2015-01-28 - EE5393 Circuits Computation and Biology Computing with Probabilities a = 6\/8 A B 1,1,0,1,0,1,1,1 1,1,0,0,1,0,1,0 AND b = 4\/8 c = 3\/8

2015-01-28 - EE5393 Circuits Computation and Biology...

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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
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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
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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
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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
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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
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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
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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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