Equivalence Verification of Polynomial Datapaths Using Ideal Membership Testing

This paper addresses the equivalence verification problem of register-transfer level (RTL) descriptions that implement arithmetic computations (such as add, mult) over bit vectors with finite widths. A bit vector of size represents integer values from 0 to 2
^m
-1, implying that the corresponding integer values are reduced modulo 2
^m
(%2
^m
). This suggests that bit-vector arithmetic can be efficiently modeled as algebra over finite integer rings, where the bit-vector size (m) dictates the cardinality of the ring (Z
₂

^m
). This paper models the arithmetic datapath verification problem as the equivalence testing of polynomial functions from Z
₂

ⁿ

₁
timesZ
₂

ⁿ

₂
times...timesZ
₂

ⁿ

_d
rarrZ
₂

^m
. We formulate the equivalence problem into that of proving whether f-gequiv0%2
^m
. Fundamental concepts and results from ldquonumber,rdquo ldquoring,rdquo and ldquoideal theoryrdquo are subsequently employed to develop systematic complete algorithmic procedures to solve the problem. We demonstrate the application of the proposed theoretical concepts to high-level (behavioral/RTL) verification of bit-vector arithmetic within practical computer-aided design settings. Using our approach, we verify a set of arithmetic datapaths at RTL, where contemporary verification approaches prove to be infeasible.