Gröbner Basis Construction Algorithms Based on Superposition Loops

We present novel Gröbner basis algorithms based on saturation loops used by modern superposition theorem provers. By combining (i) top-level Gröbner basis construction strategies based on the OTTER and DISCOUNT saturation loops, and (ii) sophisticated term indexing techniques derived both from ATP literature and from superfluous Spolynomial criteria in Gröbner basis theory, we are able to compute Gröbner bases for large, largely linear nonlinear systems of polynomial equations which are beyond the reach of previously available methods. These types of systems are typical of those arising from the use of SMT solvers in reasoning about industrial-strength software artifacts with nonlinear arithmetical components. Proving the correctness of these new Gröbner basis procedures is nontrivial, and to do so we utilise the recently introcued theory of Abstract Gröbner Bases. We illustrate the practical value of the algorithms through an experimental implementation within the Z3 SMT solver.