Efficient Decision Procedure for Bounded Integer Non-linear Operations Using SMT($\mathcal{LIA}$)

Abstract

For the verification of complex designs, one often needs to solve decision problems containing integer non-linear constraints. Due to the undecidability of the problem, one usually considers bounded integers and then either linearizes the problem into a SMT($\mathcal{LIA}$) problem (i.e., the theory of linear integer arithmetic with Boolean constraints) or bit-blasts into a SAT problem. We present a novel way of linearizing those constraints, and then show how the proposed encoding to a SMT($\mathcal{LIA}$) problem can be integrated into an incremental lazy bounding and refinement procedure (LBR) that leverages on the success of the state-of-the-art SMT($\mathcal{LIA}$) solvers. The most important feature of our LBR procedure is that the formula need not be re-encoded at every step of the procedure but rather, only bounds on variables need to be asserted/retracted, which are very efficiently supported by the recent SMT($\mathcal{LIA}$) solvers. In a series of controlled experiments, we show the effectiveness of our linearization encoding and LBR procedure in reducing the SMT solve time. We observe similar effectiveness of LBR procedure when used in a software verification framework applied on industry benchmarks.