Equivalence checking using Gröbner bases

Abstract

Motivated by the recent success of the algebraic computation technique in formal verification of large and optimized gate-level multipliers, this paper proposes algebraic equivalence checking for handling circuits that contain both complex arithmetic components as well as control logic. These circuits pose major challenges for existing proof techniques. The basic idea of Algebraic Combinational Equivalence Checking (ACEC) is to model the two compared circuits in form of Gröbner bases and combine them into a single algebraic model. It generates bit and word relationship candidates between the internal variables of the two circuits and tests their membership in the combined model. Since the membership testing does not scale for the described setting, we propose reverse engineering to extract arithmetic components and to abstract them to canonical representations. Further we propose arithmetic sweeping which utilizes the abstracted components to find and prove internal equivalences between both circuits. We demonstrate the applicability of ACEC for checking the equivalence of a floating point multiplier (including full IEEE-754 rounding scheme) against several optimized and diversified implementations.