Fuzzy symmetric threshold functions and applications

A fuzzy symmetric threshold (ST) function is defined to be a fuzzy set over the set of functions. All ST functions have full memberships in this fuzzy set. For n variables, there are (2n+2) ST functions. A distance measure between a nonsymmetric threshold function and the set of all ST functions is defined and investigated. An explicit expression for the membership function of a fuzzy ST function is defined through the use of this distance measure. An algorithm for obtaining this distance measure is presented with illustrative examples. It is also shown that any function and its complement always have the same grade of membership in the class of fuzzy ST functions. Applications to concise function representation and simple function implementation are also presented with examples. In addition, most inseparable unsymmetric functions are defined and investigated. Fuzzy ST functions are relevant to the development of practical applications of fuzzy methods and might contribute to the state of the art in the implementations of fuzzy methods in the areas requiring utilization of ST functions.

[1]  Michael L. Dertouzos,et al.  Threshold Logic: A Synthesis Approach , 1965 .

[2]  Edward T. Lee Algorithms for finding most unsymmetrical chromosome images , 1983 .

[3]  David Levy,et al.  Book review: Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence by Bart Kosko (Prentice Hall 1992) , 1992, CARN.

[4]  Edward T. Lee Boolean Algebra from the Lattice Point of View , 1994 .

[5]  Michael A. Harrison,et al.  Introduction to switching and automata theory , 1965 .

[6]  and C.L. Coates Lewis,et al.  Threshold Logic , 1967 .

[7]  Bart Kosko,et al.  Fuzzy Engineering , 1996 .

[8]  D.G. Schwartz,et al.  Fuzzy logic flowers in Japan , 1992, IEEE Spectrum.

[9]  Te-Shun Chou,et al.  Fuzzy threshold functions and applications , 1995 .

[10]  Edward T. Lee,et al.  Fuzzy Neural Networks , 1975 .

[11]  Edward T. Lee Fuzzy Symmetric Functions with Don't‐care Conditions and Applications , 1993 .

[12]  Edward J. McCluskey,et al.  Detection of group invariance or total symmetry of a Boolean function , 1956 .

[13]  Mitchell P. Marcus,et al.  The Detection and Identification of Symmetric Switching Functions with the Use of Tables of Combinations , 1956, IRE Trans. Electron. Comput..

[14]  Lotfi A. Zadeh,et al.  Note on fuzzy languages , 1969, Inf. Sci..

[15]  Edward T. Lee On Average Turnaround Time , 1990 .

[16]  Robert McNaughton,et al.  Unate Truth Functions , 1961, IRE Trans. Electron. Comput..

[17]  E. T. Lee,et al.  The shape-oriented dissimilarity of polygons and its application to the classification of chromosome images , 1974, Pattern Recognit..

[18]  Saburo Muroga,et al.  Threshold logic and its applications , 1971 .

[19]  Edward T. Lee Representations of logic functions , 1995 .

[20]  E. T. Lee,et al.  Shape-oriented storage and retrieval of geometric figures and chromosome images , 1976, Inf. Process. Manag..

[21]  E. T. Lee,et al.  Algorithms for finding most dissimilar images with possible applications to chromosome classification , 1976 .

[22]  E. Leet Proximity Measures for the Classification of Geometric Figures , 1972 .

[23]  Bart Kosko,et al.  Neural networks and fuzzy systems: a dynamical systems approach to machine intelligence , 1991 .

[24]  Te-Shun Chou,et al.  Fuzzy monotone functions and applications , 2001 .

[25]  J. David Irwin,et al.  An Introduction to Computer Logic , 1974 .

[26]  Bart Kosko,et al.  Fuzzy function approximation with ellipsoidal rules , 1996, IEEE Trans. Syst. Man Cybern. Part B.

[27]  Bart Kosko,et al.  Fuzzy function learning with covariance ellipsoids , 1993, IEEE International Conference on Neural Networks.

[28]  C L Sheng,et al.  Threshold Logic , 1969 .

[29]  Edward T. Lee Fuzzy Tree Automata and Syntactic Pattern Recognition , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[30]  Sankar K. Pal,et al.  Fuzzy models for pattern recognition : methods that search for structures in data , 1992 .