A Probabilistic Approach to Nano- computing J. Chen, J. Mundy, Y. Bai, S.-M. C. Chan, P. Petrica and R. I. Bahar Division of Engineering Brown University Acknowledgements: NSF

BROWN UNIVERSITY Jie Chen, Division of Engineering 2 Motivation Silicon-based techniques are approaching practical limits

BROWN UNIVERSITY Jie Chen, Division of Engineering 3 Nanotechnology Quantum transistors Computing with molecules, carbon nanotube arrays, pure quantum computing DNA-based computation, …

BROWN UNIVERSITY Jie Chen, Division of Engineering 4 Carbon-Nanotube Devices We use carbon nanotubes as the basis for our initial study, which provides good transistor behaviors (However, our approach is not specific to these devices !!)

BROWN UNIVERSITY Jie Chen, Division of Engineering 5 Why DNA for Self-assembling? Cee Dekar, “Nature 2002” Are there other ways and other molecules that can do it too? Yes, there are. But, DNA is the best understood, plentiful, easy to handle, robust, near-perfect and near-infinite specificity

BROWN UNIVERSITY Jie Chen, Division of Engineering 6 Non-silicon Approaches Nano-scale devices are attractive but have high probability of failure Defects may fluctuate in time

BROWN UNIVERSITY Jie Chen, Division of Engineering 7 Nano-architecture Approaches Nanofabrics [Goldstein-Budiu] Architecture detects faults and reconfigures using redundant components Array-based approach [DeHon] “PLA” logic arrays connected by conventional logic Neural Nets [Likharev] Builds neural networks from single-electron switches Needs a training stage for proper operation

BROWN UNIVERSITY Jie Chen, Division of Engineering 8 Our Probabilistic-based Approach Our Probabilistic-based Design Dynamically defects tolerant Adapts to errors as a natural consequence of probability maximization Removes need to actually detect faults “ Device failure should not cause computing systems to malfunction if they have been designed from the beginning to tolerate faults” --- Von Neumann

BROWN UNIVERSITY Jie Chen, Division of Engineering 9 Why Markov Random Fields? MRF has been widely used in pattern recognition & comm. Its operation does not depend on perfect devices or perfect connections. MRF can express arbitrary circuits and logic operation is achieved by maximizing state probability. or Minimizing a form of energy that depends on neighboring nodes in the network  low-power design 1 st Order Clique 2 nd Order Clique Neighborhood of S i

BROWN UNIVERSITY Jie Chen, Division of Engineering 10 A Half-adder Example

BROWN UNIVERSITY Jie Chen, Division of Engineering 11 Rules to Formulate Clique Energy Clique energy is the negative sum of all valid states: We use Boolean ring conversion to express each minterm representing a valid state (i.e. ‘000’):

BROWN UNIVERSITY Jie Chen, Division of Engineering 12 Clique Energy for the Summation Sum over the valid states (000, 011, 101, 110) Lemma: The energy of correct logic state is always less than that of invalid logic state by a constant. x0x1x2U

BROWN UNIVERSITY Jie Chen, Division of Engineering 13 Structural and Signal Errors Our implementation does not distinguish between devices and connections. Instead, we have structural-based and signal-based faults. -- Structural-based error: Nano-scale devices contain a large number of defects or structural errors, which fluctuate on time scales comparable to the computation cycle. The error will result in variation in the clique energy coefficients. -- The second type of error is directly accounted for process noise that affects the signals.

BROWN UNIVERSITY Jie Chen, Division of Engineering 14 Take Device Errors into Design Sum over the valid states (000, 011, 101, 110) If we take the device error into consideration, the energy can be rewritten as: In the error-free case, A=B=C=D=E=F=G=1

BROWN UNIVERSITY Jie Chen, Division of Engineering 15 Take Structural Error into Design x0x1x2U

BROWN UNIVERSITY Jie Chen, Division of Engineering 16 The Inequalities for Correct Logic We have 16 inequality relations total for this function

BROWN UNIVERSITY Jie Chen, Division of Engineering 17 Constraints on Clique Coefficients We obtain the following constraints on the coefficients: 2G>D 2F>C 2E>A 2D>B 2G>F 2F>B 2E>C 2D>A 2G>E Constraints form a polytope High order coefficients constraints the lower order ones Reliability of high order connections determine design

BROWN UNIVERSITY Jie Chen, Division of Engineering 18 Gibbs distribution for an inverter is: The conditional probability is: Take Signal Errors into Design

BROWN UNIVERSITY Jie Chen, Division of Engineering 19 Continuous Errors in Signal We model signal noise using Gaussian process Design choice 1 -- Inputs around “0” & “1” Design choice 2 -- Inputs around “-1” & “1”

BROWN UNIVERSITY Jie Chen, Division of Engineering 20 Tolerance to Temperature Variation By taking input around ‘1’, we get marginalized probability:

BROWN UNIVERSITY Jie Chen, Division of Engineering 21 Error Rate Calculation Proposed design favors for low T and small σ.

BROWN UNIVERSITY Jie Chen, Division of Engineering 22 Signal Error in NAND Design Gibbs distribution for a NAND is: The marginalized probability P(x c ) is:

BROWN UNIVERSITY Jie Chen, Division of Engineering 23 Tolerance to Temperature Variation Apply inputs “01” Apply inputs “11”

BROWN UNIVERSITY Jie Chen, Division of Engineering 24 Error Rate Calculation Proposed design works better at low energy state.

BROWN UNIVERSITY Jie Chen, Division of Engineering 25 Conclusions Proposed design doesn’t depends on specific techniques!! Propose a probabilistic approach based on MRF Dynamically defect tolerant Adapts to errors as a natural consequence of probability maximization Removes need to actually detect fault For correct operation, energy of valid states must be less than invalid states The proposed design favors for lower power operation

BROWN UNIVERSITY Jie Chen, Division of Engineering 26 Future Works We are currently investigating how this approach can be extended to more complex logic Implement design using different Nanotechnologies

Thank you “ Device failure should not cause computing systems to malfunction if they have been designed from the beginning to tolerate faults” --- Von Neumann