Improving Bit-Blasting for Nonlinear Integer Constraints

Nonlinear integer constraints are common and difficult in the verification and analysis of software/hardware. SMT(QF_NIA) generalizes such constraints, which is a boolean combination of nonlinear integer arithmetic constraints. A classical method to solve SMT(QF_NIA) is bit-blasting, which reduces them to boolean satisfiability problems. Currently, the existing pure bit-blasting based solvers are noncompetitive with other state-of-the-art SMT solvers. The bit-blasting based methods have some problems: First, the bit-blasting method is hampered by nonlinear multiplication operations; second, it sometimes does not search in a proper search space; and third, it contains some redundancy. In this paper, we focus on improving the efficiency of bit-blasting based method. To decide on a proper search space, we proposed an adaptive function for hard nonlinear multiplications, and heuristic strategies to analyze specific constraints. We also found that different orders in successive additions will result in bit vectors with different bit-widths. We proposed an optimal order decision algorithm to save redundancy in successive additions. We implement a solver with the proposed methods named BLAN. Experiments demonstrate that BLAN outperforms other state-of-the-art SMT solvers (APROVE, CVC5, MATHSAT, YICES2, Z3) on the satisfiable SMT(QF_NIA) instances in SMT-LIB. We provide an outlook of BLAN on solving unsatisfiable instances via combining with other solvers. Sensitivity analysis also demonstrates the effectiveness of the proposed methods.