The Complexity of Theory Revision

Abstract

A knowledge-based system uses its database (a.k.a. its "theory") to produce answers to the queries it receives. Unfortunately, these answers may be incorrect if the underlying theory is faulty. Standard "theory revision" systems use a given set of "labeled queries" (each a query paired with its correct answer) to transform the given theory, by adding and/or deleting either rules and/or antecedents, into a related theory that is as accurate as possible. After formally defining the theory revision task and bounding its sample complexity, this paper addresses the task's computational complexity. It first proves that, unless P = NP, no polynomial time algorithm can identify the optimal theory, even given the exact distribution of queries, except in the most trivial of si.t~a tions. It also shows that, except in such tnvial situations, no polynomial-time algorithm can produce a theory whose inaccuracy is even close (i.e., within a particular polynomial factor) to optimal. These results justify the standard practice of hill-climbing to a locally-optimal theory, based on a given set of labeled samples.