FENA poster

1
Abstract
The Synthesis of Stochastic Logic to Perform
Multivariate Polynomial Arithmetic
As the feature size of integrated circuits
continues scaling down, maintaining the paradigm
of deterministic Boolean computation is
increasingly challenging. Indeed, mounting
concerns over noise and uncertainty in signal
values motivate a new approach the design of
stochastic logic, that is to say, digital
circuitry that processes signals
probabilistically, and so can cope with errors
and uncertainty. In this work, we present a
general methodology for synthesizing stochastic
logic for the computation of multivariate
polynomial, a category that is important for
applications such as digital signal processing.
The method is based on converting polynomials
into a particular mathematical form
multivariate Bernstein polynomials -- and then
implementing the computation with stochastic
logic. The resulting logic processes serial or
parallel streams that are random at the bit
level. In the aggregate, the computation becomes
accurate, since the results depend only on the
precision of the statistics. Experiments show
that our method produces circuits that are highly
tolerant of errors in the input stream, while the
area-delay product of the circuit is comparable
to that of deterministic implementations.
Marc D. Riedel
Weikang Qian
Assistant Professor, University of Minnesota
Ph.D. Student, University of Minnesota
Mathematical Model
Example Multiplexer
Motivation
Stochastic Bit Streams
probabilistic interpretation
bit stream X each bit with prob. x to be 1

• The traditional IC design is based on
  deterministic sequence of zeroes and ones.
• Pros precise
• Cons complex design to handle variability/noise

X1
Independent Random Boolean Variable
Also a Random Boolean Variable!
real value x
A FunctionF
X2
Y
Xn
How about using stochastic input bit streams to
do computation?
Implement multivariate Bernstein polynomial
F is integer-coefficient multivariate
polynomial, with bounded degree!
Bernstein Polynomial
Problem
Stochastic Logic Implementing Bernstein Polynomial
Decoding Block (8-Input Example )
Can we implement arbitrary multivariate
polynomial?
Derive Bernstein Coefficient from Power-Basis
Coefficient (d2)
Univariate Bernstein Basis Polynomial
Possible!
Consider
Multivariate Bernstein Basis Polynomial
Degree Elevation (d2)
Multivariate Bernstein Polynomial
Probability Assignment
Experimental Result Performance with Noisy Input
Data
Synthesize Stochastic Logic to Compute
Multivariate Power-Form Polynomial
Experimental Result Hardware Comparison
Ratio of Area-Time Product of Stoch. Impl. Over
Deter. Impl.
M n1 n2 n2 n2
M n1 4 5 6
8 4 0.570 0.630 0.639
8 5 0.630 0.631 0.612
8 6 0.639 0.612 0.593
9 4 0.782 0.865 0.877
9 5 0.865 0.866 0.840
9 6 0.877 0.840 0.814
10 4 1.120 1.238 1.255
10 5 1.238 1.240 1.203
10 6 1.255 1.203 1.166
11 4 1.658 1.833 1.859
11 5 1.833 1.836 1.781
11 6 1.859 1.781 1.726

• Synthesizing Step
• Perform linear transform on the original
  polynomial, so that the polynomial satisfies the
  condition in the above theorem.
• Compute Bernstein coefficients from power-basis
  coefficients.
• While there exists one Bernstein coefficient that
  is not in the unit interval, perform degree
  elevation to obtain next set of Bernstein
  coefficients.
• Build stochastic logic to implement the Bernstein
  polynomial with all coefficients in the unit
  interval.

• Stochastic Implementation bit stream of length
  2M. Implemented either in serial or in parallel.