Approximate Reasoning in Fuzzy Resolution

Affiliation(s)

Department of Mathematics, Visva-Bharati University, Santiniketan, West Bengal, India.

Department of Mathematics, Visva-Bharati University, Santiniketan, West Bengal, India.

ABSTRACT

Resolution is an useful tool for mechanical theorem proving in modelling the refutation proof procedure, which is mostly used in constructing a “proof” of a “theorem”. An attempt is made to utilize approximate reasoning methodology in fuzzy resolution. Approximate reasoning is a methodology which can deduce a specific information from general knowledge and specific observation. It is dependent on the form of general knowledge and the corresponding deductive mechanism. In ordinary approximate reasoning, we derive from *A*→*B* and by some mechanism. In inverse approximate reasoning, we conclude from *A*→*B* and using an altogether different mechanism. An important observation is that similarity is inherent in fuzzy set theory. In approximate reasoning methodology-similarity relation is used in fuzzification while, similarity measure is used in fuzzy inference mechanism. This research proposes that similarity based approximate reasoning-modelling generalised modus ponens/generalised modus tollens—can be used to derive a resolution—like inference pattern in fuzzy logic. The proposal is well-illustrated with artificial examples.

Cite this paper

B. Mondal and S. Raha, "Approximate Reasoning in Fuzzy Resolution,"*International Journal of Intelligence Science*, Vol. 3 No. 2, 2013, pp. 86-98. doi: 10.4236/ijis.2013.32010.

B. Mondal and S. Raha, "Approximate Reasoning in Fuzzy Resolution,"

References

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[5] D. Dubois, J. Lang and H. Prade, “Fuzzy Sets in Approximate Reasoning, Part 2: Logical Approaches,” Fuzzy Sets and Systems, Vol. 40, No. 1, 1991, pp. 203-244. doi:10.1016/0165-0114(91)90051-Q

[6] Z. Shen, L. Ding and M. Mukaidono, “Fuzzy Resolution Principle,” Proceedings of the 18th International Symposium on Multivalued Logic, Palma de Mallorca, 24-26 May 1988, pp. 210-215. doi:10.1109/ISMVL.1988.5176

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[8] D. Dubois and H. Prade, “Necessity and the Resolution Principle,” IEEE Transactions on Systems, Man and Cybernetics, Vol. 17, No. 3, pp. 474-478.

[9] C. S. Kim, D. S. Kim and J. Park, “A New Fuzzy Resolution Principle Based on the Antonym,” Fuzzy Sets and Systems, Vol. 113, No. 2, 2000, pp. 299-307. doi:10.1016/S0165-0114(98)00063-3

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[11] F. A. Fontana and F. Formato, “A Similarity-Based Resolution Principle,” International Journal of Intelligent Systems, Vol. 17, No. 9, 2002, pp. 853-872. doi:10.1002/int.10067

[12] S. Raha and K. S. Ray, “Approximate Reasoning Based on Generalised Disjunctive Syllogism,” Fuzzy Sets and Systems, Vol. 61, No. 2, 1994, pp. 143-151.

[13] B. Mondal and S. Raha, “Similarity-Based Inverse Approximate Reasoning,” IEEE Transaction on Fuzzy Systems, Vol. 19, No. 6, 2011, pp. 1058-1071. doi:10.1109/TFUZZ.2011.2159981

[14] B. Lazzerini and F. Marcelloni, “Some Considerations on Input and Output Partitions to Produce Meaningful Conclusions in Fuzzy Inference,” Fuzzy Sets and Systems, Vol. 113, No. 2, 2000, pp. 221-235. doi:10.1016/S0165-0114(98)00096-7

[15] B. Bouchon-Meunier, M. Rifqi and S. Bothorel, “Towards General Measures of Comparison of Objects,” Fuzzy Sets and Systems, Vol. 84, No. 2, 1996, pp. 143-153. doi:10.1016/0165-0114(96)00067-X

[16] R. Zwick, E. Carlstein and D. V. Budescu, “Measures of Similarity among Fuzzy Concepts: A Comparative Analysis,” International Journal of Approximate Reasoning, Vol. 1, No. 2, 1987, pp. 221-242. doi:10.1016/0888-613X(87)90015-6

[17] B. Mondal, D. Mazumdar and S. Raha, “Similarity in Approximate Reasoning,” International Journal of Computational Cognition, Vol. 4, No. 3, 2006, pp. 46-56.

[18] S. Raha, N. R. Pal and K. S. Ray, “Similarity Based Approximate Reasoning: Methodology and Application,” IEEE Transactions on Systems, Man and Cybernatics, Part A: Systems and Humans, Vol. 32, No. 4, 2002, pp. 541-547. doi:10.1109/TSMCA.2002.804787

[19] N. Mellouli and B. Bouchon-Meunier, “Fuzzy Approaches of Abductive Inference,” Proceedings of the 8th International Workshop Non-Monotonic Reasoning, Breikenridge, 2000.

[20] N. Mellouli and B. Bouchon-Meunier, “Abductive Reasoning and Measure of Similitude in the Presence of Fuzzy Rules,” Fuzzy Sets and Systems, Vol. 137, No. 1, 2003, pp. 177-188. doi:10.1016/S0165-0114(02)00439-6

[21] F. Klawonn and J. L. Castro, “Similarity in Fuzzy Reasoning,” Mathware and Soft Computing, Vol. 3, 1995, pp. 197-228.

[22] L. Ughetto, D. Dubois and H. Prade, “Implicative and Conjunctive Fuzzy Rule—A Tool for Reasoning from Knowledge and Examples,” Proceedings of the 16th National Conference on Artificial Intelligence and the 11th Innovative Applications of Artificial Intelligence Conference, American Association for Artificial Intelligence, Menlo Park, 1999, pp. 214-219.

[23] L. A. Zadeh, “A Theory of Approximate Reasoning,” In: J. E. Hayes, D. Michie and L. I. Mikulich, Eds., Machine Intelligence, Vol. 9, Elsevier, New York, 1979, pp. 149-194.

[1] J. A. Robinson, “A Machine Oriented Logic Based on the Resolution Principle,” Journal of the ACM, Vol. 12, No. 1, 1965, pp. 23-41. doi:10.1145/321250.321253

[2] J. J. Kelly, “The Essence of Logic,” Prentice-Hall, New Delhi, 1997.

[3] R. C. T. Lee and C. L. Chang, “Some Properties of Fuzzy Logic,” Information and Control, Vol. 19, No. 1, 1971, pp. 417-431. doi:10.1016/S0019-9958(71)90684-X

[4] R. C. T. Lee, “Fuzzy Logic and the Resolution Principle,” Journal of the ACM, Vol. 19, No. 1, 1972, pp. 109-119. doi:10.1145/321679.321688

[5] D. Dubois, J. Lang and H. Prade, “Fuzzy Sets in Approximate Reasoning, Part 2: Logical Approaches,” Fuzzy Sets and Systems, Vol. 40, No. 1, 1991, pp. 203-244. doi:10.1016/0165-0114(91)90051-Q

[6] Z. Shen, L. Ding and M. Mukaidono, “Fuzzy Resolution Principle,” Proceedings of the 18th International Symposium on Multivalued Logic, Palma de Mallorca, 24-26 May 1988, pp. 210-215. doi:10.1109/ISMVL.1988.5176

[7] M. Mukaidono, “Fuzzy Inference of Resolution Style,” In: R. R. Yager, Ed., Fuzzy Set and Possibility Theory, Per- gamon Press, New York, 1988, pp. 224-231.

[8] D. Dubois and H. Prade, “Necessity and the Resolution Principle,” IEEE Transactions on Systems, Man and Cybernetics, Vol. 17, No. 3, pp. 474-478.

[9] C. S. Kim, D. S. Kim and J. Park, “A New Fuzzy Resolution Principle Based on the Antonym,” Fuzzy Sets and Systems, Vol. 113, No. 2, 2000, pp. 299-307. doi:10.1016/S0165-0114(98)00063-3

[10] M. A. C. Viedma, R. M. Morales and I. N. Sanchez, “Fuzzy Temporal Constraint Logic: A Valid Resolution Principle,” Fuzzy Sets and Systems, Vol. 117, No. 2, 2001, pp. 231-250. doi:10.1016/S0165-0114(99)00099-8

[11] F. A. Fontana and F. Formato, “A Similarity-Based Resolution Principle,” International Journal of Intelligent Systems, Vol. 17, No. 9, 2002, pp. 853-872. doi:10.1002/int.10067

[12] S. Raha and K. S. Ray, “Approximate Reasoning Based on Generalised Disjunctive Syllogism,” Fuzzy Sets and Systems, Vol. 61, No. 2, 1994, pp. 143-151.

[13] B. Mondal and S. Raha, “Similarity-Based Inverse Approximate Reasoning,” IEEE Transaction on Fuzzy Systems, Vol. 19, No. 6, 2011, pp. 1058-1071. doi:10.1109/TFUZZ.2011.2159981

[14] B. Lazzerini and F. Marcelloni, “Some Considerations on Input and Output Partitions to Produce Meaningful Conclusions in Fuzzy Inference,” Fuzzy Sets and Systems, Vol. 113, No. 2, 2000, pp. 221-235. doi:10.1016/S0165-0114(98)00096-7

[15] B. Bouchon-Meunier, M. Rifqi and S. Bothorel, “Towards General Measures of Comparison of Objects,” Fuzzy Sets and Systems, Vol. 84, No. 2, 1996, pp. 143-153. doi:10.1016/0165-0114(96)00067-X

[16] R. Zwick, E. Carlstein and D. V. Budescu, “Measures of Similarity among Fuzzy Concepts: A Comparative Analysis,” International Journal of Approximate Reasoning, Vol. 1, No. 2, 1987, pp. 221-242. doi:10.1016/0888-613X(87)90015-6

[17] B. Mondal, D. Mazumdar and S. Raha, “Similarity in Approximate Reasoning,” International Journal of Computational Cognition, Vol. 4, No. 3, 2006, pp. 46-56.

[18] S. Raha, N. R. Pal and K. S. Ray, “Similarity Based Approximate Reasoning: Methodology and Application,” IEEE Transactions on Systems, Man and Cybernatics, Part A: Systems and Humans, Vol. 32, No. 4, 2002, pp. 541-547. doi:10.1109/TSMCA.2002.804787

[19] N. Mellouli and B. Bouchon-Meunier, “Fuzzy Approaches of Abductive Inference,” Proceedings of the 8th International Workshop Non-Monotonic Reasoning, Breikenridge, 2000.

[20] N. Mellouli and B. Bouchon-Meunier, “Abductive Reasoning and Measure of Similitude in the Presence of Fuzzy Rules,” Fuzzy Sets and Systems, Vol. 137, No. 1, 2003, pp. 177-188. doi:10.1016/S0165-0114(02)00439-6

[21] F. Klawonn and J. L. Castro, “Similarity in Fuzzy Reasoning,” Mathware and Soft Computing, Vol. 3, 1995, pp. 197-228.

[22] L. Ughetto, D. Dubois and H. Prade, “Implicative and Conjunctive Fuzzy Rule—A Tool for Reasoning from Knowledge and Examples,” Proceedings of the 16th National Conference on Artificial Intelligence and the 11th Innovative Applications of Artificial Intelligence Conference, American Association for Artificial Intelligence, Menlo Park, 1999, pp. 214-219.

[23] L. A. Zadeh, “A Theory of Approximate Reasoning,” In: J. E. Hayes, D. Michie and L. I. Mikulich, Eds., Machine Intelligence, Vol. 9, Elsevier, New York, 1979, pp. 149-194.