Equational Bit-Vector Solving via Strong Gr\"obner Bases

Abstract

Bit-vectors, which are integers in a finite number of bits, are ubiquitous in software and hardware systems. In this work, we consider the satisfiability modulo theories (SMT) of bit-vectors. Unlike normal integers, the arithmetics of bit-vectors are modular upon integer overflow. Therefore, the SMT solving of bit-vectors needs to resolve the underlying modular arithmetics. In the literature, two prominent approaches for SMT solving are bit-blasting (that transforms the SMT problem into boolean satisfiability) and integer solving (that transforms the SMT problem into integer properties). Both approaches ignore the algebraic properties of the modular arithmetics and hence could not utilize these properties to improve the efficiency of SMT solving. In this work, we consider the equational theory of bit-vectors and capture the algebraic properties behind them via strong Gr\"obner bases. First, we apply strong Gr\"obner bases to the quantifier-free equational theory of bit-vectors and propose a novel algorithmic improvement in the key computation of multiplicative inverse modulo a power of two. Second, we resolve the important case of invariant generation in quantified equational bit-vector properties via strong Gr\"obner bases and linear congruence solving. Experimental results over an extensive range of benchmarks show that our approach outperforms existing methods in both time efficiency and memory consumption.