Boolean Gröbner Basis Reductions on Finite Field Datapath Circuits Using the Unate Cube Set Algebra

Abstract

Recent developments in formal verification of arithmetic datapaths make efficient use of symbolic computer algebra algorithms. The circuit is modeled as an ideal in polynomial rings, and Grobner basis (GB) reductions are performed over these polynomials to derive a canonical representation. As they model logic gates of the circuit, the ideals comprise largely of Boolean (or pseudo-Boolean) polynomials. This paper considers a logic synthesis analogue of GB reductions over Boolean polynomials, by interpreting symbolic algebra as the unate cube set algebra over characteristic sets. By representing Boolean polynomials as characteristic sets using zero-suppressed binary decision diagrams (ZDDs), implicit algorithms are efficiently designed for GB-reduction for datapath circuits. We show that the imposition of circuit-topology-based monomial orders exposes a special structure on the ZDD representation of the polynomials. The subexpressions employed in the GB-reduction are readily visible as subgraphs on the ZDDs, which are directly used to compose the result. Our division algorithms effectively cancel multiple monomials implicitly in one-step, simplify the search for divisors, and avoid intermediate size explosion. Experiments performed over various finite field arithmetic architectures demonstrate the efficiency of our algorithms and implementations; our approach is orders of magnitude faster as compared to conventional methods.