Document Type: Research Paper


Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran


In this paper, we first define the notion of a complete general fuzzy
automaton with threshold c and construct an $H_{nu}$- group, as well as commutative
hypergroups, on the set of states of a complete general fuzzy automaton
with threshold c. We then define invertible general fuzzy automata, discuss
the notions of “homogeneity, “separation, “thresholdness connected, “thresholdness
inner irreducible and “principal and strongly connected, as applied
to them and use these concepts to construct a quasi-order hypergroup on an
invertible general fuzzy automaton. Finally, we derive relationships between
the properties of an invertible general fuzzy automaton and the induced hypergroup.


[1] R. Ameri, Fuzzy hypervector spaces over valued fields, Iranian Journal of Fuzzy Systems,
2(1) (2005), 37-47.
[2] M. A. Arbib, From automata theory to brain theory, International Journal of Man-Machin
Studies, 7(3) (1975), 279-295.
[3] W. R. Ashby, Design for a brain, Chapman and Hall, London, 1954.
[4] D. Ashlock, A. Wittrock and T. Wen, Training finite state machines to improve PCR primer
design, in: Proceedings of the 2002 Congress on Evolutionary Computation (CEC) 20, 2002.
[5] P. Corsini, Prolegomena of hypergroup theory, Aviani Editore, 1993.
[6] P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Kluwer Academic Publishers,
Advances in Mathematics, 2003.
[7] P. Corsini and I. Cristea, Fuzzy grade I.P.S hypergroups of order 7, Iranian Journal of Fuzzy
Systems, 1(2) (2004), 15-32.
[8] M. Doostfatemeh and S. C. Kremer, New directions in fuzzy automata, International Journal
of Approximate Reasoning, 38 (2005), 175-214.
[9] B. R. Gaines and L. J. Kohout, The logic of automata, International Journal of General
Systems, 2 (1976), 191-208.
[10] Y. M. Li and W. Pedrycz, Fuzzy finite automata and fuzzy regular expressions with membership
valuesin lattice-ordered monoids, Fuzzy Sets and Systems, 156 (2005), 68-92.
[11] R. Maclin and J. Shavlik, Refing domain theories expressed as finite-state automata, in:
L.B.G. Collins (Ed.), Proceedings of the 8th International Workshop on Machine Learning
(ML’91), Morgan Kaufmann, San Mateo CA, 1991.
[12] R. Maclin and J. Shavlik, Refing algorithm with knowledge-based neural networks: improving
the choufasma algorithm for protein folding, in: S. Hanson, G. Drastal and R. Rivest (Eds.),
Computational Learning Theory and Natural Learning Systems, MIT Press, Cambridge, MA,
[13] J. N. Mordeson and D. S. Malik, Fuzzy automata and languages, Theory and Applications,
Chapman and Hall/CRC, London/Boca Raton, FL, 2002.
[14] W. Omlin, K. K. Giles and K. K. Thornber, Equivalence in knowledge representation: automata,
rnns, and dynamical fuzzy systems, Proceeding of IEEE, 87(9) (1999), 1623-1640.
[15] W. Omlin, K. K. Thornber and K. K. Giles, Fuzzy finite-state automata can be deterministically
encoded into recurrent neural networks, IEEE Transactions on Fuzzy Systems, 5(1)
(1998), 76-89.
[16] B. Tucker (Ed.), The computer science and engineering handbook, CRC Press, Boca Raton,
FL, 1997.
[17] J. Virant and N. Zimic, Fuzzy automata with fuzzy relief, IEEE Transactions on Fuzzy Systems,
3(1) (1995), 69-74.

[18] W. G. Wee, On generalization of adaptive algorithm and application of the fuzzy sets concept
to pattern classification, Ph.D. dissertation, Purdue University, Lafayette, IN, 1967.
[19] M. Ying, A formal model of computing with words, IEEE Transactions on Fuzzy Systems,
10(5) (2002), 640-652.
[20] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338-353.
[21] M. M. Zahedi, M. Horry and K. Abolpor, Bifuzzy (General) topology on max-min general
fuzzy automata, Advanced in Fuzzy Mathematics, 3(1) (2008), 51-68.