Computing Ground Reducability and Inductively Complete Positions

We provide the extended ground-reducibility test which is essential for induction with term-rewriting systems based on [Kuc89]: Given a term, determine at which sets of positions it is ground-reducible by which subsets of rules. The core of our method is a new parallel cover algorithm based on recursive decomposition. From this we obtain a separation algorithm which determines constructors and defined function symbols in a term-algebra presented by a rewrite system. We then reduce our main problem of extended ground-reducibility to separation and cover. Furthermore, using the knowledge of algebra separation, we refine the bounds of [JK86] for the size of ground reduction test-sets. Both our cover algorithm and our extended ground-reducibility test are engineered to be adaptive to actual problem structure, i.e., to allow for lower than the worst case bounds for test-sets on well conditioned problems, including well conditioned subproblems of difficult cases.