Logic Synthesis from Polynomials with Coefficients in the Field of Rationals

Abstract

This paper introduces a novel concept of performing logic synthesis from multivariate polynomials with coefficients in the field of rationals $(\mathbb{Q})$, where the variables take only Boolean values. Such polynomials are encountered during synthesis and verification of arithmetic circuits using computer algebra and algebraic geometry based techniques. The approach takes as input a polynomial f over $\mathbb{Q}$ with binary variables, and derives a corresponding polynomial $\tilde f$ over the finite field $\left( {{\mathbb{F}_2}} \right)$ of two elements, such that f has the same variety (zero-set) as f. As ${\mathbb{F}_2}$ is isomorphic to Boolean algebra, $\tilde f$ can be translated to a Boolean network by mapping the products and sums as AND and XOR operators, respectively. We prove the correctness of our algebraic transformation, and present a recursive algorithm for the same. The translated $\tilde f \in {\mathbb{F}_2}$ resultingly corresponds to a positive Davio decomposition, and is computed using both explicit and implicit representations. The approach is used to synthesize subfunctions of arithmetic circuits, under the partial synthesis framework. The efficacy of our approach is demonstrated over various integer multiplier architectures, where other contemporary approaches are infeasible.