Variations on backtracking for TMS

Abstract

We review the backtracking technique of Doyle's Truth Maintenance System (TMS). Backtracking in principle is to recover from a failure state by retracting some assumptions. We discuss three possible applications for the TMS: (1) Transforming a contradiction proof into a direct proof if a false-node is labelled with In. (2) Selecting another extension if an odd loop failure arises. (3) Doing theory revision if no extension exists. In this paper, we elaborate the first and the second alternative. If the TMS is used in an autoepistemic or default prover (as in [15]) it needs sufficient justifications for first-order proofs. We show how backtracking allows to complete this set incrementally by applying a special case of the deduction theorem. For finding extensions, we describe Doyle's label methods as meta rules deriving statements of the form In(p) and ¬In(q). If a node q of an odd loop is labelled with In and Out we obtain a contradiction in this meta reasoning system. Hence, we can again apply backtracking to identify and retract the label assumptions being responsible for the failure. Thus, we obtain an efficient TMS that detects an extension whenever one exists.