applications
Frame Difference-Based Image Segmentation
if
P
Xt
 PX t1  PThreshold

then PY  0; else PY  1;
Sensing Applications
Image Contrast Enhancing
510  a  b
PK 
,
510  a  b
ab
PC 
510
Sensing Applications
Kernel Density
EstimationBased Image
Segmentation
Comparison of Encoding
Binary Radix Encoding
Circuit Area
Error
Tolerance
Delay
Large
Bad
Short
(Positional,
Weighted)
(Positional)
(Compact,
Efficient)
Binary Radix Encoding
Stochastic Encoding
Small
(Uniform)
(Uniform,
Good Long Stream)
Long
(Not compact,
Long Stream)
Stochastic Encoding
Spectrum of Encoding
Future Directions
Spectrum of Encoding
Binary Radix Encoding
(Compact, Positional)
?
Stochastic Encoding
(Not compact, Uniform)
Possible encodings in the middle with the
advantages of both?
General Random Bit Generators
pdf of R
pdf of R
Random
Source
R
R
C
R
probability to be one
<
Constant
Value
C
Comparator
1,0,1,…
If R < C, output a one;
If R ≥ C, output a zero.
Types of Random Sources
• Pseudorandom Number Generator
Linear Feedback Shift Register
Random
Source
<
Constant
Value
(expensive)
• Physical Random Source
Thermal Noises
(cheap)
Challenge with
Physical Random Sources
cheap
Random
Source
<
Voltage Regulators
expensive
Constant
Value
expensive
Suppose many different probabilities are needed:
{0.2, 0.78, 0.2549, 0.43, 0.671, 0.012, 0.82, …}.
It is costly to generate them directly.
(many expensive constant values required.)
pdf of R
C1 C C2 R
Opportunity with
Physical Random Sources
cheap
expensive
Random
Source
Random
Random
Source
Source
<
<
<
Constant
Value
• Independent
• Same probability
pdf of R
C
1,1,0,0,0, …
0,1,0,1,0, …
0,0,1,0,1, …
R
Solution
When we need many different probabilities:
{0.2, 0.78, 0.2549, 0.43, 0.671, 0.012, 0.82, …}
• Generate a few source probabilities directly
from random bit generators.
• Synthesize combinational logic to generate
other probabilities.
Probability: Probability of a signal being logical one
Basic Problem
Set S of Input
Probabilities
{p11 , p22}
…
p2
…
p1
p1
Random Bit p1
Independent
Generators p2
p2
Other
Probabilities
Needed
p1
p1
p1 Logic
p2 Circuit
p2
q1
q2
q3
q4
p2
• Synthesize Logic Circuit?
• Choose Set S ? (|S| small)
Example
…
…
0.4
0.4
0.5
0.5
Logic
Circuit
0.6
0.2
P(x = 1) = 0.4
0,1,0,1,0,0,1,1,0,0
P(z = 1) = 0.2
x
P(z
= 1) = 0.6
P(x = 1) = 0.4
z
yx
z
0,0,0,1,0,0,1,0,0,0
1,0,1,1,0,1,0,0,0,0
0,1,0,0,1,0,1,1,1,1
1,0,1,1,0,0,1,0,0,1
AND
P(y = 1) = 0.5
P(z = 1) = P(x = 0)
P(z = 1) = P(x = 1) P(y = 1)
Generating Decimal Probabilities
Choose Set S = {p1, p2, p3}
… …
|S| Small!
…
Independent
p1
p1
p2
p2
p3
p3
Arbitrary Decimal
Probabilities
Logic
Circuit
• Found Set S for
 |S| = 2
 |S| = 1
q1
q2
q3
q4
Generating Decimal Probabilities
Theorem:
With S = {0.4, 0.5}, we can synthesize arbitrary
decimal output probabilities.
• Constructive proof.
• Derived a synthesis algorithm.
Algorithm
Example: Synthesize q = 0.757 from S = {0.4, 0.5}
0.4
0.757
1−
1−
1−
×0.5
×0.5
×0.4 0.14 1 − 0.86 ×0.5
0.6
0.3
0.7
0.35
1−
1−
×0.5
×0.5
0.243 ×0.40.6075
0.43
0.3925
0.785
0.215
(Black dots
are inverters)
0.4
0.5
0.5
0.4
0.5
0.5
0.5
0.4
0.6
AND
0.7
AND
0.35
0.86
AND
AND
0.43
AND
0.785
AND
0.6075
AND
0.757
For a probability value with n digits, need at most 3n AND gates.