Document Type : Research Paper
Authors
1 Department of Mathematics, Graduate University of Advanced Technology, Kerman, Iran
2 [Automata, Fuzzy automata] Department of Mathematics, Kerman Graduate University of Advanced Technology.
Abstract
In this article, after the definitions of the reduced hesitant L-fuzzy automaton (RHLFA) and the minimal hesitant L-fuzzy automaton; we convert a hesitant L-fuzzy automaton (HLFA) to an RHLFA by reducing the number of its states such that its language is equal to the original HLFA language. Then, by defining an equivalence relation on the monoid X*,we construct an HLFA whose language is equal to the language of the transformed RHLFA, and we show that this HLFA is minimal. In conclusion, we delineate the criteria under which, the number of states in the minimal HLFA is equal to the number of states in the RHLFA
Keywords
Main Subjects
[1] N. C. Basak, A. Gupta, On quotient machines of a fuzzy automata and the minimal machine, Fuzzy Sets and
Systems, 125(2) (2002), 223-229. https://doi.org/10.1016/S0165-0114(01)00064-1
Systems, 125(2) (2002), 223-229. https://doi.org/10.1016/S0165-0114(01)00064-1
[2] B. Bedregal, R. Reiser, H. Bustince, C. Lopez-Molina, V. Torra, Aggregation functions for typical hesitant fuzzy
elements and the action of automorphisms, Information Sciences, 255 (2014), 82-99. https://doi.org/10.1016/
j.ins.2013.08.024
elements and the action of automorphisms, Information Sciences, 255 (2014), 82-99. https://doi.org/10.1016/
j.ins.2013.08.024
[3] W. Cheng, Z. W. Mo, Minimization algorithm of fuzzy finite automata, Fuzzy Sets and Systems, 141 (2004), 439-448.
https://doi.org/10.1016/S0165-0114(02)00607-3
https://doi.org/10.1016/S0165-0114(02)00607-3
[4] A. Choubey, K. Ravi, Intuitionistic fuzzy automata and intuitionistic fuzzy regular expressions, Journal of Applied
Mathematics and Informatics, 27(1-2) (2009), 409-417.
Mathematics and Informatics, 27(1-2) (2009), 409-417.
[5] V. S. Costa, B. Bedregal, On typical hesitant fuzzy automata, Soft Computing, 24(12) (2020), 8725-8736. https:
//doi.org/10.1007/s00500-020-04896-7
//doi.org/10.1007/s00500-020-04896-7
[6] A. H. Dehmiry, M. Mashinchi, R. Mesiar, Hesitant L-fuzzy sets, International Journal of Intelligent Systems, 33
(2018), 1027-1042. https://doi.org/10.1002/int.21910
(2018), 1027-1042. https://doi.org/10.1002/int.21910
[7] J. R. Gonzalez de Mendivil, Conditions for minimal fuzzy deterministic finite automata via brzozowski’s procedure,
IEEE Transactions on Fuzzy Systems, 26 (2018), 2409-2420. https://doi.org/10.1109/TFUZZ.2017.2775601
IEEE Transactions on Fuzzy Systems, 26 (2018), 2409-2420. https://doi.org/10.1109/TFUZZ.2017.2775601
[8] A. G. D. M. Grau, S. Stanimirovic, F. Farina, Minimal determinization algorithm for fuzzy automata, IEEE Transactions
on Fuzzy Systems, 31 (2023), 3812-3822. https://doi.org/10.1109/TFUZZ.2023.3268406
on Fuzzy Systems, 31 (2023), 3812-3822. https://doi.org/10.1109/TFUZZ.2023.3268406
[9] S. Halamish, O. Kupferman, Minimizing deterministic lattice automata, ACM Transactions on Computational Logic,
6604 (2015), 199-213. https://doi.org/10.1007/978-3-642-19805-2_14
6604 (2015), 199-213. https://doi.org/10.1007/978-3-642-19805-2_14
[10] J. Ignjatovic, M. Ciric, S. Bogdanovic, T. Petkovic, Myhill-Nerode type theory for fuzzy languages and automata,
Fuzzy Sets and Systems, 161 (2010), 1288-1324. https://doi.org/10.1016/j.fss.2009.06.007
Fuzzy Sets and Systems, 161 (2010), 1288-1324. https://doi.org/10.1016/j.fss.2009.06.007
[11] Z. Jancic, M. Ciric, Brzozowski type determinization for fuzzy automata, Fuzzy Sets and Systems, 249 (2014),
73-82. https://doi.org/10.1016/j.fss.2014.02.021
73-82. https://doi.org/10.1016/j.fss.2014.02.021
[12] Y. B. Jun, Quotient structures of intuitionistic fuzzy finite state machines, Information Sciences, 177 (2007),
4977-4986. https://doi.org/10.1016/j.ins.2007.06.008
4977-4986. https://doi.org/10.1016/j.ins.2007.06.008
[13] V. Ladimir, V. Topencharov, K. Peeva, Equivalence, reduction and minimization on finite fuzzy automata, Journal
of Mathematica Analysis and Applications, 84 (1981), 270-281. https://doi.org/10.1016/0022-247X(81)
90165-7
of Mathematica Analysis and Applications, 84 (1981), 270-281. https://doi.org/10.1016/0022-247X(81)
90165-7
[14] H. S. Lee, Minimizing fuzzy finite automata, 2000. Fuzzy Systems, FUZZ IEEE 2000. The Ninth IEEE International
Conference on, 1 (2000), 65-70. https://doi.org/10.1109/FUZZY.2000.838635
Conference on, 1 (2000), 65-70. https://doi.org/10.1109/FUZZY.2000.838635
[15] E. T. Lee, L. A. Zadeh, Note on fuzzy languages, Information Sciences, 1 (1969), 421-434. https://doi.org/10.
1016/0020-0255(69)90025-5
1016/0020-0255(69)90025-5
[16] Y. Li, W. Pedrycz, Minimization of lattice finite automata and its application to the decomposition of lattice
languages, Fuzzy Sets and Systems, 158 (2007), 1423-1436. https://doi.org/10.1016/j.fss.2007.03.003
languages, Fuzzy Sets and Systems, 158 (2007), 1423-1436. https://doi.org/10.1016/j.fss.2007.03.003
[17] L. Li, D. Qiu. On the state minimization of fuzzy automata, IEEE Transactions on Fuzzy Systems, 23 (2015),
434-443. https://doi.org/10.1109/TFUZZ.2014.2315620
434-443. https://doi.org/10.1109/TFUZZ.2014.2315620
[18] A. Mateescu, A. Salomaa, K. Salomaa, S. Yu, Lexical analysis with a simple finite fuzzy automaton model, Journal
of Universal Computer Science, 1 (1995), 292-311. https://doi.org/10.1007/978-3-642-80350-5_28
of Universal Computer Science, 1 (1995), 292-311. https://doi.org/10.1007/978-3-642-80350-5_28
[19] S. Moghari, M. M. Zahedi, Minimization of deterministic fuzzy tree automata, Journal of Fuzzy Set-Valued Analysis,
2014 (2014), 1-18. https://doi.org/10.5899/2014/jfsva-00191
2014 (2014), 1-18. https://doi.org/10.5899/2014/jfsva-00191
[20] J. N. Mordeson, D. S. Malik, Fuzzy automata and languages theory and applications, Chapman Hall, CRC Boca
Raton, London, New York, Washington DC, 2002.
Raton, London, New York, Washington DC, 2002.
[21] D. Qiu, Supervisory control of fuzzy discrete event systems: A formal approach, IEEE Transactions on Systems,
Man and Cybernetics Part B, 35 (2005), 72-88. https://doi.org/10.1109/TSMCB.2004.840457
Man and Cybernetics Part B, 35 (2005), 72-88. https://doi.org/10.1109/TSMCB.2004.840457
[22] K. Ravi, A. Choubey, Interval-valued fuzzy regular language, Journal of Applied Mathematics and Informatics,
28(3-4) (2010), 639-649.
28(3-4) (2010), 639-649.
[23] K. Ravi, A. Choubey, Myhill-Nerode theorem for interval-valued fuzzy regular language, In: AIP Conference
Proceedings. American Institute of Physics, Chandigarh, 1324(1) (2010), 30-33. https://doi.org/10.1063/1.
3526220
Proceedings. American Institute of Physics, Chandigarh, 1324(1) (2010), 30-33. https://doi.org/10.1063/1.
3526220
[24] M. Shamsizadeh, M. M. Zahedi, Intuitionistic general fuzzy automata, Soft Computing, 20 (2016), 3505-3519.
https://doi.org/10.1007/s00500-015-1969-x
https://doi.org/10.1007/s00500-015-1969-x
[25] M. Shamsizadeh, M. M. Zahedi, Bisimulation of type 2 for BL-general fuzzy automata, Soft Computing, 23 (2019),
9843-9852. https://doi.org/10.1007/s00500-019-03812-y
9843-9852. https://doi.org/10.1007/s00500-019-03812-y
[26] M. Shamsizadeh, M. M. Zahedi, On reduced fuzzy multiset finite automata, Soft Computing, 22 (2022), 13381-
13390. https://doi.org/10.1007/s00500-022-07549-z
13390. https://doi.org/10.1007/s00500-022-07549-z
[27] M. Shamsizadeh, M. M. Zahedi, Kh. Abolpour, Reduction of BL-general L-fuzzy automata, Iranian Journal of
Mathematical Sciences and Informatics, 18(1) (2023), 131-144. https://doi.org/10.52547/ijmsi.18.1.131
Mathematical Sciences and Informatics, 18(1) (2023), 131-144. https://doi.org/10.52547/ijmsi.18.1.131
[28] M. Shamsizadeh, M. M. Zahedi, M. J. Agheli Goki, Minimization of hesitant L-fuzzy automata, Fuzzy Set and
System, 477 (2024), 108793. https://doi.org/10.1016/j.fss.2023.108793
System, 477 (2024), 108793. https://doi.org/10.1016/j.fss.2023.108793
[29] A. K. Srivastava, S. P. Tiwari, IF-topologies and IF-automata, Soft Computing, 14 (2010), 571-578. https://doi.
org/10.1007/s00500-009-0427-z
org/10.1007/s00500-009-0427-z
[30] S. P. Tiwari, Sh. Sharan, Fuzzy automata based on lattice-ordered monoid with algebraic and topological aspects,
Fuzzy Information and Engineering, 4 (2012), 155-164. https://doi.org/10.1007/s12543-012-0108-y
Fuzzy Information and Engineering, 4 (2012), 155-164. https://doi.org/10.1007/s12543-012-0108-y
[31] S. P. Tiwari, A. K. Srivastava, On a decomposition of fuzzy automata, Fuzzy Sets and Systems, 151 (2005), 503-511.
https://doi.org/10.1016/j.fss.2004.06.014
https://doi.org/10.1016/j.fss.2004.06.014
[32] V. Torra, Hesitant fuzzy sets, International Journal of Intelligent Systems, 25 (2010), 529-539. https://doi.org/
10.1002/int.20418
10.1002/int.20418
[33] V. Torra, Y. Narukawa, On hesitant fuzzy sets and decision, 2009 IEEE International Conference on Fuzzy Systems.
IEEE, (2009), 1378-1382. https://doi.org/10.1109/FUZZY.2009.5276884
IEEE, (2009), 1378-1382. https://doi.org/10.1109/FUZZY.2009.5276884
[34] A. Tripathi, R. Kaur, A review of state minimization and state reduction techniques in fuzzy automata, AIP
Conference Proceedings, 2061 (2019), 020-033. https://doi.org/10.1063/1.5086655
Conference Proceedings, 2061 (2019), 020-033. https://doi.org/10.1063/1.5086655
[35] W. G. Wee, On generalization of adaptive algorithm and application of the fuzzy sets concept to pattern classification,
Ph.D. Thesis, Purdue University, Lafayette, IN, 1967.
Ph.D. Thesis, Purdue University, Lafayette, IN, 1967.
[36] L. Wu, D. Qiu, Automata theory based on complete residuated lattice-valued logic: Reduction and minimization,
Fuzzy Sets and Systems, 161 (2010), 1635-1656. https://doi.org/10.1016/j.fss.2009.12.011
Fuzzy Sets and Systems, 161 (2010), 1635-1656. https://doi.org/10.1016/j.fss.2009.12.011
[37] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965,) 338-353. https://doi.org/10.1016/S0019-9958(65)
90241-X
90241-X
)