An algebraic method for managing reliability in propositional logics

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Google Scholar
Research Projects
Organizational Units
Journal Issue
The aim of this paper is to expound an original algebraic model for managing information with different degrees of unreliability. This model is conceived as an extension of classical propositional logics in which each propositional formula is associated with an unreliability value, and results in an approach quite different from that of probabilistic logics. In our model, the logical notions of tautological consequence and consistency of a set of formulae are reformulated on behalf of the foreseen unreliability values. The core of the paper is related to the discovery of a remarkable relation between these unreliability values associated to tautological consequence and the calculation of the reduced Groebner basis of a polynomial ideal. In this way, our method for assigning these unreliability values to information and reasoning turns out to have a straightforward translation into algebraic terms.
J. A. Alonso, E. Briales, Lógicas Polivalentes y Bases de Gröbner. In: C. Martin (ed.), Actas del V Congreso de Lenguajes Naturales y Lenguajes Formales. University of Seville, Seville, 1995, pp. 307-315. T. Becker, V. Weisspfenning, Gröbner bases. A computational approach to commutative algebra, Berlin, Graduate Studies in Mathematics-Springer, 1993. B. Buchberger, Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elementals of the residue class ring of a zero dimensional polynomial ideal, Journal of Symbolic Computation 41/3-4 (2006) 475-511. B. Buchberger, Applications of Gröbner Bases in Non-Linear Computational Geometry. In: J. R. Rice (ed.), Mathematical Aspects of Scientific Software. Springer-Verlag, New York, 1988, pp. 60-88. J. Chazarain, A. Riscos, J. A. Alonso, E. Briales, Multivalued Logic and Gröbner Bases with Applications to Modal Logic, Journal of Symbolic Computation 11 (1991) 181-194. G. de Cooman, F. Hermans, Imprecise probability trees: Bridging two theories of imprecise probability, Artificial Intelligence 172 (2008) 1400-1427. D. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, New York, 1992. F. G. Cozman, C. P. de Campos, J. C. Ferreira da Rocha, Probabilistic logic with independence, Int. Journal Approximate Reasoning 49 (2008) 3-17. J. C. Faugère, A new efficient algorithm for computing Gröbner bases, Journal of Pure and Applied Algebra 139/1 (1999) 61-88. J. C. Faugère, A new efficient algorithm for computing Gröbner bases without reduction to zero. In: T. Mora (ed.), Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation (ISSAC 2002), ACM Press, 2002, pp. 75-83. M. Garcia-Remesal, V. Maojo, L.M. Laita, E. Roanes-Lozano, J. Crespo, An Algebraic Approach to Detect Logical Inconsistencies in Medical Appropriateness Criteria. In: Engineering in Medicine and Biology Society, (EMBS 2007), 29th Annual International Conference of the IEEE, Lyon, 2007, pp. 5148-5151. V. P. Gerdt, M. V. Zinin, A Pommaret Division Algorithm for Computing Gröbner Bases in Boolean Rings. In: J. R. Sendra, L. Gonzlez-Vega (eds.), Symbolic and Algebraic Computation, International Symposium ISSAC 2008, ACM Press, 2008, pp. 95-102. G. Gerla, Inferences in Probability Logic, Artificial Intelligence 70(1-2) (1994) 33-52. P. Hansen, B. Jaumard, Probabilistic satisfiability, Report G-96-31. Les Cahiers du GERAD, École Polytechnique de Montréal, 1996. J. Hsiang, Refutational Theorem Proving using Term-Rewriting Systems, Artificial Intelligence 25 (1985) 255-300. D. Kapur, P. Narendran, 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, 1985, pp. 1146-1153. P. Krause, D. Clark, Representing Uncertain Knowledge, Kluwer, Dordrecht, 1993. L. M. Laita, L. de Ledesma, E. Roanes-Lozano, E. Roanes-Maćas, An Interpretation of the Propositional Boolean Algebra as a k-algebra. Effective Calculus. In: J. Campbell, J. Calmet (eds.): Proceedings of the Second International Workshop/Conference on Artificial Intelligence and Symbolic Mathematical Computing (AISMC-2). Lecture Notes in Computer Science 958, Springer-Verlag, 1995, pp. 255-263. L.M. Laita, E. Roanes-Lozano, L. de Ledesma, J.A. Alonso, A computer algebra approach to verification and deduction in many valued knowledge systems. Soft Computing 3/1 (1999) 7-19. L.M. Laita, E. Roanes-Lozano, V. Maojo, L. de Ledesma, L. Laita: An Expert System for Managing Medical Appropriateness Criteria Based on Computer Algebra Techniques. Computers and Mathematics with Applications 51/5 (2000) 473-481. L.M. Laita, G. González-Paez, E. Roanes-Lozano, V. Maojo, L. de Ledesma, L. Laita, A methodology for constructing expert systems for medical diagnosis, in: Lecture notes in computer science vol. 2199, Springer, Berlin-Heidelberg, 2001. S. L. Lauritzen, D. J. Spiegelhalter, Local computations with probabilities on graphical structures and their application to expert systems (with discussion), J. Roy. Stat. Soc. Ser. B 50 (1988) 157-224. M. Lourdes Jimenez, J. M. Santamaría, R. Barchino, L. Laita, L. M. Laita, L. A. González, A. Asenjo: Knowledge representation for diagnosis of care problems through an expert system: Model of the auto-care deficit situations, Expert System with Applications 34 (2008) 2847-2857. T. Lukasiewicz, Probabilistic deduction with conditional constraints over basic events, Journal of Artificial Intelligence Research 10 (1999) 199-241. T. Lukasiewicz, Weak nonmonotonic probabilistic logics, Artificial Intelligence 168 (2005) 119-161. T. Lukasiewicz, Expressive probabilistic description logics, Artificial Intelligence 172 (2008) 852-883. N. J. Nilsson, Probabilistic logic, Artificial Intelligence 28 (1986) 71-87. J. Pearl, Probabilistic Reasoning in Intelligent Systems, Morgan Kaufman, San Mateo (CA), 1988. C. Pérez-Carretero, L.M. Laita, E. Roanes-Lozano, L. Lázaro, J. González-Cajal, L. Laita, A Logic and Computer Algebra-Based Expert System for Diagnosis of Anorexia, Mathematics and Computers in Simulation 58 (2002) 183-202. E. Roanes-Lozano, L. M. Laita: Verification of Knowledge Based Systems with Commutative Algebra and Computer Algebra Techniques, Proceedings of the 1st International Conference on Applications of Computer Algebra (IMACS), New Mexico University, USA, 1995 (electronic book). E. Roanes-Lozano, L. M. Laita: Verification of Knowledge Based Systems: An Algebraic Interpretation, Proceedings of the International Conference on Artifiical Intelligence (IJCAI-95) (Workshop on Verification and Validation of Knowledge Based Systems), MCGill University, Montreal, Canada, 1995, pp. 91-95. E. Roanes-Lozano, L. M. Laita and E. Roanes-Maćas, A Polynomial Model for Multivalued Logics with a Touch of Algebraic Geometry and Computer Algebra, Mathematics and Computers in Simulation 45/1 (1998) 83-99. C. Rodr?́guez-Solano, L.M. Laita, E. Roanes Lozano, L. López Corral, L. Laita, A Computational System for Diagnosis of Depressive Situations, Expert System with Applications 31 (2006) 47-55. G. Shafer, A Mathematical Theory of Evidence, Princeton (NJ), Princeton Univ. Press, 1976. G. Shafer, Belief functions and parametric models (with discussion), Journal Roy. Stat. Soc. Ser. B 44 (1982) 322-352. G. Shafer, R. Logan, Implementing Dempster's rule for hierarchical evidence, Artificial Intelligence 33 (1987) 271-298. G. Shafer, V. Vovk, Probability and Finance: It's Only a Game!, Wiley, New York, 2001. G. Shafer, P. R. Gillet, R. B. Scherl, A new understanding of subjective probability and its generalization to lower and upper prevision, Int. Journal Approximate Reasoning 33 (2003) 1-49. D. J. Spiegelhalter, A. P. Dawid, S. L. Lauritzen, R. G. Cowell, Bayesian analysis in expert systems (with discussion), Stat. Sci. 8 (1993) 219-283. P. Walley, T. L. Fine, Towards a frequentist theory of upper and lower probability, Annals of Statistics, 10(3), 741-761, 1982 P. Walley, Statistical Reasoning with Imprecise Probabilities, Chapman and Hall, London, 1991. P. Walley, Measures of uncertainty in expert systems, Artificial Intelligence 83 (1996) 1-58. P. Walley, Towards a unified theory of imprecise probability, Int. Journal Approximate Reasoning 24 (2000) 125-148. F. Winkler, Polynomial algorithms in computer algebra, Springer, Vienna, 1996.