TOPOLOGICAL CHARACTERIZATION FOR FUZZY REGULAR LANGUAGES

Document Type: Research Paper

Authors

1 College of Computer Science, Shaanxi Normal University, Xi'an 710062, P.R. China and College of Mathematics and Computation, Anqing Normal University, Anqing 246013, P.R. China

2 College of Computer Science, Shaanxi Normal University, Xi'an 710062, P.R. China

Abstract

We present a topological characterization for fuzzy regular languages:
we show that there is a bijective correspondence between fuzzy regular languages
and the set of all clopen fuzzy subsets with finite image in the induced fuzzy topological space of Stone space (Profinite space), and then we give a representation of closed fuzzy subsets in the induced fuzzy topological space via fuzzy regular languages. 
Moreover, we prove that the induced fuzzy topological space has a basis consisting of leveled characteristic functions of the closure of cut languages of fuzzy regular languages.

Keywords


[1] J. Almeida, Residually nite congrences and quasi-regular subsets in uniform algebras, Par-
tugalie Mathematicae, 46 (1989), 313{328.
[2] J. Almeida, Finite Semigroups and Universal Algebra, World Scientfii c , Singapore, 1994.

[3] C. L. Chang, Fuzzy topological space, Journal of Mathematical Analysis and Applications,
24 (1968), 182{190.
[4] M. Ciric, J. Ignjatovic, I. Jancic and N. Damljanovic, Computation of the greatest simulations
and bisimulations between fuzzy automata, Fuzzy Sets and Systems, 208 (2012), 22{42.
[5] M. Gehrke, S. Grigorieff and J. E. Pin, Duality and equational theory of regular languages,
Lect. Notes Comp.Sci. (ICALP), 5152 (2008), 246{257.
[6] M. Gehrke, Stone duality and the recognisable languages over an algebra, Algebra and Coal-
gebra in Computer Science Lecture Notes in Computer Science, 5728 (2009), 236{250.
[7] M. Gehrke, Duality and recognization, 36th International Conference on Mathematical Foun-
dations of Computer Science, Spinger-Verlag, 6907 (2011), 3{18.
[8] M. Hall, A topology for free groups and related groups, Annals of Mathematics, 52 (1950),
127{139.
[9] J. Ignjatovic, M. Ciric and S. Bogdanovic, Determinization of fuzzy automata with member-
ship values in complete residuated lattices, Information Sciences, 178 (2008), 164{180.
[10] J. Ignjatovic, M. ciric, S. Bogdanovic and T. Petkovic, Myhill-Nerode type theory for fuzzy
languages and automata, Fuzzy Sets and Systems, 161 (2010), 1288{1324.
[11] Z. Jancic and M. Ciric, Brzozowski type determinization for fuzzy automata, Fuzzy Sets and
Systems, 249 (2014), 73{82.
[12] Y. M. Liu and M. K. Luo, Fuzzy Topology, World Scientifi c Press, Sigapore, 1997.
[13] Y. M. Liu and D. X. Zhang, Lowen Spaces, Math. Anal. Appl., 241 (2000), 30{38.
[14] R. Lowen, Fuzzy topological spaces and fuzzy compactness, Math. Anal. Appl., 56 (1976),
621{633.
[15] E. T. Lee and L. A. Zadeh, Note on fuzzy languages, Information Sciences, 1 (1969), 421{434.
[16] Y. M. Li and W. Pedrycz, Fuzzy fi nite automata and fuzzy regular expressions with member-
ship values in lattice-ordered monoids, Fuzzy Sets and Systems, 156 (2005), 68{92.
[17] J. N. Mordeson and D. S. Malik, Fuzzy Automata and Languages: Theory and Applications,
Boca Raton, London: Chapman & Hall/CRC, 2002.
[18] J. E. Pin, Profi nite methods in automata theory, 26th International Symposium on Theoret-
ical Aspects of Computer Science, (2009), 31{50.
[19] J. E. Pin and H. Straubing, Some results on C-varieties, Theoret. Informatics Appl., 39
(2005), 239{262.
[20] H. Y. Pan, Y. M. Li, Y. Z. Cao and P. Li, Nondeterministic fuzzy automata with membership
values in complete residuated lattices, International Journal of Approximate Reasoning, 82
(2017), 22{38.
[21] N. Pippenger, Regular languages and stone duality, Theory of Computing Systems, 30 (1989),
121{134.
[22] D. W. Qiu, Automata theory based on complete residuated lattice-valued logic (I), Science in
China, 44 (2001), 419{429.
[23] D. W. Qiu, Automata theory based on complete residuated lattice-valued logic (II), Science
in China, 45 (2002), 442{452.
[24] D. W. Qiu, Pumping lemma in automata theory based on complete residuated lattice-value
logic: A note, Fuzzy Sets and Systems, 157 (2006), 2128{2138.
[25] E. S. Santos, Maximin automata, Inf. Control., 12 (1968), 367{377.
[26] E. S. Santos, On reduction of max-min machines, J. Math. Anal. Appl., 37 (1972), 677{686.
[27] S. Shirali and H. L. Vasudeva, Metric Spaces, Springer, 2006.
[28] A. K. Srivastava and S. P. Tiwari, A topology for fuzzy automata, in: Proc. 2002 AFSS
Internat. Conf. on Fuzzy Systems, Lecture Notes in Artfii cial Intelligence, Springer, Berlin,
2275 (2002), 485{490.
[29] A. K. Srivastava and S. P. Tiwari, On relationships among fuzzy approximation operators,
fuzzy topology, and fuzzy automata, Fuzzy Sets and Systems, 138 (2003), 197{204.
[30] S. P. Tiwari and A. K. Singh and S. Sharan and V. K. Yadav, Bifuzzy core of fuzzy automata,
Iranian Journal of Fuzzy Systems, 12(2) (2015), 63-73.

[31] S. Torunczyk, Languages of Pro nite Words and the Limitedness Problem, PhD thesis, Uni-
versity of Warsaw, 2011.
[32] G. J. Wang, Theory of L-Fuzzy Topology Spaces, Shaanxi Normal University Press, 1988.