1. Weikang Qian The Synthesis of Stochastic Logic to Perform Multivariate Polynomial Arithmetic Abstract Ph.D. Student, University of Minnesota Marc D. Riedel Assistant Professor, University of Minnesota Motivation 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. The traditional IC design is based on deterministic sequence of zeroes and ones. Pros: precise Cons: complex design to handle variability/noise How about using stochastic input bit streams to do computation? Stochastic Bit Streams bit stream X: each bit with prob. x to be 1 real value x probabilistic interpretation combinational logic 1,1,0,1,0,0,1,1 5/8 0,1,0,1,1,0,1,0 4/8 1,0,0,0,0,1,0,1 3/8 Mathematical Model combinational logic X2X2 X1X1 XnXn Independent Random Boolean Variable Y Also a Random Boolean Variable! Suppose A Function F ( is integer coefficient) F is integer-coefficient multivariate polynomial, with bounded degree! Example: Multiplexer Problem Can we implement arbitrary multivariate polynomial? Possible! Then, we have Consider Bernstein Polynomial Set,,, Univariate Bernstein Basis Polynomial Multivariate Bernstein Basis Polynomial Multivariate Bernstein Polynomial is Bernstein Coefficient Derive Bernstein Coefficient from Power-Basis Coefficient (d=2) Degree Elevation (d=2) Stochastic Logic Implementing Bernstein Polynomial Decoding Block (8-Input Example ) Probability Assignment Implement multivariate Bernstein polynomial with all coefficients in the unit interval Synthesize Stochastic Logic to Compute Multivariate Power-Form Polynomial then, it can be converted into a multivariate Bernstein polynomial with all coefficients in the unit interval. Theorem: If a multivariate polynomial for any Synthesizing Step: 1.Perform linear transform on the original polynomial, so that the polynomial satisfies the condition in the above theorem. 2.Compute Bernstein coefficients from power-basis coefficients. 3.While there exists one Bernstein coefficient that is not in the unit interval, perform degree elevation to obtain next set of Bernstein coefficients. 4.Build stochastic logic to implement the Bernstein polynomial with all coefficients in the unit interval. Experimental Result: Performance with Noisy Input Data Implement Bivariate Polynomial Deterministic Implementation: binary radix encoding with M bits. Need one multiplier and adder and a iterations. Stochastic Implementation: bit stream of length 2 M. Implemented either in serial or in parallel. Ratio of Area-Time Product of Stoch. Impl. Over Deter. Impl. Mn1n1 n2n2 456 8 40.5700.6300.639 50.6300.6310.612 60.6390.6120.593 9 40.7820.8650.877 50.8650.8660.840 60.8770.8400.814 10 41.1201.2381.255 51.2381.2401.203 61.2551.2031.166 11 41.6581.8331.859 51.8331.8361.781 61.8591.7811.726 Experimental Result: Hardware Comparison Butterworth Polynomial, used in signal processing, is We implement and, for n = 2, 3, 4 and 5.