Abstract
The aim of this paper is to present theoretically a new algebraic method for detecting potentially dangerous states in a Rule Based Expert System whose knowledge is represented by propositional Boolean logic. Given a dangerous state which does not happen at present, our method is able to detect a possible input fact such that, if it also occurred, the dangerous situation really would happen. This method, inspired by automatic discovery of geometric theorems, is based on calculating just one reduced Groebner basis of a polynomial ideal representing the system’s knowledge. An implementation in the computer algebra system Maple is included.
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Alonso, J.A., Briales, E.: Lógicas polivalentes y bases de Gröbner. In: Martin, C. (ed.) Actas del V Congreso de Lenguajes Naturales y Lenguajes Formales, pp. 307–315. University of Seville, Seville (1995)
Buchberger, B.: Bruno Buchberger’s PhD thesis 1965: an algorithm for finding the basis elementals of the residue class ring of a zero dimensional polynomial ideal. J. Symb. Comput. 41(3–4), 475–511 (2006)
Buchberger, B.: Applications of Gröbner bases in non-linear computational geometry. In: Rice, J.R. (ed.) Mathematical Aspects of Scientific Software, IMA vol. 14, pp. 60–88. Springer, New York (1988)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, New York (1992)
Chazarain, J., Riscos, A., Alonso, J. A., Briales, E.: Multivalued logic and Gröbner bases with applications to modal logic. J. Symb. Comput. 11, 181–194 (1991)
Chou, S.-C.: Mechanical Geometry Theorem Proving. Kluwer, Dordrecht (1987)
Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Commun. ACM 5(7), 394–397 (1962)
Faugère, J.C.: A new efficient algorithm for computing Gröbner bases. J. Pure Appl. Algebra 139(1), 61–88 (1999)
Faugère, J.C.: A new efficient algorithm for computing Gröbner bases without reduction to zero. In: Mora, T. (ed.) Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation ISSAC 2002, pp. 75–83. ACM, New York (2002)
Gerdt, V.P., Zinin, M.V.: A Pommaret division algorithm for computing Gröbner bases in Boolean rings. In: Sendra, J.R., González-Vega, L. (eds.) Symbolic and Algebraic Computation, International Symposium, ISSAC 2008, pp. 95–102. ACM, New York (2008)
Hsiang, J.: Refutational theorem proving using term-rewriting systems. Artif. Intell. 25, 255–300 (1985)
Kapur, D., Mundy, J.L.: Wu’s method and its application to perspective viewing. In: Kapur, D., Mundy, J.L. (eds.) Geometric Reasoning, pp. 15–36. MIT, Cambridge (1989)
Kapur, D., Narendran, P.: An equational approach to theorem proving in first-order predicate calculus. In: Proceedings of the 9th International Joint Conference on Artificial Intelligence (IJCAI-85), vol. 2, pp. 1146–1153 (1985)
Laita, L.M., de Ledesma, L., Roanes-Lozano, E., Roanes-Macías, E.: An interpretation of the propositional Boolean algebra as a k-algebra. Effective calculus. In: Campbell, J., Calmet, J. (eds.) Proceedings of the Second International Workshop/Conference on Artificial Intelligence and Symbolic Mathematical Computing (AISMC-2). Lecture Notes in Computer Science, vol. 958, pp. 255–263. Springer, New York (1995)
Laita, L.M., Roanes-Lozano, E., Maojo, V., de Ledesma, L., Laita, L.: An expert system for managing medical appropriateness criteria based on computer algebra techniques. Comput. Math. Appl. 51(5), 473–481 (2000)
Lourdes-Jiménez, M., Santamaría, J.M., Barchino, R., Laita, L., Laita, L.M., González, L.A., Asenjo, A.: Knowledge representation for diagnosis of care problems through an expert system: model of the auto-care deficit situations. Expert Syst. Appl. 34, 2847–2857 (2008)
Pérez-Carretero, C., Laita, L.M., Roanes-Lozano, E., Lázaro, L., González-Cajal, J., Laita, L.: A logic and computer algebra-based expert system for diagnosis of Anorexia. Math. Comput. Simul. 58, 183–202 (2002)
Recio, T., Vélez, M.P.: Automatic discovery of theorems in elementary geometry. J. Autom. Reason. 23, 63–82 (1999)
Roanes-Lozano, E., Laita, L.M., Roanes-Macías, E.: Maple V in A.I.: the Boolean algebra associated to a KBS. CAN Nieuwsbrief 14, 65–70 (1995)
Roanes-Lozano, E., Laita, L.M., Roanes-Macías, E.: A polynomial model for multivalued logics with a touch of algebraic geometry and computer algebra. Math. Comput. Simul. 45(1), 83–99 (1998)
Roanes-Lozano, E., Laita, L.M., Roanes-Macías, E.: A Groebner bases based many-valued modal logic implementation in Maple. In: Autexier, S., et al. (eds.) Determination of Geometric Loci. 3D-Extension of Simson-Steiner. AISC / Calculemus / MKM 2008. Lecture Notes in Artificial Intelligence, vol. 5144, pp. 170–183. Springer, Berlin (2008)
Rodríguez-Solano, C., Laita, L.M., Roanes-Lozano, E., López-Corral, L., Laita, L.: A computational system for diagnosis of depressive situations. Expert Syst. Appl. 31, 47–55 (2006)
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Roanes-Lozano, E., Hernando, A., Laita, L.M. et al. A Groebner bases-based approach to backward reasoning in rule based expert systems. Ann Math Artif Intell 56, 297–311 (2009). https://doi.org/10.1007/s10472-009-9147-4
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DOI: https://doi.org/10.1007/s10472-009-9147-4