1. 5/31/07IWLS 20071 Computing Beyond CMOS Intense research into novel materials and devices: Carbon Nanotubes… Molecular Switches… Biological Processes…

2. 5/31/07IWLS 20072 Computing Beyond CMOS Many technologies still in exploratory phase: ! 

3. 5/31/07IWLS 20073 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

4. Circuit Modeling logic 0 1 0 0 1 Characterize probability of outcomes. inputsoutputs Model defects, variations, uncertainty, etc.:

5. Circuit Modeling logic Functional description is Boolean: inputsoutputs

6. Consider a probabilistic interpretation: logic stochastic logic inputsoutputs Circuit Modeling

7. stochastic logic Stochastic Logic inputsoutputs 0 1 0 0,1,1,0,1,0,1,1,0,1,… 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:

8. stochastic logic Stochastic Logic inputsoutputs 0 1 0 Consider a probabilistic interpretation:

9. stochastic logic Stochastic Logic 0 1 0 0 1 0 0 1 0 1 0 0 0 p 1 = Prob(one) p 2 = Prob(one) parallel bit streams Consider a probabilistic interpretation:

10. stochastic logic Stochastic Logic 0 1 0 parallel bit streams Consider a probabilistic interpretation:

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

12. 5/31/07IWLS 200712 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:

13. 5/31/07IWLS 200713 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 0 1 0 0 1 x X

14. 5/31/07IWLS 200714 Arithmetic Operations Multiplication(Scaled) Addition ba BPAP CPc    )()( )( ) )1( ()](1[)()( )( bsas BPSPAPSP CPc   

15. 5/31/07IWLS 200715 Nanowire Crossbar (idealized)

16. 5/31/07IWLS 200716 Nanowire Crossbar (idealized) Randomized connections, yet nearly one-to-one.

17. 5/31/0717 Shuffled AND

18. 5/31/07IWLS 200718 Takes the AND of randomly chosen pairs. Multiplication Shuffled AND

19. 5/31/0719 Bundleplexing

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

21. 5/31/07IWLS 200721 Stochastic Logic Shuffled ANDs, Bundleplexers { { A 0 A 1... { A n } B

22. 5/31/07IWLS 200722 Stochastic Logic Shuffled ANDs, Bundleplexers { { { }... 1 0 1