Algebraic properties of \(L\)-fuzzy finite automata. (English) Zbl 1284.68421
Summary: Fuzzy automata theory on lattice-ordered monoids was introduced by Li and Pedrycz. Dropping the distributive laws, fuzzy finite automata (L-FFAs for short) based on a more generalized structure \(L\), named a po-monoid, are presented and investigated from the view of algebra in this paper. The notions of (strong) successor and source operators, fuzzy successor and source operators which are shown to be closure operators on certain conditions are introduced and discussed in detail. Using the weak primary submachines, a unique decomposition theorem of a fuzzy finite automaton based on a lattice-ordered monoid is obtained. Taking \(L\) as a quantale, fuzzy subsystems are proved to be the same as fuzzy submachines of an L-FFA. In particular, intrinsic connections between algebraic properties of \(L\) and properties of some operators of an L-FFA are discovered. It is shown that the join-preserving property of fuzzy successor and source operators can be fully characterized by the right and left distributive laws respectively, and the idempotence of successor operator can be characterized equivalently by the nonexistence of zero divisors when \(L\) is a lattice-ordered monoid.
MSC:
| 68Q70 | Algebraic theory of languages and automata |
Keywords:
\(L\)-fuzzy finite automata; lattice-ordered monoid; successor; sub-machines; decomposition; equivalent characterizationsReferences:
| [1] | Asveld, P. R.J., Algebraic aspects of families of fuzzy languages, Theoret. Comput. Sci., 293, 417-445 (2003) · Zbl 1026.68078 |
| [2] | Bailador, G.; Trivino, G., Pattern recognition using temporal fuzzy automata, Fuzzy Sets Syst., 61, 37-55 (2009) · Zbl 1185.68588 |
| [3] | Bavel, Z., Introduction to the Theory of Automata (1983), Reston Publishing Company, Inc.: Reston Publishing Company, Inc. Reston, Virginia · Zbl 0517.68063 |
| [4] | Be˘lohlávek, R., Determinism and fuzzy automata, Inform. Sci., 143, 1, 205-209 (2002) · Zbl 1018.68040 |
| [5] | Gupta, M. M.; Saridis, G. N.; Gaines, B. R., Fuzzy Automata and Decision Processes (1977), North-Holland: North-Holland New York, pp. 133-175 · Zbl 0378.68035 |
| [6] | Holcombe, W. M.L., Algebraic Automata Theory (1982), Cambridge University Press: Cambridge University Press New York · Zbl 0489.68046 |
| [7] | Ignjatović, J.; Ćirić, M.; Bogdanović, S., Determinization of fuzzy automata with membership values in complete residuated lattices, Inform. Sci., 178, 164-180 (2008) · Zbl 1128.68047 |
| [8] | Kandel, A.; Lee, S. C., Fuzzy Switching and Automata: Theory and Applications (1980), Arnold: Arnold London |
| [9] | Kuroki, N.; Mordeson, J. N., Successor and source functions, J. Fuzzy Math., 5, 173-182 (1997) · Zbl 0868.68082 |
| [10] | Lei, H. X.; Li, Y. M., Minimization of states in automata theory based on finite lattice-ordered monoids, Inform. Sci., 177, 6, 1413-1421 (2007) · Zbl 1109.68058 |
| [11] | Li, Y. M.; Shi, Z. K., Remarks on uninorm aggregation operators, Fuzzy Sets Syst., 114, 377-380 (2000) · Zbl 0962.03052 |
| [12] | Li, Y. M.; Pedrycz, W., Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids, Fuzzy Sets Syst., 156, 68-92 (2005) · Zbl 1083.68059 |
| [13] | Li, Y. M.; Pedrycz, W., Minimization of lattice finite automata and its application to the decomposition of lattice languages, Fuzzy Sets Syst., 158, 1423-1436 (2007) · Zbl 1123.68063 |
| [14] | Li, Y. M., Analysis of Fuzzy Systems (2005), Science Press: Science Press Beijing |
| [15] | Li, Ping; Li, Y. M., Algebraic properties of LA-languages, Inform. Sci., 176, 21, 3232-3255 (2006) · Zbl 1110.68078 |
| [16] | Li, H. Z.; Li, P.; Li, Y. Y., The relationships among several types of fuzzy automata, Inform. Sci., 176, 15, 2208-2226 (2006) · Zbl 1110.68065 |
| [17] | Liu, F. C.; Qiu, D. W., Diagnosability of fuzzy discrete-event systems: a fuzzy approach, IEEE Trans. Fuzzy Syst., 17, 2, 372-384 (2009) |
| [18] | Malik, D. S.; Mordeson, J. N.; Sen, M. K., Submachines of a fuzzy finite state machine, J. Fuzzy Math., 2, 781-792 (1994) · Zbl 0824.68077 |
| [19] | Malik, D. S.; Mordeson, J. N.; Sen, M. K., On subsystems of a fuzzy finite state machine, Fuzzy Sets Syst., 68, 83-92 (1994) · Zbl 0846.68036 |
| [20] | Malik, D. S.; Mordeson, J. N.; Sen, M. K., Products of fuzzy finite state machines, Fuzzy Sets Syst., 92, 95-102 (1997) · Zbl 0935.68064 |
| [21] | Malik, D. S.; Mordeson, J. N., Structure of upper and lower approximation spaces of infinite sets, (Lin, T. Y.; Yao, Y. Y.; Zadeh, L. A., Data Mining, Rough Sets and Granular Computing. Data Mining, Rough Sets and Granular Computing, Studies in Fuzziness and Soft Computing, vol. 95 (2002), Physica-Verlag: Physica-Verlag Heidelberg, New York), 461-472 · Zbl 1017.68116 |
| [22] | Mordeson, J. N.; Malik, D. S., Fuzzy Automata and Languages: Theory and Applications (2002), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, London · Zbl 1046.68068 |
| [23] | Orlowska, E., Semantic analysis of inductive reasoning, Theoret. Comput. Sci., 43, 81-89 (1986) · Zbl 0601.68059 |
| [24] | Pawlak, Z., Rough Sets, Theoretical Aspects about Data (1991), Kluwer Academic Publisher: Kluwer Academic Publisher Dordrecht · Zbl 0758.68054 |
| [25] | Peeva, K.; Zahariev, Zl., Computing behavior of finite fuzzy machines - algorithm and its application to reduction and minimization, Inform. Sci., 178, 21, 4152-4165 (2008) · Zbl 1170.68506 |
| [26] | Qiu, D. W., Automata theory based on complete residuated lattice-valued logic(I), Sci. China Ser. F, 44, 6, 419-429 (2001) · Zbl 1125.68383 |
| [27] | Qiu, D. W., Automata theory based on complete residuated lattice-valued logic(II), Sci. China Ser. F, 45, 6, 442-452 (2002) · Zbl 1161.68549 |
| [28] | Qiu, D. W., Characterizations of fuzzy finite automata, Fuzzy Sets Syst., 141, 391-414 (2004) · Zbl 1059.68069 |
| [29] | Qiu, D. W., Notes on automata theory based on quantum logic, Science in China Series F: Inform. Sci., 50, 2, 154-169 (2007) · Zbl 1121.68068 |
| [30] | Rosenthal, K. L., Quantales and their Applications (1990), Longman Scientific and Technical: Longman Scientific and Technical London · Zbl 0703.06007 |
| [31] | Santos, E. S., Maximin automata, Inform. Control, 12, 367-377 (1968) · Zbl 0174.03601 |
| [32] | Santos, E. S., On reduction of max-min machines, J. Math. Anal. Appl., 37, 677-686 (1972) · Zbl 0245.94040 |
| [33] | Santos, E. S., Fuzzy automata and languages, Inform. Sci., 10, 193-197 (1976) · Zbl 0336.94040 |
| [34] | Shukla, W.; Srivastava, A. K., A topology for automata: a note, Inform. Control, 32, 163-168 (1976) · Zbl 0338.94028 |
| [35] | Tiwari, S. P.; Srivastava, Arun K., On a decomposition of fuzzy automata, Fuzzy Sets Syst., 151, 503-511 (2005) · Zbl 1064.68060 |
| [36] | Wee, W. G.; Fu, K. S., A formulation of fuzzy automata and its application as a model of learning systems, IEEE Trans. Systems Man Cybernet., 5, 215-223 (1969) · Zbl 0188.33203 |
| [37] | Young, B. S., Intuitionistic fuzzy transformation semigroups, Inform. Sci., 179, 24, 4284-4291 (2009) · Zbl 1183.68356 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
