Verification of arithmetic datapaths using polynomial function models and congruence solving

Abstract

Abstract-This paper addresses the problem of solving finite word-length (bit-vector) arithmetic with applications to equivalence verification of arithmetic datapaths. Arithmetic datapath designs perform a sequence of add, mult, shift, compare, concatenate, extract, etc., operations over bit-vectors. We show that such arithmetic operations can be modeled, as constraints, using a system of polynomial functions of the type f :This enables the use of modulo-arithmetic based decision procedures for solving such problems in one unified domain. We devise a decision procedure using Newton's p-adic iteration to solve such arithmetic with composite moduli, while properly accounting for the word-sizes of the operands. We describe our implementation and show how the basic p-adic approach can be improved upon. Experiments are performed over some communication and signal processing designs that perform non-linear and polynomial arithmetic over word-level inputs. Results demonstrate the potential and limitations of our approach, when compared against SAT-based approaches.

Rectification of Arithmetic Circuits with Craig Interpolants in Finite Fields

“…, f 10 , f spec , f m } in Example 5.1 are already written according to the term order of Eqn. (6). Note also that the leading terms of the generators of the ideal J L are the same as the leading terms of polynomials in { f 1 , .…”

“…Recent techniques have investigated the use of polynomial algebra and algebraic geometry techniques for their verification. These include verification of integer arithmetic circuits [1] [2] [3], integer modulo-arithmetic circuits [4], word-level RTL models of polynomial datapaths [5] [6], finite field combinational circuits [7] [8] [9], and also sequential designs [10]. A common theme among the above approaches is that designs are modeled as sets of polynomials in rings with coefficients from integers Z, finite integer rings Z 2 k , finite fields F 2 k , and more recently also from the field of fractions Q.…”