Foundations of Assumption-based Truth Maintenance Systems: Preliminary Report

In this paper we (1) define the concept of a Clause Managetnent System (CMS) — a generaizatiou of de Kleer’s ATMS, (2) motivate such systems in terms of efficiency of search and abductive reasoning, and (3) characterize the computation affected by a CMS in terms of the concept of prime implicants.’ 1. A Problem-Solving Architecture Figure 1 illustrates an architecture for a problem solving system consisting of a domain dependent Reasoner coupled to a domain independent Clause Management System (CMS). For our present purposes, the Reasoner is a black box which, m the process of doing whatever it does, occasionally transmits a propositional clause2 to the CMS. The Reasoner is also permitted to query the CMS any time it feels so inclined. A query takes the form of a propositional clause C. The CMS is expected to respond with every shortest clause S for which the clause S V C is a logical consequence, but S is not a logical consequence, of the clauses thus far transmitted to the CMS by the Reasoner. In Section 2 we show why obtaining such S’s is important for many Al systems. For example, for abductive reasoning -‘S will be an hypothesis, which, if known, sanctions the conclusion C. For efficient search -~Sdefines a most general context in which C holds. A traditional ATMS/TMS is a restricted CMS in which (1) the clauses transmitted to the CMS are limited to be either Horn (i.e., justifications) or negative (i.e., nogoods), Fellow of the Canadian Institute for Advanced Reserch. This research was funded by the Canadian National Science and Engineering Research Council under grant A0044. 2 In actual fact, the reasoner may transmit an arbitrary predicate calculus clause (containing variables for example), but this clause would be treated propositionally by the CMS. In other words, different atomic formulas are treated as different propositional symbols by the CMS. and (2) the queries (C) are always literals. The fundamental TMS problem is to identify the contexts in which a given singleton clause C holds — this is equivalent to querying the CMS for the shortest clauses S of the preceding paragraph, as the negation of each such S implies C. CLAUSE MANAGEMENT SYSTEM 1: A problem-solving architecture 2. Motivation and Formal Preliminaries We shall assume a propositional language with countably infinitely many propositional symbols and with the logical connectives V, —~ The connectives A, j are defined in terms of v, -, in the usual way, as are the formulas of the language. The definition of the entailment relation, ~, is also standard: If S is a set of formulas and w a formula then S ~ w just in case every assignment of truth values to the propositional symbols of the language which makes each formula of S true also makes w true. REASONER