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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Balas and R.G. Jeroslow, Canonical cuts on the unit hypercube, SIAM J. Appl. Math. 23(1972)61.
E. Balas and S.M. Ng, On the set covering polytope: I. All the facets with coefficients in {0, 1, 2}, Working paper 43-85-86, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, PA; revised February 1986.
C.E. Blair, R.G. Jeroslow and J.K. Lowe, Some results and experiments in programming techniques for propositional logic, Working paper, College of Management, Georgia Institute of Technology, Atlanta, GA; revised July 1985.
G. Boole,The Mathematical Analysis of Logic (Cambridge University Press, 1847, reprinted by Oxford University Press, 1948).
V. Chvátal, Edmonds polytopes and a hierarchy of combinatorial problems, Discr. Math. 4(1973)305.
S.A. Cook, The complexity of theorem-proving procedures,Proc. 3rd ACM Symp. on the Theory of Computing (1971) pp. 151–158.
S.A. Cook and R.A. Reckhow, The relative efficiency of propositional proof systems, J. Symbolic Logic 44(1979)36.
W. Cook, C.R. Coullard and G. Turán, On the complexity of cutting-plane proofs, Working paper, Cornell University, Ithaca, NY (1985).
A. Haken, The intractability of resolution, Theor. Comp. Sci. 39(1985)297.
P.L. Hammer, E.L. Johnson and U.N. Peled, Facets of regular 0–1 polytopes, Math. Progr. 8(1975)179.
J.N. Hooker, Resolution vs. cutting plane solution of inference problems: Some computational experience, Oper. Res. Lett., to appear.
R.G. Jeroslow, An extension of mixed-integer programming models and techniques to some database and artificial intelligence settings, Working paper, College of Management, Georgia Institute of Technology, Atlanta, GA (1985).
R.G. Jeroslow, Computation-oriented reductions of predicate to propositional logic, Working paper, College of Management, Georgia Institute of Technology, Atlanta, GA (1985).
R.G. Jeroslow, On monotone chaining procedures of the CF type, Working paper, College of Management, Georgia Institute of Technology, Atlanta, GA (1985).
R.G. Jeroslow, The theory of cutting planes, in:Combinatorial Optimization, ed. N. Christofides et al. (Wiley, New York, 1979) pp. 21–72.
W. Kneale and M. Kneale,The Development of Logic (Oxford University Press, London, 1962).
L.B. Kovács, Extended set covering problem and logical programming, Computer and Automation Institute, Hungarian Academy of Sciences, Budapest, presented at the 5th Bonn Workshop on Combinatorial Optimization (1984).
J.K. Lowe, Modelling with integer variables, Ph.D. thesis, Georgia Institute of Technology (1984).
N.J. Nilsson,Principles of Artificial Intelligence (Tioga Publ. Co., Palo Alto, CA, 1980).
M.W. Padberg, Characterisation of totally unimodular, balanced and perfect matrices, in:Combinatorial Programming: Methods and Applications, ed. B. Roy (1975) pp. 275–284.
M.W. Padberg, Perfect zero-one matrices, Math. Progr. 6(1974)180.
W.V. Quine, The problem of simplifying truth functions, Amer. Math. Monthly 59(1952)521.
W.V. Quine, A way to simplify truth functions, Amer. Math. Monthly 62(1955)627.
J.A. Robinson, A machine-oriented logic based on the resolution principle, J. ACM 12(1965)23.
G.S. Tseitin, On the complexity of derivations in the propositional calculus, in:Structures in Constructive Mathematics and Mathematical Logic, Part II, ed. A.O. Slisenko (Consultants Bureau, New York, 1968) pp. 115–125 (translated from Russian).
H.P. Williams, Linear and integer programming applied to the propositional calculus, Int. J. Syst. Res. and Inf. Sci. 2(1987)81.
Author information
Authors and Affiliations
Additional information
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.
Rights and permissions
About this article
Cite this article
Hooker, J.N. Generalized resolution and cutting planes. Ann Oper Res 12, 217–239 (1988). https://doi.org/10.1007/BF02186368
Issue Date:
DOI: https://doi.org/10.1007/BF02186368