Polynomial Word-Level Verification of Arithmetic Circuits

Verifying the functional correctness of a circuit is often the most time-consuming part of the design process. Recently, world-level formal verification methods, e.g., Binary Moment Diagram (BMD) and Symbolic Computer Algebra (SCA) have reported very good results for proving the correctness of arithmetic circuits. However, these techniques still frequently fail due to memory or time requirements. The unknown complexity bounds of these techniques make it impossible to predict before invoking the verification tool whether it will successfully terminate or run for an indefinite amount of time. In this paper, we formally prove that for integer arithmetic circuits, the entire verification process requires at most linear space and quadratic time with respect to the size of the circuit function. This is shown for the two main word-level verification methods: backward construction using BMD and backward substitution using SCA. We support the architectures which are used in the imnlementation of integer polynomial operations, e.g., $X^{3}-XY^{2}+XY$. Finally, we show in practice that the required space and run times of the word-level methods match the predicted results in theory when it comes to the verification of different arithmetic circuits, including exponentiation circuits with different power values $(X^{P}$: $2\leqslant P\leqslant 7)$ and more complicated circuits (e.g., $X^{2}+XY+X$.