5/31/07IWLS Computing Beyond CMOS Intense research into novel materials and devices: Carbon Nanotubes… Molecular Switches… Biological Processes…

5/31/07IWLS Computing Beyond CMOS Many technologies still in exploratory phase: ! 

5/31/07IWLS Nanoscale Circuits Topological constraints. Inherent randomness. High defect rates. Features: Challenges: High density of bits. Identify general traits that impinge upon logic synthesis: carbon nanowire crossbar

Circuit Modeling logic Characterize probability of outcomes. inputsoutputs Model defects, variations, uncertainty, etc.:

Circuit Modeling logic Functional description is Boolean: inputsoutputs

Consider a probabilistic interpretation: logic stochastic logic inputsoutputs Circuit Modeling

stochastic logic Stochastic Logic inputsoutputs ,1,1,0,1,0,1,1,0,1,… 1,0,0,0,1,0,0,0,0,0,… p 1 = Prob(one) p 2 = Prob(one) serial bit streams Consider a probabilistic interpretation:

stochastic logic Stochastic Logic inputsoutputs Consider a probabilistic interpretation:

stochastic logic Stochastic Logic p 1 = Prob(one) p 2 = Prob(one) parallel bit streams Consider a probabilistic interpretation:

stochastic logic Stochastic Logic parallel bit streams Consider a probabilistic interpretation:

stochastic logic Stochastic Logic Interpret outputs according to fractional weighting: 0 1 0

5/31/07IWLS Synthesis of Stochastic Logic Circuit that computes a probability distribution corresponding to a logical specification. Given a technology characterized by: Synthesize: High degree of structural parallelism. Inherent randomness in logic/interconnects. Cast problem in terms of arithmetic operations. Perform synthesis with binary moment diagrams. Strategy:

5/31/07IWLS A real value x in [ 0, 1 ] is encoded as a stream of bits X. For each bit, the probability that it is one is: P( X=1 ) = x. Probabilistic Bundles x X

5/31/07IWLS Arithmetic Operations Multiplication(Scaled) Addition ba BPAP CPc    )()( )( ) )1( ()](1[)()( )( bsas BPSPAPSP CPc   

5/31/07IWLS Nanowire Crossbar (idealized)

5/31/07IWLS Nanowire Crossbar (idealized) Randomized connections, yet nearly one-to-one.

5/31/0717 Shuffled AND

5/31/07IWLS Takes the AND of randomly chosen pairs. Multiplication Shuffled AND

5/31/0719 Bundleplexing

Scaled Addition Randomly selection of wires from different bundles,. Randomly selection of wires from different bundles, according to a fixed ratio. ¾ Bundleplexer

5/31/07IWLS Stochastic Logic Shuffled ANDs, Bundleplexers { { A 0 A 1... { A n } B

5/31/07IWLS Stochastic Logic Shuffled ANDs, Bundleplexers { { { }