EE5393, Circuits, Computation, and Biology Computing with Probabilities 1,1,0,0,0,0,1,0 1,1,0,1,0,1,1,1 1,1,0,0,1,0,1,0 a = 6/8 c = 3/8 b = 4/8

Sequential Constructs What about complex functions such as tanh, exp, and abs ?

Sequential Constructs

Sensing Applications Median Filter-Based Image Noise Reduction

Sensing Applications Frame Difference-Based Image Segmentation

Sensing Applications Image Contrast Enhancing

Sensing Applications Kernel Density Estimation- Based Image Segmentation

Comparison of Encoding Spectrum of Encoding Binary Radix EncodingStochastic Encoding Binary Radix EncodingStochastic Encoding Circuit AreaLargeSmall Error Tolerance BadGood DelayShortLong (Positional) (Uniform, Long Stream) (Not compact, Long Stream) (Compact, Efficient) (Positional, Weighted) (Uniform)

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 C probability to be one R If R < C, output a one; If R ≥ C, output a zero. 1,0,1,… Comparator

Types of Random Sources Pseudorandom Number Generator Physical Random Source Linear Feedback Shift Register (expensive) Thermal Noises (cheap)

Challenge with Physical Random Sources cheap Voltage Regulators expensive Suppose many different probabilities are needed: {0.2, 0.78, , 0.43, 0.671, 0.012, 0.82, …}. It is costly to generate them directly. (many expensive constant values required.) expensive C1C1 C2C2

Opportunity with Physical Random Sources cheap expensive 1,1,0,0,0, … 0,1,0,1,0, … 0,0,1,0,1, … Independent Same probability

Solution When we need many different probabilities: {0.2, 0.78, , 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 Random Bit Generators Logic Circuit q2q2 q1q1 q3q3 q4q4 Set S of Input Probabilities {p 1, p 2 } Other Probabilities Needed Synthesize Logic Circuit? Choose Set S ? … … p1p1 p1p1 p1p1 p2p2 p2p2 p2p2 Set S of Input Probabilities {p 1, p 2 } p1p1 p1p1 p1p1 p2p2 p2p2 p2p2 Independent ( |S| small)

Example Logic Circuit P(x = 1) = 0.4 P(z = 1) = 0.6 1,0,1,1,0,1,0,0,0,00,1,0,0,1,0,1,1,1,1 P(z = 1) = P(x = 0) P(x = 1) = 0.4 P(z = 1) = 0.2 0,1,0,1,0,0,1,1,0,0 0,0,0,1,0,0,1,0,0,0 P(z = 1) = P(x = 1) P(y = 1) 1,0,1,1,0,0,1,0,0,1 P(y = 1) = … …

Generating Decimal Probabilities Logic Circuit q2q2 q1q1 q3q3 q4q4 Arbitrary Decimal Probabilities |S| Small! Choose Set S = {p 1, p 2, p 3 } Found Set S for  |S| = 2  |S| = 1 p1p1 p2p2 p1p1 p2p2 p3p3 p3p3 Independent … … …

Generating Decimal Probabilities Theorem: With S = {0.4, 0.5}, we can synthesize arbitrary decimal output probabilities. Constructive proof. Derived a synthesis algorithm.

Algorithm (Black dots are inverters) × − 0.14 × × −× − 0.4 × − × −× − Example: Synthesize q = from S = {0.4, 0.5} For a probability value with n digits, need at most 3n AND gates.