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Generalized resolution and cutting planes

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Abstract

This paper illustrates how the application of integer programming to logic can reveal parallels between logic and mathematics and lead to new algorithms for inference in knowledge-based systems. If logical clauses (stating that at least one of a set of literals is true) are written as inequalities, then the resolvent of two clauses corresponds to a certain cutting plane in integer programming. By properly enlarging the class of cutting planes to cover clauses that state that at least a specified number of literals are true, we obtain a generalization of resolution that involves both cancellation-type and circulant-type sums. We show its completeness by proving that it generates all prime implications, generalizing an early result by Quine. This leads to a cutting-plane algorithm as well as a generalized resolution algorithm for checking whether a set of propositions, perhaps representing a knowledge base, logically implies a given proposition. The paper is intended to be readable by persons with either an operations research or an artificial intelligence background.

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This report was prepared as part of the activities of the Management Sciences Research Group, Carnegie-Mellon University. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.

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Hooker, J.N. Generalized resolution and cutting planes. Ann Oper Res 12, 217–239 (1988). https://doi.org/10.1007/BF02186368

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  • DOI: https://doi.org/10.1007/BF02186368

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