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Abstract

We present a variant of the CONEstrip algorithm for checking whether the origin lies in a finitely generated convex cone that can be open, closed, or neither. This variant is designed to deal efficiently with problems where the rays defining the cone are specified as linear combinations of propositional sentences. The variant differs from the original algorithm in that we apply row generation techniques. The generator problem is WPMaxSAT, an optimization variant of SAT; both can be solved with specialized solvers or integer linear programming techniques. We additionally show how optimization problems over the cone can be solved by using our propositional CONEstrip algorithm as a preprocessor. The algorithm is designed to support consistency and inference computations within the theory of sets of desirable gambles. We also make a link to similar computations in probabilistic logic, conditional probability assessments, and imprecise probability theory.

Keywords

sets of desirable gambles linear programming row generation satisfiability SAT PSAT WPMaxSAT consistency coherence inference natural extension 

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References

  1. 1.
    Baioletti, M., Capotorti, A., Tulipani, S., Vantaggi, B.: Simplification rules for the coherent probability assessment problem. Annals of Mathematics and Artificial Intelligence 35(1-4), 11–28 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bansal, N., Raman, V.: Upper bounds for MaxSat: Further improved. In: Aggarwal, A.K., Pandu Rangan, C. (eds.) ISAAC 1999. LNCS, vol. 1741, pp. 247–258. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. 3.
    Biazzo, V., Gilio, A., Lukasiewicz, T., Sanfilippo, G.: Probabilistic logic under coherence: complexity and algorithms. Annals of Mathematics and Artificial Intelligence 45(1-2), 35–81 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press (2009)Google Scholar
  5. 5.
    Cozman, F.G., di Ianni, L.F.: Probabilistic satisfiability and coherence checking through integer programming. In: van der Gaag, L.C. (ed.) ECSQARU 2013. LNCS (LNAI), vol. 7958, pp. 145–156. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  6. 6.
    Georgakopoulos, G., Kavvadias, D., Papadimitriou, C.H.: Probabilistic satisfiability. Journal of Complexity 4, 1–11 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Gomes, C.P., Kautz, H., Sabharwal, A., Selman, B.: Satisfiability solvers. In: van Harmelen, F., Lifschitz, V., Porter, B. (eds.) Handbook of Knowledge Representation, ch. 2, pp. 89–134. Elsevier (2008)Google Scholar
  8. 8.
    Hansen, P., Jaumard, B., de Aragão, M.P., Chauny, F., Perron, S.: Probabilistic satisfiability with imprecise probabilities. International Journal of Approximate Reasoning 24(2-3), 171–189 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Manquinho, V., Marques-Silva, J., Planes, J.: Algorithms for weighted boolean optimization. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 495–508. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Prestwich, S.: CNF encodings. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, ch. 2, pp. 75–98. IOS Press (2009)Google Scholar
  11. 11.
    Pretolani, D.: Probability logic and optimization SAT: The PSAT and CPA models. Annals of Mathematics and Artificial Intelligence 43(1-4), 211–221 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Quaeghebeur, E.: The CONEstrip algorithm. In: Kruse, R., Berthold, M.R., Moewes, C., Gil, M.A., Grzegorzewski, P., Hryniewicz, O. (eds.) Synergies of Soft Computing and Statistics. AISC, vol. 190, pp. 45–54. Springer, Heidelberg (2013), http://hdl.handle.net/1854/LU-3007274 CrossRefGoogle Scholar
  13. 13.
    Quaeghebeur, E.: Desirability. In: Coolen, F.P.A., Augustin, T., de Cooman, G., Troffaes, M.C.M. (eds.) Introduction to Imprecise Probabilities. Wiley (2014)Google Scholar
  14. 14.
    Quaeghebeur, E., de Cooman, G., Hermans, F.: Accept & reject statement-based uncertainty models (submitted), http://arxiv.org/abs/1208.4462
  15. 15.
    Walley, P.: Statistical reasoning with imprecise probabilities. Monographs on Statistics and Applied Probability, vol. 42. Chapman & Hall (1991)Google Scholar
  16. 16.
    Walley, P.: Towards a unified theory of imprecise probability. International Journal of Approximate Reasoning 24(2-3), 125–148 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Walley, P., Pelessoni, R., Vicig, P.: Direct algorithms for checking consistency and making inferences from conditional probability assessments. Journal of Statistical Planning and Inference 126(1), 119–151 (2004)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Erik Quaeghebeur
    • 1
  1. 1.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands

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