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An Algebraic Foundation for Truth Maintenance

We have recast the problem of truth maintenance in a setting of algebraic equations over Boolean lattices. If a method of labeling propositions to justify them according to some reasoning agent's constraints of belief happens to conform to the postulates of Boolean lattices, the labeling system can be reformulated as an algebraic equation solving system. All truth maintenance systems known to us can be so reformulated. This note summarizes our investigations into the existence and structure of solutions of these algebraic systems. Our central result is a unique factorization theorem for lattice equational systems and their solutions. Our theoretical results are interpreted to compare various styles of truth maintenance and to reveal certain computational difficulties implicit in the algebraic structure of truth maintenance.