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Fuzzy Kolmogorov Complexity Based on Fuzzy Decompression Algorithms and Its Application to Fuzzy Data Mining

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13087))

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Abstract

We propose a new practical fuzzification of classical Kolmogorov complexity to measure the minimum amount of fuzzy algorithmic information needed to produce generic fuzzy data and we then cultivate the foundation of such a fuzzification. In this work, we view Kolmogorov complexity as a measure indicating the size of the maximally compressed data, from which the best possible decompressor algorithmically recovers the original data. As such decompressors, we use a generalized model of deterministic fuzzy Turing machines of Yamakami [SCIS 2014 & ISIS 2014, pp. 29–35] and introduce the notion of fuzzy Kolmogorov complexity of generic fuzzy data based on this computational model. As a direct application to data-mining issues, such as clustering and classification, we provide a set of new practical information distances induced by our notion of fuzzy Kolmogorov complexity.

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Notes

  1. 1.

    Fuzzy strings, which were called fuzzy languages in [6], are limited to n-valued fuzzy subsets. The reader should not be confused with the terminology of this work and the ones in [6, 18, 21].

  2. 2.

    The reader should not be confused with the terminology of this work and the terminology used in [6, 18, 21].

  3. 3.

    A nonnegative function \(g:\varSigma ^*\times \varSigma ^*\rightarrow \mathbb {R}\) is called a metric if (i) g satisfies the identity property (i.e., \(g(x,y)=0\) iff \(x=y\)), (ii) g is symmetric, and (iii) g satisfies the triangle inequality.

References

  1. Bedregal, B.C., Figueira, S.: On the computing power of fuzzy Turing machines. Fuzzy Sets Syst. 159, 1072–1083 (2008)

    Article  MathSciNet  Google Scholar 

  2. Bernstein, E., Vazirani, U.: Quantum compelxity theory. SIAM J. Comput. 26, 1411–1473 (1997)

    Article  MathSciNet  Google Scholar 

  3. Berthiaume, A., van Dam, W., Laplante, S.: Quantum Kolmogorov complexity. J. Comput. System Sci. 63, 201–221 (2001)

    Article  MathSciNet  Google Scholar 

  4. Castro, J.L., Delgado, M., Mantas, C.J.: A new approach for the execution and adjustment of a fuzzy algorithm. Fuzzy Sets Syst. 121, 491–503 (2001)

    Article  MathSciNet  Google Scholar 

  5. Chaitin, G.: On the length of programs for computing finite binary sequences. J. ACM 13, 547–569 (1966)

    Article  MathSciNet  Google Scholar 

  6. Dai, S.: Fuzzy Kolmogorov complexity based on a classical description. Entropy 22, article 66 (2020)

    Google Scholar 

  7. Doostfatemeh, M., Kremer, S.: New directions in fuzzy automata. Int. J. Approx. Reason. 38, 175–214 (2005)

    Article  MathSciNet  Google Scholar 

  8. Doty, D., Moser, P.: Feasible depth. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) CiE 2007. LNCS, vol. 4497, pp. 228–237. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73001-9_24

    Chapter  Google Scholar 

  9. Elkan, C.: Using the triangle inequality to accelerate k-means. In: Proceedings of ICML 2003, pp. 147–153. AAAI Press (2003)

    Google Scholar 

  10. Faloutsos, C., Megalooikonomou, V.: On data mining, compression, and Kolmogorov complexity. Data Min. Knowl. Disc. 15, 3–20 (2007)

    Article  MathSciNet  Google Scholar 

  11. Gács, P.: Quantum algorithmic entropy. J. Phys. A: Math. Gen. 34, 6859–6880 (2001)

    Article  MathSciNet  Google Scholar 

  12. Hanss, M.: Applied Fuzzy Arithmiteic: An Introduction with Engineering Applications. Springer, Berlin (2010)

    Google Scholar 

  13. Jordon, L., Moser, P.: On the difference between finite-state and pushdown depth. In: Chatzigeorgiou, A., et al. (eds.) SOFSEM 2020. LNCS, vol. 12011, pp. 187–198. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-38919-2_16

    Chapter  MATH  Google Scholar 

  14. Keogh, E., Lonardi, S., Wei, L., Ratanamahatana, C.A., Lee, S.H., Handley, J.: Compression-based data mining of sequential data. Data Min. Knowl. Disc. 14, 99–129 (2007)

    Article  MathSciNet  Google Scholar 

  15. Kolmogorov, A.: Three approaches to the quantitative definition of information. Probl. Inform. Transm. 1, 1–7 (1965)

    Google Scholar 

  16. Lee, E.T., Zadeh, L.A.: Note on fuzzy languages. Inf. Sci. 1, 421–431 (1969)

    Google Scholar 

  17. Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn. Springer-Science, New York (2008)

    Google Scholar 

  18. Li, Y.: Fuzzy Turing machines: variants and universality. IEEE Trans. Fuzzy Syst. 16, 1491–1502 (2008)

    Article  Google Scholar 

  19. Luca, A.D., Termini, S.: Entropy of L-fuzzy set. Inform. Control 24, 55–73 (1974)

    Article  MathSciNet  Google Scholar 

  20. Moniri, M.: Fuzzy and intuitionistic fuzzy Turing machines. Fundamenta Informaticae 123, 305–315 (2013)

    Article  MathSciNet  Google Scholar 

  21. Mordenson, J.N., Malik, D.S.: Fuzzy Automata and Languages: Theory and Applications. Chapman and Hall, London, U.K. (2002)

    Book  Google Scholar 

  22. Needham, S., Dowe, D.: Message length as an effective Ockham’s razor in decision tree induction. In: Proceedings of AISTATS 2001, pp. 216–223 (2001)

    Google Scholar 

  23. Pal, N., Pal, S.: Higher order fuzzy entropy and hybrid entropy of a fuzzy set. Inf. Sci. 61, 211–221 (1992)

    Article  MathSciNet  Google Scholar 

  24. Rissanen, J.J.: Modeling by the shortest data description. Automatica 14, 465–471 (1978)

    Article  Google Scholar 

  25. Santos, E.S.: Fuzzy algorithms. Inform. Control 17, 326–339 (1970)

    Article  MathSciNet  Google Scholar 

  26. Solomonoff, R.A.: A formal theory of inductive inference, part I. Inform. Control 7, 1–22 (1964)

    Article  MathSciNet  Google Scholar 

  27. Vitányi, P.M.: Quantum Kolmogorov compelxity based on classical descriptions. IEEE Trans. Inform. Theory 47, 2464–2479 (2001)

    Article  MathSciNet  Google Scholar 

  28. Wallace, C.S., Boulton, D.M.: An information measure for classification. Comput. J. 11, 185–195 (1968)

    Article  Google Scholar 

  29. Wang, H., Qiu, D.: Computing with words via Turing machines: a formal approach. IEEE Trans. Fuzzy Syst. 11, 742–753 (2003)

    Article  Google Scholar 

  30. Wiedermann, J.: Characterizing the super-turing computing power and efficiency of classical fuzzy Turing machines. Theor. Comput. Sci. 317, 61–69 (2004)

    Article  MathSciNet  Google Scholar 

  31. Yager, R.: On the measure of fuzziness and negation, part I: membership in the unit interval. Int. J. Gen. Syst. 5, 221–229 (1979)

    Google Scholar 

  32. Yamakami, T.: The world of combinatorial fuzzy problems and the efficiency of fuzzy approximation algorithms. In: Proceedings of the Joint Conference of SCIS 2014 and ISIS 2014, pp. 29–35. IEEE (2014). arXiv: 1509.03057

  33. Yamakami, T.: Quantum logical depth and shallowness of streaming data by one-way quantum finite-state transducers (preliminary report). In: Proceedings of UCNC 2021. LNCS, vol. 12984, pp. 177–193. Springer (2021)

    Google Scholar 

  34. Zadeh, L.A.: Fuzzy sets. Inform. Control 8, 338–353 (1965)

    Article  Google Scholar 

  35. Zadeh, L.A.: Fuzzy algorithms. Inform. Control 12, 94–102 (1968)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Engineering, University of Fukui, 3-9-1 Bunkyo, Fukui, 910-8507, Japan

    Tomoyuki Yamakami

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  1. Tomoyuki Yamakami
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Editor information

Editors and Affiliations

  1. Nanjing University of Aeronautics and Astronautics, Nanjing, China

    Bohan Li

  2. University of Queensland, Brisbane, QLD, Australia

    Lin Yue

  3. University of Technology Sydney, Sydney, NSW, Australia

    Jing Jiang

  4. University of Queensland, Brisbane, QLD, Australia

    Weitong Chen

  5. University of Queensland, Brisbane, QLD, Australia

    Xue Li

  6. University of Technology Sydney, Sydney, NSW, Australia

    Guodong Long

  7. Carnegie Mellon University, Pittsburgh, USA

    Fei Fang

  8. Nanyang Technological University, Singapore, Singapore

    Han Yu

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Yamakami, T. (2022). Fuzzy Kolmogorov Complexity Based on Fuzzy Decompression Algorithms and Its Application to Fuzzy Data Mining. In: Li, B., et al. Advanced Data Mining and Applications. ADMA 2022. Lecture Notes in Computer Science(), vol 13087. Springer, Cham. https://doi.org/10.1007/978-3-030-95405-5_30

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  • DOI: https://doi.org/10.1007/978-3-030-95405-5_30

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-95404-8

  • Online ISBN: 978-3-030-95405-5

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