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Motivation Silicon-based techniques are approaching practical A - - PowerPoint PPT Presentation

Aug 26, 2022 463 likes •548 views

Motivation Silicon-based techniques are approaching practical A Probabilistic Approach to Nano- limits computing J. Chen, J. Mundy, Y. Bai, S.-M. C. Chan, P. Petrica and R. I. Bahar Division of Engineering Brown University Acknowledgements:

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SLIDE 1

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

Jie Chen, Division of Engineering 2 BROWN UN BROWN UNIVERSITY IVERSITY

Motivation

Silicon-based techniques are approaching practical limits

http://www.intel.com/research/silicon/mooreslaw.htm

Jie Chen, Division of Engineering 3 BROWN UN BROWN UNIVERSITY IVERSITY

Nanotechnology

Quantum transistors Computing with molecules, carbon nanotube arrays, pure quantum computing DNA-based computation, …

Jie Chen, Division of Engineering 4 BROWN UN BROWN UNIVERSITY IVERSITY

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 !!)

http://www.ibm.com

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SLIDE 2

Jie Chen, Division of Engineering 5 BROWN UN BROWN UNIVERSITY IVERSITY

Why DNA for Self Why DNA for Self-

  • assembling?

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

Jie Chen, Division of Engineering 6 BROWN UN BROWN UNIVERSITY IVERSITY

Non-silicon Approaches

Nano-scale devices are attractive but have high probability of failure Defects may fluctuate in time

Jie Chen, Division of Engineering 7 BROWN UN BROWN UNIVERSITY IVERSITY

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

Jie Chen, Division of Engineering 8 BROWN UN BROWN UNIVERSITY IVERSITY

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
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SLIDE 3

Jie Chen, Division of Engineering 9 BROWN UN BROWN UNIVERSITY IVERSITY

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.

  • r

Minimizing a form of energy that depends on neighboring nodes in the network low-power design

i

s

2 n

s

1 n

s

3 n

s

i

N

1st Order Clique 2nd Order Clique Neighborhood of Si

1 ( )

1 ( | )

c c C

U s T i i

P s e Z

∈

− ∑

Ν =

Jie Chen, Division of Engineering 10 BROWN UN BROWN UNIVERSITY IVERSITY

A Half-adder Example

x x

1 2

x x

3

x

1

x0

2

x x

3

i

1 2 3 4 5 6 7

x x x

1 2 1 1 1 1 1 1 1 1 1 Valid

State

Invalid Invalid Invalid Valid Invalid Valid Valid 1 1 1

i

1 2 3 4 5 6 7

x x x

1 3 1 1 1 1 1 1 1 1 1 Valid

State

Invalid Valid Valid Invalid Valid Invalid Invalid 1 1 1

(a) For Summation (b) For Carrier

Jie Chen, Division of Engineering 11 BROWN UN BROWN UNIVERSITY IVERSITY

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’):

1 2 1 2

( , , ) ( , , ), where 1

i i i

U x x x f x x x f = − =

∑

' ' ' 0 1 2 1 2 1 0 1 2 1 2 0 1 2 1 2 0 1 2

(1 )(1 )(1 ) (1 )(1 ) 1 x x x x x x x x x x x x x x x x x x x x x x x = − − − = − − − − = − − − + + + −

Jie Chen, Division of Engineering 12 BROWN UN BROWN UNIVERSITY IVERSITY

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.

1 2 1 2 1 2 1 2

1 2 2 2 4 U x x x x x x x x x x x x = − + + + − − − +

1 1 1

  • 1

1 1

  • 1

1 1 1

  • 1

1 1 1 1

  • 1

U x2 x1 x0

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SLIDE 4

Jie Chen, Division of Engineering 13 BROWN UN BROWN UNIVERSITY IVERSITY

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.

Jie Chen, Division of Engineering 14 BROWN UN BROWN UNIVERSITY IVERSITY

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

1 2 1 2 1 2 1 2

1 2 2 2 4 U x x x x x x x x x x x x = − + + + − − − +

1 2 0 1 2 1 2 0 1 2

2 2 2 4 U Ax Bx Cx Dx x Ex x Fx x Gx x x κ = − + + + − − − +

Jie Chen, Division of Engineering 15 BROWN UN BROWN UNIVERSITY IVERSITY

Take Structural Error into Design

1 2 0 1 2 1 2 0 1 2

2 2 2 4 U Ax Bx Cx Dx x Ex x Fx x Gx x x κ = − + + + − − − +

011

2 U B C F κ = − + + −

100

U A κ = − +

011 100

U U <

1 1 1

  • 1

1 1

  • 1

1 1 1

  • 1

1 1 1 1

  • 1

U x2 x1 x0

Jie Chen, Division of Engineering 16 BROWN UN BROWN UNIVERSITY IVERSITY

The Inequalities for Correct Logic

We have 16 inequality relations total for this function

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SLIDE 5

Jie Chen, Division of Engineering 17 BROWN UN BROWN UNIVERSITY IVERSITY

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

1 2 0 1 2 1 2 0 1 2

2 2 2 4 U Ax Bx Cx Dx x Ex x Fx x Gx x x κ = − + + + − − − + High order coefficients constraints the lower order ones

Reliability of high order connections determine design

Jie Chen, Division of Engineering 18 BROWN UN BROWN UNIVERSITY IVERSITY

Gibbs distribution for an inverter is: The conditional probability is:

1 1

( , ) ( | ) ( ) P x x P x x P x =

1 1

1 ( ) 1

2 1 ( , )

T

x x x x P x x e Z

−

− − =

Take Signal Errors into Design

x

1

x

Jie Chen, Division of Engineering 19 BROWN UN BROWN UNIVERSITY IVERSITY

Continuous Errors in Signal

We model signal noise using Gaussian process

2 2

( ) 2

1 2

gaussian

x

e

P

µ σ

πσ

− −

=

Design choice 1 -- Inputs around “0” & “1” Design choice 2 -- Inputs around “-1” & “1”

Jie Chen, Division of Engineering 20 BROWN UN BROWN UNIVERSITY IVERSITY

Tolerance to Temperature Variation

By taking input around ‘1’, we get marginalized probability:

−1 −0.5 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x‘1 p(x‘1) probability T=0.1 T=0.25 T=0.5 −1 −0.5 0.5 1 0.2 0.4 0.6 0.8 1 x‘1 p(x‘1) probability σ=sqrt(2) σ=sqrt(0.2) σ=sqrt(0.02)

2

1 2 1 2 2 1 1 1 2

( 1) ( ) ( | ) x P x P x x e dx

σ σ σ

η

− + − −

− = ∫

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SLIDE 6

Jie Chen, Division of Engineering 21 BROWN UN BROWN UNIVERSITY IVERSITY

Error Rate Calculation

0.5 1 1.5 2 2.5 3 3.5 4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Temperature variation (T) of Gaussian Process Error rate

incorrect probability Error rate correct incorrect probability = +

1 1.2 1.4 1.6 1.8 2 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Standard deviation (σ) of Gaussian Process Error rate

Proposed design favors for low T and small σ.

Jie Chen, Division of Engineering 22 BROWN UN BROWN UNIVERSITY IVERSITY

Signal Error in NAND Design

Gibbs distribution for a NAND is: The marginalized probability P(xc) is:

−1 −0.5 0.5 1 0.2 0.4 0.6 0.8 1 xc p(x1) probability T=0.1 T=0.25 T=0.5

1 (2 )

1 ( , , )

a b c a b c

x x x x x x T a b c

P x x x e Z

− − −

=

xa xb xc

Jie Chen, Division of Engineering 23 BROWN UN BROWN UNIVERSITY IVERSITY

Tolerance to Temperature Variation

−1 −0.5 0.5 1 0.2 0.4 0.6 0.8 1 x‘c p(x‘c) probability T=0.1 T=0.25 T=0.5 T=1 T=4

Apply inputs “01”

−1 −0.5 0.5 1 0.2 0.4 0.6 0.8 1 x‘c p(x‘c) probability T=0.1 T=0.25 T=0.5 T=1 T=4

Apply inputs “11”

Jie Chen, Division of Engineering 24 BROWN UN BROWN UNIVERSITY IVERSITY

Error Rate Calculation

0.5 1 1.5 2 2.5 3 3.5 4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 T Error rate xa = 1, xb = 1 xa = 0, xb = 1

Proposed design works better at low energy state.

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SLIDE 7

Jie Chen, Division of Engineering 25 BROWN UN BROWN UNIVERSITY IVERSITY

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

Jie Chen, Division of Engineering 26 BROWN UN BROWN UNIVERSITY IVERSITY

Future Works

We are currently investigating how this approach can be extended to more complex logic Implement design using different Nanotechnologies

Thank you

Jie_Chen@Brown.Edu http://binary.engin.brown.edu “ Device failure should not cause computing systems to malfunction if they have been designed from the beginning to tolerate faults”

  • -- Von Neumann