, Volume 21, Issue 1, pp 59–76 | Cite as

Projection, consistency, and George Boole

  • J. N. HookerEmail author


Although best known for his work in symbolic logic, George Boole made seminal contributions in the logic of probabilities. He solved the probabilistic inference problem with a projection method, leading to the insight that inference (as well as optimization) is essentially a projection problem. This unifying perspective has applications in constraint programming, because consistency maintenance is likewise a form of inference that can be conceived as projection. Viewing consistency in this light suggests a concept of J-consistency, which is achieved by projection onto a subset J of variables. We show how this projection problem can be solved for the satisfiability problem by logic-based Benders decomposition. We also solve it for among, sequence, regular, and all-different constraints. Maintaining J-consistency for global constraints can be more effective than maintaining traditional domain and bounds consistency when propagating through a richer structure than a domain store, such as a relaxed decision diagram. This paper is written in recognition of Boole’s 200th birthday.


Projection Consistency Optimization Inference Satisfiability Logic-based Benders decomposition Boole 


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Carnegie Mellon UniversityPittsburghUSA

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