Heuristic search and pruning in polynomial constraints satisfaction

Abstract

In this paper, we report how certain AI techniques can be used to speed up an algebraic algorithm for deciding the satisfiability of a system of polynomial equations, disdequations, and inequalities. We begin by viewing the algebraic algorithm (Cylindrical Algebraic Decomposition) as a search procedure which searches for a solution of the given system. When viewed in this way, the algebraic algorithm is nonddeterministic, in the sense that it can often achieve the same goal, but following different search paths requiring different amounts of computing times. Obviously one wishes to follow the least timedconsuming path. However, in practice it is not possible to determine such an optimal path. Thus it naturally renders itself to the heuristic search techniques of AI. In particular we experimented with BestdFirst strategy. We also study a way to prune the search space, which proves to be useful especially when the given polynomial system is not satisfiable. The experimental results indicate that such AI techniques can often help in speeding up the algebraic method, sometimes dramatically.