Journal of Information Technology Services (한국IT서비스학회지)

Korea Society of IT Services (한국IT서비스학회)
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Improved Computation of L-Classes for Efficient Computation of J Relations

효율적인 J 관계 계산을 위한 L 클래스 계산의 개선

  • 한재일 (국민대학교 전자정보통신대학 컴퓨터공학부) ;
  • 김영만 (국민대학교 전자정보통신대학 컴퓨터공학부)
  • Received : 2010.10.24
  • Accepted : 2010.12.16
  • Published : 2010.12.31
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Abstract

The Green's equivalence relations have played a fundamental role in the development of semigroup theory. They are concerned with mutual divisibility of various kinds, and all of them reduce to the universal equivalence in a group. Boolean matrices have been successfully used in various areas, and many researches have been performed on them. Studying Green's relations on a monoid of boolean matrices will reveal important characteristics about boolean matrices, which may be useful in diverse applications. Although there are known algorithms that can compute Green relations, most of them are concerned with finding one equivalence class in a specific Green's relation and only a few algorithms have been appeared quite recently to deal with the problem of finding the whole D or J equivalence relations on the monoid of all $n{\times}n$ Boolean matrices. However, their results are far from satisfaction since their computational complexity is exponential-their computation requires multiplication of three Boolean matrices for each of all possible triples of $n{\times}n$ Boolean matrices and the size of the monoid of all $n{\times}n$ Boolean matrices grows exponentially as n increases. As an effort to reduce the execution time, this paper shows an isomorphism between the R relation and L relation on the monoid of all $n{\times}n$ Boolean matrices in terms of transposition. introduces theorems based on it discusses an improved algorithm for the J relation computation whose design reflects those theorems and gives its execution results.

Keywords

Acknowledgement

Supported by : 국민대학교

References

  1. 김창범, 한재일, "Classifications of D-classes in the semigroup $M_n$(F) of all nxn Boolean matrices over F = {0, 1}", 퍼지 및 지능시스템학회논문지, 제16권, 제3호(2006), pp.338-348.
  2. 한재일, "모든 lxn, nxm, rnxk 불리언 행렬 사이의 중첩곱셈에 대한 연구", 한국IT서비스학회지, 제5권, 제1호(2006), pp.191-198.
  3. 한재일, "효율적인 D-클래스 계산을 위한 알고리즘", 한국IT서비스학회지, 제6권, 제1호(2007), pp.151-158.
  4. 한재일, "불리언 행렬의 모노이드에서의 J관계 계산 알고리즘", 한국IT서비스학회지, 제7권, 제4호(2008), pp.221-230.
  5. Howie, J. M., An Introduction to semigroup theory, Oxford University Press, 1995.
  6. Linton, S. A., G. Pfeiffer, E. F. Robertson, and N. Ruskuc, "Computing Transformation Semigroups", Journal of Symbolic Computation, Vol.33(2002), pp.145-162. https://doi.org/10.1006/jsco.2000.0406
  7. Wall, J. R., "Green's relations for stochastic matrices", Czechoslovak Mathematical Journal, Vol.25, No.2(1975), pp.247-260.
  8. Heyworth, A., "One-Sided Noncommutative Grobner Bases with Applications to Computing Green's Relations", Journal of Algebra, Vol.242(2001), pp.401-416. https://doi.org/10.1006/jabr.2001.8801
  9. Lee, L., "Fast context-free grammar parsing require fast Boolean matrix multiplication", JACM, Vol.49, No.1(2002), pp.1-15. https://doi.org/10.1145/505241.505242
  10. Yi, X. et al., "Fast Encryption for Multimedia", IEEE Transactions on Consumer Electronics, Vol.47, No.1(2001), pp.101-107. https://doi.org/10.1109/30.920426
  11. Martin, D. F., "A Boolean matrix method for the computation of linear precedence functions", CACM, Vol.15, No.6(1972), pp.448-454. https://doi.org/10.1145/361405.361413
  12. Nakamura, Y. and T. Yoshimura, "A partitioning- based logic optimization method for large scale circuits with Boolean matrx", Proceedings of the 32nd ACM/IEEE conference on Design automation, (1995), pp.653-657.
  13. Pratt, V. R., "The Power of Negative Thinking in Multiplying Boolean matrices", Proceedigs of the annual ACM symposium on Theory of computing, (1974), pp.80-83.
  14. Rim, D. S. and J. B. Kim, "Tables of D-Classes in the semigroup B of the binary relations on a set X with n-elements", Bull. Korea Math Soc., Vol.20, No.1(1983) , pp.9-13.
  15. Atkinson, D. M., N. Santoro, and J. Urrutia, "On the integer complexity of Boolean matrix multiplication", ACM SIGACT News, Vol.18 No.1(1986), p.53 https://doi.org/10.1145/8312.8316
  16. Yelowitz, L., "A Note on the Transitive Closure of a Boolean Matrix", ACM SIGMOD Record, Vol.25, No.2(1978), p.30.
  17. Comstock, D. R., "A note on multiplying Boolean matrices II", CACM, Vol.7 No.1 (1964), p.13. https://doi.org/10.1145/363872.363891
  18. Macii, E., "A Discussion of Explicit Methods for Transitive Closure Computation Based on Matrix Multiplication", 29th Asilom Conference on Signals, Systems and Computers, Vol.2(1995), pp.799-801.
  19. Angluin, D., "The four Russians' algorithm for boolean matrix multiplication is optimal in its class", ACM SIGACT News, Vol.8, No.1(1976), pp.29-33. https://doi.org/10.1145/1008591.1008593
  20. Booth, K. S., "Boolean matrix multiplication using only bit operations", ACM SIGACT News, Vol.9, No.3(1977), p.23. https://doi.org/10.1145/1008361.1008362
  21. Cousineau, G., J. F. Perrot, and J. M. Rifflet, "APL programs for direct computation of a finite semigroup", APL Congres, Vol.73(1973), pp.67-74.
  22. Lallement, G., Semigroups and Combinatorial Applications, John Wiley and Sons, New York, 1979.
  23. Champarnaud, J. M. and G. Hansel, "AUTOMATE, a computing package for automata and finite semigroups", Journal of Symbolic Computation, Vol.12(1991), pp.197-220. https://doi.org/10.1016/S0747-7171(08)80125-3
  24. Sutner, K., "Finite State Machines and Syntatic Semigroups", The Mathematica Journal, Vol.2(1991), pp.78-87.
  25. Lallement, G. and R. McFadden, "On the determination of Green's relations in finite transformation semigroups," Journal of Symbolic Computation, Vol.10(1990), pp.481-498. https://doi.org/10.1016/S0747-7171(08)80057-0
  26. Konieczny, J., "Green's equivalences in finite semigroups of binary relations", Semigroup Form, Vol.48(1994), pp.235-252. https://doi.org/10.1007/BF02573672
  27. Plemmons, R. J. and M. T. West, "On the semigroup of binary relations", Pacific Journal of Mathematics, Vol.35(1970), pp.743-753. https://doi.org/10.2140/pjm.1970.35.743
  28. Simon, I., "On semigroups of matrices over the tropical semiring", Informatique Theorique et Applications, Vol.28(1994), pp.277-294. https://doi.org/10.1051/ita/1994283-402771
  29. Straubing, H., "The Burnside problem for semigroups of matrices", in Combinatorics on Words, Progress and Perspectives, L. J. Cummings(ed.), Academic Press, (1983), pp.279-295.
  30. Froidure, V. and J. Pin, "Algorithms for computing finite semigroups", Foundations of Computational Mathematics, (1997), pp.112-126.