An Analogue of the Myhill-Nerode Theorem and Its Use in Computing Finite-Basis Characterizations (Extended Abstract)

Advances in t,he theory of well-partially-ordered sets now make it possible to prove the existence of low-degree polynomial-time decision algorithms for a vast assortment of natural problems, many of which seem to resist more traditional means of complexity classification. Surprisingly, these proofs are nonconstructive, based on the promise of an iiiiknown but finite obstruction set. Recent progress has yielded constructivization strategies that, for most applications, allow the desired decision (and search) algorithms to be known [FL3], despite the nonconstructive nature of the underlying mathematical tools on which the existence of these algorithms is based. These constructivizations produce algorithms that rely on the finiteness of an obstruction set, yet they ensure no means for computing or even verifying a candidate set. The main purpose of this paper is to prove a theorem that is a graph-theoretic analogue of the Myhill-Nerode characterization of regular languages. We employ this result to establish that, for many applications, obstruction sets are computable by known algorithms. * This research is supported in part by the National Science Foundation under grant MIP8603879, by the Office of Naval Research under contracts N00014-88-Ii-0343 and N00014-88-Ii0456, and by the National Aeronautics and Space Administration under engineering research center grant NAGWI1406. Michael A . Langston Dept of Computer Science University of Tennessee Knoxville, T N 37996