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7.7. Provided is a verbal description of the system

… the concentration of substance depends on its previous level and is inhibitively affected by substrate x1. The other substrate whose concentration is denoted by x2 has a positive influence on the concentration of this substance…

Convert this description into a logical model using fuzzy neurons. Experiment with the network. In particular, analyze it for several triangular norms and selected values of the connections. What is the steady state of the concentration? Can you determine this value analytically not resorting to simulations. Verify your findings experimentally.

7.8. The network shown in Fig. 7.33 can be regarded as a logic oscillator. Write down its formal description in the form of a fuzzy neural network. Assuming some values of the connections (a, b, c, d), analyze the behavior of the network for time (k) approaching infinity. Run some experiments to confirm your findings.


Figure 7.33  Logic oscillator

7.9. Consider the Boolean functions given in the canonical form of sums of minterms

where the don’t conditions are denoted by “d”. Propose a suitable topology of the fuzzy neural network, complete its training and derive the simplified versions of the functions. Hint: think how to treat the don’t conditions occurring in these functions.

7.10. Consider the problem of sensor fusion involving three sensors where now the dependency relationship is formulated in terms of similarity between the indications of sensors, namely we require that the outputs of the sensors need to be similar. (Hint: augment the system with some matching neurons.) How to generalize this problem to “n” sensors?

7.11. Interpret the fuzzy neural network shown in Fig. 7.34. Consider several values of the threshold levels.


Figure 7.34  Example fuzzy neural network to be interpreted

7.12. The control protocol is composed of four rules each of them with two antecedents (subconditions)

Considering that the membership function of all fuzzy sets are given, propose a fuzzy neural network. Discuss the role of the connections of this network.

7.8. References

1.  J.A. Anderson, E. Rosenfeld, (eds.) Neurocomputing: Foundations of Research, MIT Press, Cambridge, MA, 1988.
2.  L.B. Booker, D.E. Goldberg, J. H. Holland, Classifier systems and genetic algorithms, Artificial Intelligence, 40, 1989, 235 - 282.
3.  D. Butnariu, E. P. Klement, Triangular Norm-based Measures and Games with Fuzzy Coalitions, Kluwer Academic Publishers, Dordrecht, Boston, 1993.
4.  L. Davis, Handbook of Genetic Algorithms, van Nostrand Reinhold, New York, 1991.
5.  A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy Relational Equations and Their Applications in Knowledge Engineering, Kluwer Academic Press, Dordrecht, 1989.
6.  L.A. Feldkamp, G.V. Puskorius, F. Yuan, L.I. Davis, Jr., Architecture and training of a hybrid neural-fuzzy system, Proc. 2nd Int. Conf. on Fuzzy Logic and Neural Networks, Iizuka, Japan, 1992, pp. 131-134.
7.  M. Figueiredo, F. Gomide, W. Pedrycz, Fuzzy neurons and networks: models and learning, Proc. EUFIT-95, August 28 - 31, 1995, Aachen, vol. I, pp. 332 - 335.
8.  S. Grossberg, Neural Networks and Natural Intelligence, The MIT Press, Cambridge, MA, 1988.
9.  R. Hecht-Nielsen, Neurocomputing, Addison-Wesley, Reading, MA, 1991.
10.  F. J. Hill, G. R. Peterson, Introduction to Switching Theory and Logical Design, 3rd edition, J. Wiley & Sons, New York, 1981.
11.  K. Hirota, W. Pedrycz, OR/AND neuron in modeling fuzzy set connectives, IEEE Trans. on Fuzzy Systems, 2, 1994, 151-161.
12.  IEEE Trans. on Neural Networks - Special issue on fuzzy logic and neural networks, 3, 1992.
13.  N. Ikoma, W. Pedrycz, K. Hirota, Estimation of fuzzy relational matrix by using probabilistic descent method, Fuzzy Sets and Systems, 57, 1993, 335-349.
14.  J. Kacprzyk, Multistage Decision-Making under Fuzziness, Verlag TUV Rheinland, Cologne, 1983.
15.  J. Kacprzyk, Group decision-making with a fuzzy majority via linguistic quantifiers, parts I and II, Cybernetics and Systems, 1985, 16, 119-129, 131-144.
16.  J. Kacprzyk, Group decision making with a fuzzy linguistic majority, Fuzzy Sets and Systems, 18, 1986, 105-118.
17.  R.J. Machado, A.F. Rocha, The combinatorial neural network: a connectionist model for knowledge based systems, 3rd Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-based Systems, Paris, France, July 2-6, 1990, 9-11.
18.  S. Muroga, Threshold Logic and Its Applications, J. Wiley, New York, 1971.
19.  W. Pedrycz, Direct and inverse problem in comparison of fuzzy data, Fuzzy Sets and Systems, 34, 1990, 223-236.
20.  W. Pedrycz, Neurocomputations in relational systems, IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 13, 1991, 289-296.
21.  W. Pedrycz, Fuzzy neural networks and neurocomputations, Fuzzy Sets and Systems, 56, 1993, 1-28.
22.  W.Pedrycz, Fuzzy Control and Fuzzy Systems, 2nd edition, Research Studies Press/J.Wiley, Taunton/New York, 1993.
23.  W. Pedrycz, A. F. Rocha, Fuzzy-set based models of neurons and knowledge-based networks, IEEE Trans. on Fuzzy Systems, 1, 1993, 254-266.
24.  A.F. Rocha, Neural Nets: A Theory for Brain and Machine. Lecture Notes in Artificial Intelligence, vol. 638, Springer-Verlag, Berlin, Heidelberg, New York, 1992.
25.  W.G. Schneeweiss, Boolean Functions with Engineering Applications, Springer-Verlag, Berlin, 1989.
26.  A. Stoica, Synaptic and sematic operators for fuzzy neurons: which t- norms to choose, Proc. NAFIPS - 96, June 19 - 22, 1996, Berkeley, CA, pp. 55-58.
27.  R.R. Yager, Quantifiers in the formulation of multiple objective decision functions, Information Sciences, 31, 1983, 107-139.
28.  R. R. Yager, On ordered weighted averaging aggregation operators, IEEE Trans. on Systems, Man and Cybernetics, 18, 1988, 183-190.
29.  L.A.Zadeh, A computational approach to fuzzy quantifiers in natural languages, Computers and Mathematics with Applications, 9, 1983, 149-184.
30.  H.J. Zimmermann, P. Zysno, Latent connectives in human decision making, Fuzzy Sets and Systems, 4, 1980, 37-51.
31.  G. Weisbuch, Complex Systems Dynamics, Addison-Wesley, Redwood City, CA, 1991.
32.  S. Wolfram, Cellular Automata and Complexity. Collected Papers, Addison-Wesley, Reading, MA, 1994.
33.  J. von Neumann, The Computer and the Brain, Yale University Press, New Haven, 1958, pp. 66-82.


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