The grade of membership function for fuzzy monotone functions is defined and investigated. An algorithm for finding the membership value is presented. A minterm even function and minterm odd function are defined and studied. It is found that these two functions are the two most alternation functions. The relationships with threshold functions are also presented. In addition, three ways to implement a fuzzy monotone increasing function are investigated. Applications to function representation, data compression and error detection are illustrated. The results have useful applications in fuzzy logic, expert systems, fuzzy expert systems, and also management of uncertainty.
Line-oriented two-dimensional grammars (LOTDGs), region-oriented two-dimensional grammars (ROTDGs) and parallel productions are introduced. The relationships between LOTDGs and ROTDGs are stated. Examples of LOTDGS for generating all possible 458 rightangled triangles, all possible squares, all possible 458 isosceles trapezoids, and all possible 458 parallelograms using parallel productions are presented. A new concise representation of a derivation chain is also introduced and illustrated by examples. LOTDGs and ROTDGs are compared. Generally speaking, LOTDGs require less terminal variables and non-terminal variables, require less storage space, and require less derivation steps. Seven challenging problems for future research are also included. In addition, parallel production is an effective tool to model parallel computers as well as parallel processing. The results have useful applications in robot vision interpretation, robot pictorial communication, artificial intelligence, visual languages, software engineering, medical expert systems, and fuzzy logic functions.
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.
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