Paper:

# Fuzzy Concept Lattices Constrained by Hedges

## Radim Belohlavek^{*,**} and Vilem Vychodil^{**}

^{*}Dept. Systems Science and Industrial Engineering, T. J. Watson School of Engineering and Applied Science Binghamton University - SUNY, Binghamton, NY 13902-6000, USA

^{**}Dept. Computer Science, Palacky University, Olomouc, Tomkova 40, CZ-779 00 Olomouc, Czech Republic

We study concept lattices constrained by hedges. The principal aim is to control, in a parameterical way, the size of concept lattices, i.e. the number of conceptual clusters extracted from data. The paper presents theoretical insight, comments, and examples. We introduce new, parameterized, concept-forming operators and study their properties. We obtain an axiomatic characterization of the concept-forming operators. Then, we show that a concept lattice with hedges is indeed a complete lattice which is isomorphic to an ordinary concept lattice. We describe the isomorphism and its inverse. These mappings serve as translation procedures. As a consequence, we obtain a theorem characterizing the structure of concept lattices with hedges which generalizes the well-known main theorem of ordinary concept lattices. Furthermore, the isomorphism and its inverse enable us to compute a concept lattice with hedges using algorithms for ordinary concept lattices. Further insight is provided for boundary choices of hedges. We demonstrate by experiments that the size reduction using hedges as parameters is smooth.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.11, No.6, pp. 536-545, 2007.

- [1] R. Belohlavek, “Fuzzy Galois connections,” Math. Logic Quarterly, 45, (4), pp. 497-504, 1999.
- [2] R. Belohlavek, “Reduction and a simple proof of characterization of fuzzy concept lattices,” Fundamenta Informaticae, 46, (4), pp. 277-285, 2001.
- [3] R. Belohlavek, “Fuzzy Relational Systems: Foundations and Principles,” Kluwer, Academic/Plenum Publishers, New York, 2002.
- [4] R. Belohlavek, “Concept lattices and order in fuzzy logic,” Ann. Pure Appl. Logic, 128, pp. 277-298, 2004.
- [5] R. Belohlavek and T. Funiokova, “Fuzzy interior operators,” Int. J. General Systems, 33, (4), pp. 315-330, 2004.
- [6] R. Belohlavek, T. Funioková, and V. Vychodil, “Galois connections with hedges,” In: Proc. 11th IFSA Congress 2005, pp. 1250-1255, Springer.
- [7] R. Belohlavek, V. Sklenar, and J. Zacpal, “Crisply Generated Fuzzy Concepts,” In: B. Ganter and R. Godin (Eds.), ICFCA 2005, LNCS, 3403, pp. 268-283, Springer-Verlag, Berlin/Heidelberg, 2005.
- [8] R. Belohlavek and V. Vychodil, “Reducing the size of fuzzy concept lattices by hedges,” In: FUZZ-IEEE 2005, The IEEE International Conference on Fuzzy Systems, May 22-25, 2005, Reno (Nevada, USA), pp. 663-668.
- [9] R. Belohlavek and V. Vychodil, “What is a fuzzy concept lattice?,” In: Proc. CLA 2005, 3rd Int. Conference on Concept Lattices and Their Applications, September 7-9, 2005, Olomouc, Czech Republic, pp. 34-45,

URL: http://ceur-ws.org/Vol-162/. - [10] R. Belohlavek and V. Vychodil, “Attribute implications in a fuzzy setting,” In: R. Missaoui and J. Schmid (Eds.), ICFCA 2006, LNAI, 3874, pp. 45-60, 2006.
- [11] S. B. Yahia and A. Jaoua, “Discovering knowledge from fuzzy concept lattice,” In: A. Kandel, M. Last, and H. Bunke, “Data Mining and Computational Intelligence,” pp. 167-190, Physica-Verlag, 2001.
- [12] A. Burusco and R. Fuentes-Gonzáles, “The study of the L-fuzzy concept lattice,” Mathware & Soft Computing, 3, pp. 209-218, 1994.
- [13] C. Carpineto and G. Romano, “Concept Data Analysis,” Theory and Applications, J. Wiley, 2004.
- [14] B. Ganter and R. Wille, “Formal Concept Analysis,” Mathematical Foundations, Springer, Berlin, 1999.
- [15] G. Gerla, “Fuzzy Logic,” Mathematical Tools for Approximate Reasoning, Kluwer, Dordrecht, 2001.
- [16] J. A. Goguen, “The logic of inexact concepts,” Synthese, 18, pp. 325-373, 1968-9.
- [17] P. Hájek, “Metamathematics of Fuzzy Logic,” Kluwer, Dordrecht, 1998.
- [18] P. Hájek, “On very true,” Fuzzy Sets and Systems, 124, pp. 329-333, 2001.
- [19] G. J. Klir and B. Yuan, “Fuzzy Sets and Fuzzy Logic,” Theory and Applications, Prentice Hall, 1995.
- [20] S. Krajči, “Cluster based efficient generation of fuzzy concepts,” Neural Network World, 5, pp. 521-530, 2003.
- [21] S. Krajči, “A generalized concept lattice,” Logic Journal of IGPL, 13, (5), pp. 543-550, 2005.
- [22] D. Maier, “The Theory of Relational Databases,” Computer Science Press, Rockville, 1983.
- [23] O. Ore, “Galois connections,” Trans. Amer. Math. Soc., 55, pp. 493-513, 1944.
- [24] S. Pollandt, “Fuzzy Begriffe,” Springer, Berlin, 1997.
- [25] G. Snelting and F. Tip, “Understanding class hierarchies using concept analysis,” ACM Trans. Program. Lang. Syst., 22, (3), pp. 540-582, May 2000.
- [26] G. Takeuti and S. Titani, “Globalization of intuitionistic set theory,” Annals of Pure and Applied Logic, 33, pp. 195-211, 1987.
- [27] L. A. Zadeh, “The concept of a linguistic variableand its application to approximate reasoning I, II, III,” Inf. Sci., 8, (3), pp. 199-251, 1975; pp. 301-357; 9, pp. 43-80, 1975.