Abraham Kandel

On Minimization of Fuzzy Functions

The study of fuzziness in combinational switching systems by means of a suitable fuzzy algebra is discussed. The insufficiency of the methods for simplification of fuzzy functions as presented by Lee and Chang [9] and by Siy and Chen [10] is discussed. A new technique for minimization of fuzzy functions is developed. Special properties of fuzzy functions are discussed and their relationships to two-valued logic are investigated.

On the minimization of incompletely specified fuzzy functions

This article is concerned with the study of fuzziness in incompletely specified combinational networks by means of a suitable fuzzy algebra. For completely specified fuzzy functions, minimal sets have been obtained using the definition of fuzzy consensus. Using the basic properties of fuzzy logic, the concepts of fuzzy prime implicant, essential fuzzy prime implicant, and fuzzy iterated consensus are used to find the minimal sum-of-products expression of an incompletely specified fuzzy function.

Computer science—a vicious circle

In many computer science departments throughout the country, including some interdisciplinary departments, the curriculum has placed increasing emphasis upon applied mathematics and the fundamental nature of computational systems. A set of core courses which usually are fitted into such a program can be found in the report of the ACM curriculum committee on computer science, Curriculum 68 [1]. Some computer science departments have done such a magnificent job of de-emphasizing the importance of the experimental laboratory in their program that their graduates emerge thoroughly unprepared to tackle the intricacies associated with design work in the real-life world—both in software and hardware. This is basically because most computer science departments do not emphasize sufficiently the practical aspects of computational system design and implementation. For example, some students have serious difficulty in designing even a shift register of an 8-4-2-1 counter with ripple carry, not to mention more complicated computer systems. Perhaps even more important, computer science in the academic sense has implemented the “modus ponens” (rule of detachment) from the real-life world and has very little to do with real computers. The relationship between the abstract side of computer science and questions of cost and marketing are hardly ever mentioned in courses.

A note on the simplification of fuzzy switching functions

Abstract This note is concerned with the modification of the definition of fuzzy consensus. The modified version is used to find the set of all fuzzy prime implicants of a fuzzy switching function as defined previously. The fuzzy algebra used to derive these functions satisfies the set of axioms of a distributive lattice with unique identities under the operators of maximum and minimum as described earlier.

Inexact switching logic

In this correspondence, we are concerned with an algorithm used to find the set of all inexact prime implicants of an inexact switching function as defined with the help of fuzzy algebra. The principal advantage claimed over existing methods is that it is suitable for an efficient computer implementation. The steps are easy to apply and well adapted to being programmed without any heuristic methods.

Application of fuzzy logic to the detection of static hazards in combinational switching systems

In this paper the fuzzy set as discussed by Zadeh is viewed as a multivalued logic with a continuum of truth values in the interval [0,1]. The concept of static hazard in combinational switching systems is related to fuzzy logic and various properties of this relation are established. The paper derives the necessary and sufficient conditions for a fuzzy function to adequately describe the steady-state and static hazard behavior of a combinational system, by extending the ternary method discussed by Yoeli and Rinon and using the resolution principle of mechanical theorem-proving.

Column table approach for the minimization of fuzzy functions

Abstract An algorithm is described which finds all essential prime implicants of a fuzzy function in disjunctive form. This algorithm has two main advantages: 1. (1) It does not require expansion of the function before minimization is performed. 2. (2) Many nonessential prime implicants are not generated. It is shown, by comparison with fuzzy consensus, that this algorithm yields a function representation containing only fuzzy prime implicants of the original function. In addition, the only prime implicants not generated are those which are not essential.

Comment on an algorithm that generates fuzzy prime implicants by Lee and Chang

This correspondence points out the insufficiency of the algorithm to generate the set of all fuzzy prime implicants of a fuzzy formula as presented in Lee and Chang (1971) . Two theorems, which are the basis of a new technique that generates the complete set of fuzzy implicants, and used for the minimization of fuzzy functions, as presented in Kandel (1972) , are discussed.

Comments on "Minimization of Fuzzy Functions"

This correspondence points out the insufficiency of the method for minimization of fuzzy functions as presented in a recent short note.1 Two lemmas, which are the basis for a new minimization technique [2], are represented.

Computer science—seminars for undergraduates

In recent years a number of curricula for computer science education have been proposed. Most notable among these are [1-4]. The results from these four studies were unbalanced oriented educational programs each with a different primary emphasis on various aspects of computer science and its application. None offered a prescription within the undergraduate framework for a satisfactory synthesis of hardware, software, and abstract theory.