Abstract
This paper describes how succinct rules, which reduce the size of decision tables, can be found by employing multiple-valued logic (MVL). Two multiple-valued algebras are described, one based on level detection, and the other on literal functions. Then a decision table which had also been reduced in size using rough set theory, is now reduced using both algebras and it is seen that all three approaches lead to reductions of comparable simplicity. The new methods require coding the values of each attribute as integers. Then an MVL function that maps the coding of the condition attributes to the coding of the decision attribute is found. As the coded table is sparse only some of the basis functions for each algebra are required. Then a simple approach requiring the reduction of a matrix to row echelon form is used to finding all suitable MVL functions. By decomposing a function in terms of its variables a complete set of rules can be found. The MVL function encodes the data in a very compact form and its decomposition into subfunctions reveals a good way to slice up the table into subtables. The structure of the subfunctions can then be used to simplify each subtable until compact sets of rules emerge. Alternatively, rules can be found by substitution into the MVL function. Encoding a decision table using MVL makes the data easy to manipulate and can uncover relationships that may not become apparent when using other methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, K. J., Campbell, J. G., Maguire L. P., Webb J. A. C. (1999). State Assignment Techniques in Multiple Valued Logic. Proc. 29th Int. Symp. on Multiple-Valued Logic, 220–225. Freiburg, Germany (IEEE).
Cohn, M. (1960). Switching Function Canonical Form Over Integer Fields. PhD Thesis, Harvard University, Cambridge, Mass.
Davio, M., Deschamps, J. & Thayse, A. (1978). Discrete and Switching Functions. McGraw-Hill.
Dubrova, E. V. & Muzio, J. C. (1996). Generalized Reed-Muller Canonical Form for Multiple-Valued Algebra. Multiple-Valued Logic, An International Journal 1: 65–84. Gordon and Breach, Netherlands.
Falkowski, B. J. & Rahardja, S. (1995). Efficient Algorithm for Generation of Fixed Polarity Quaternary Reed-Muller Expansions. Proc. 25th Int. Symp. on Multiple-Valued Logic, 158–163. Bloomington, Indiana, USA (IEEE).
Fazio, A. & Bauer, M. (1997). Intel StrataFlash™ Memory Technology Development and Implementation. Intel Technology Journal 4th Quarter, 1–11 (Intel).
Freitag, K. & Moraga, C. (1999). Quaternary Coded Genetic Algorithms. Proc. 29th Int. Symp. on Multiple-Valued Logic, 194–199, Frieburg, Germany (IEEE).
Green, D. H. & Taylor, I. S. (1974). Modular Representation of Multiple-Valued Logic Systems. Proc. of the IEE 121: 424–429 (IEE).
Green, D. H. (1989). Ternary Reed-Muller Switching Functions with Fixed and Mixed Polarities. Int. J. Electronics 67: 761–775, November. Taylor and Francis group.
Green, D. H. (1990). Reed-Muller Expansions with Fixed and Mixed Polarities Over GF(4). IEE Proceedings, Vol. 137, Pt. E, No. 5, September (IEE).
Guan, J. W. & Bell, D. A. (1998). Rough Discovery for Nuclear Safety. FLINS, Third Int. Conf. On Intelligent Techniques and Soft Computing in Nuclear Science and Engineering, 203–210. Antwerp, Belgium, World Scientific, Singapore.
Guan, J. W. & Bell, D. A. (2000). Data Mining for Monitoring Loose Parts in Nuclear Power Plants. The 2nd International Conference on Rough Sets and Current Trends in Computing, 276–283. Banff, Canada, Technical Report CS-(2000)-07, Ziarko & Yao Eds., Univ. of Regina Saskatchewan Canada, ISBN 0–7731–0413–5.
Hanyu, T. & Higuchi, T. (1988). Design of a Highly Parallel AI Processor Using New Multiple-Valued MOS Devices. Proc. 18th Int. Symp. on Multiple-Valued Logic, 300–306. Palma de Mallorca, Spain (IEEE).
Hanyu, T., Kojima, Y. & Higuchi, T. (1991). A Multiple-Valued Logic Array VLSI Based on Two-Transistor Delta Literal Circuit and Its Application to Real-Time Reasoning Systems. Proc. 21st Int. Symp. on Multiple-Valued Logic, 16–23. Victoria, BC, Canada (IEEE).
Kodandapani, K. L. & Setlur, R. V. (1975). Reed-Muller Canonical Forms in Multivalued Logic. IEEE Transactions on Computers, Vol. c-24, No.6, June (IEEE).
Loris A., Gomez, J. & Roman, R. (1993). Using Decision Trees for Minimization of Multiple-Valued Functions. International Journal of Electronics 75(6): 1–35–1041, June. Taylor and Francis group.
Lu, J. J., Murray, N. V. & Rosenthal, E. (1998). Framework for Automated Reasoning in Multiple-Valued Logics. Journal of Automated Reasoning, Vol. 21, No.1, 39–67, Aug. Kluwer Academic Publishers.
Luba, T. (1995). Decomposition of Multiple-Valued Functions. Proc. 25th Int. Symp. on Multiple-Valued Logic, 256–261. Bloomington, Indiana, USA (IEEE).
Md, H., Babu, H. & Sasao, T. (1999). Shared Multi-Valued Decision Diagrams for Multiple-Output Functions. Proc. 29th Int. Symp. on Multiple-Valued Logic, 166–172. Freiburg, Germany (IEEE).
Menger, K. S. (1969). A Transform for Logic Networks. IEEE Transactions on Computers, Vol. c-18, No.3, March (IEEE).
Muller, D. E. (1954). Application of Boolean Algebra to Switching Circuit Design and to Error Detection. IRE Trans. Electron. Computers, EC-3, 6–12 (IRE).
Oh, Y. G., Hong, H. P., Han, S. J., Chun, C. S. & Kin, B. K. (1996). Fuzzy Logic Utilization for the Diagnosis of Metallic Loose Part Impact in Nuclear Power Plant. Intelligent Systems and Soft Computing for Nuclear Science Industry. Proceedings of the 2nd International FLINS Workshop, 372–378. Moi, Belgium, World Scientific, Singapore.
Pawlak, Z. (1982). Rough Sets. International Journal of Computer and Information Sciences 11: 341–356. Plenum Press.
Pawlak, Z. (1991). Rough Sets. Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht, Boston, London.
Pawlak, Z., Grzymala-Busse, J. W., Slowinski, R. & Ziarko, W. (1995). Rough Sets. Communications of the ACM 38: 88–95. Association for Computing Machinery.
Pradhan, D. K. (1974). A Multi-Valued Algebra Based on Finite Fields. Proc. 4th Int. Symp. on Multiple-Valued Logic, 95–112. Morganstown, W.V., USA (IEEE).
Reed, I. S. (1954). A Class of Multiple-Error Correcting Codes and Their Decoding Scheme. IRE Trans. on Inform. Theory 4: 38–42 (IRE).
Stankovic, R. S. (1995). Functional Decision Diagrams for Multiple-Valued Functions. Proc. 25th Int. Symp. on Multiple-Valued Logic, 284–289. Bloomington, Indiana, USA (IEEE).
Stankovic, R. S., Jankovic, D. & Moraga, C. (1998). Reed-Muller-Fourier Versus Galois Field Representation of Four-Valued Logic Functions. Proc. 28th Int. Symp. on Multiple-Valued Logic, 186–191. Fukuoka, Japan (IEEE).
Steinbach, B., Perkowski, M. A. & Lang, C. (1999). Bi-Decompositions of Multi-Valued Functions for Circuit Design and Data Mining Applications. Proc. 29th Int. Symp. on Multiple-Valued Logic, 50–58. Freiburg, Germany (IEEE).
Tang, Z., Cao, Q. & Ishizuka, O. (1998). Learning Multiple-Valued Logic Network: Algebra, Algorithm, and Applications. IEEE Transactions on Computers, Vol. 47, No.2, 247–251, Feb. (IEEE).
Wesselkamper, T. C. (1978). Divided Difference Methods in Galois Switching Functions. IEEE Transactions on Computers, C-27, No.3, March (IEEE).
Wesselkamper, T. C. & Danowitz, J. (1995). Some New Results for Multiple-Valued Genetic Algorithms. Proc. 25th Int. Symp. on Multiple-Valued Logic, 264–269. Bloomington, Indiana, USA (IEEE).
Zilic, Z. & Vranesic, Z. G. (1995). Reed-Muller Forms for Incompletely Specified Functions Via Sparse Polynomial Interpolation. Proc. 25th Int. Symp. on Multiple-Valued Logic, 36–43. Bloomington, Indiana, USA (IEEE).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Adams, K., Bell, D., Maguire, L. et al. Knowledge Discovery from Decision Tables by the Use of Multiple-Valued Logic. Artificial Intelligence Review 19, 153–176 (2003). https://doi.org/10.1023/A:1022617312306
Issue Date:
DOI: https://doi.org/10.1023/A:1022617312306