Some aspects of Special Fuzzy Boolean algebra

International Journal of Computer Applications, 2013

It has been established that the fuzzy sets do not hold the Complement laws of Boolean algebra. The aim of this article is to introduce and study a kind of family of fuzzy subsets of a finite set which holds the Complement laws. The complement operation has redefined for this purpose. Further, it has been established that the introduced fuzzy subsets can form Boolean lattice and also Boolean algebra.

The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. Abstract— The aim of this article is to study the behaviour of the fuzzy Boolean algebras formed by the fuzzy subsets of a finite set. The properties of homomorphism, isomorphism and automorphism of the fuzzy Boolean algebras have been investigated. Further, the ideal and filter of the fuzzy Boolean algebras have also been observed with their characteristics.

The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. ABSTRACT It has been established that the fuzzy sets do not hold the Complement laws of Boolean algebra. The aim of this article is to introduce and study a kind of family of fuzzy subsets of a finite set which holds the Complement laws. The complement operation has redefined for this purpose. Further, it has been established that the introduced fuzzy subsets can form Boolean lattice and also Boolean algebra. Keywords fuzzy subset, membership value, the complement operation, the complement laws, Boolean lattice, subsets of a fuzzy subset, fuzzy power set.

TheScientificWorldJournal, 2015

It is well known that an important task of the artificial intelligence is to make a computer simulate a human being in dealing with certainty and uncertainty in information. Logic gives a technique for laying the foundations of this task. Information processing dealing with certain information is based on the classical logic. Nonclassical logic including many-valued logic and fuzzy logic takes the advantage of the classical logic to handle information with various facets of uncertainty, such as fuzziness and randomness. Therefore, nonclassical logic has become a formal and useful tool for computer science to deal with fuzzy information and uncertain information.

— The set of all mappings from a finite set X into a closed interval   0,1 is the set of fuzzy sets denoted by F. This set F is closed under the binary operation absolute difference,  of fuzzy set F satisfies the axioms, closure, commutativity, identity and inverse law under the binary operation . The associative law is not satisfied by F. In this article, we wish to introduce the subset B of F with binary operation absolute difference  and fuzzy intersection  , as a special fuzzy Boolean ring briefly denoted by SFBR. Keywords— Special fuzzy Boolean ring (SFBR), absolute difference, subs SFBR, Isomorphic SFBR, Divisor of empty fuzzy set.

2013

Abstract. In this paper, we analyze the more adequate tools to solve many current logical challenges in A I. When approaching the complex practical problems, the choice of solving strategy is crucial. As we will see, the method to be used depends on many factors, as the type of problem or its applications. Sometimes, a combination of more than one approach may be used. For instance, a compound is many times possible, and may be further refined. In particular, we describe the behaviour of different fuzzy logics relative to principles as Contradiction and Excluded Middle Laws. So, we see if the class of fuzzy sets possess the character of Boolean Algebras, and the very interesting case considering Lukasiewicz Logic.

Kybernetes, 1993

General representation theorems for L-fuzzy quantities was given by Höhle[1], using lattice theoretic concepts, which include the Negoita-Ralescu set representation as a special case (see also [2, pp. 95-6] for related works).

Fuzzy Sets and Systems, 1992

The classical theory of fuzzy sets has been developed within the Cantorian framework, as a simple generalization of the indicator or characteristic function and thus as a generalization of the membership property only, paying no attention to the equality relation. The nonstandard approach to fuzzy sets on the contrary is based on a non-Cantorian framework, which essentially amounts to viewing the eternally constant Cantorian-Platonic universe through two observers: the absolute Cantorian and the local non-Cantorian [5]. The construction of such a theory is based on a generalization of both equality and membership relations and technically is based on the theory of Boolean powers and a Boolean generalization of Nonstandard Analysis based on them [5, 6, 8, 15]. For Nonstandard Analysis see [1, 13]. In this paper, which is a continuation of [5], we give some new results for the basic concepts of fuzzy sets from our point of view, along with interpretations which are more interesting to fuzzy theorists.

IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics), 2001

Multi valued relation is introduced in [13] as a table in which for certain combination of input variables values one of several specified output values can be selected. For instance, in Figure 15g in cell for z = 1, G = 0 there are two values, 0 and 1. It means that any ternary value other than value 2 can be taken for this combination of input variable values. This is called a generalized don't care and it generalizes a standard don't care concept where any set of values of a given output is allowed for given input combination. Thus, the generalized don't cares of a ternary signal are: {0,1}, {1,2} and {0,2}. The standard don't care is {0,1,2}. Let us observe that the generalized and standard don't cares correspond to the following values in fuzzy logic: {0,1} = x' i or x i x' i (when an undecided shape is between the one from Fig.2b and the one from Fig. 3a). {1,2} = x i x' i or x i (when an undecided shape is between the one from Fig. 3a and the one from Fig. 2a). {0,2} x' i or x i (when an undecided shape is between the one from Fig. 2b and the one from Fig. 2a). {0,1,2} when the shape of x i is irrelevant. There are several ways to specify the initial fuzzy relations A graphical method is illustrated in Figure 15a. The OR relations among groups of terms denote that the choice of any of the groups of terms pointed by the two arrows originating from word OR can be made. Thus the function from Fig. 15 is specified by the expression: F(x, y, z) = yz CHOICE-OF[ x' y' z z ' OR (z z' x x' y' + z z' y y' x') ] + xz. In general, a fuzzy relation can be specified by an arbitrary multi-level decision unate function on variables G I , each of these variables denoting Max of terms for a sum-of-products form of fuzzy relation. Such unate function uses functors AND and OR and variables G I corresponding to Max groups of terms. The above fuzzy relation is specified by the unate decision function: A AND B AND (C OR D) = (A AND B AND C) OR (A AND B AND D) where: A = yz , B = xz, C = x' y' z z' , D = (z z' x x' y' + z z' y y' x'). Thus, every fuzzy relation corresponds to a set of sum-of-products fuzzy functions among whichwe can freely choose.

2011 Fourth International Symposium on Computational Intelligence and Design, 2011

In several aspects of the fuzzy sets theory and its applications, it is convenient to do some manipulations on the formulas that are proposed in order to obtain some enhancements in methods such as in design and implementation. Currently, the proper fuzzy sets characteristics do not let Boolean simplification methods be applied in an effective way, and consequently, the algorithm solutions proposed so far for these methods cannot be used. This paper presents preliminary ideas in the first algorithmic implementation design of a simplification method of Boolean and fuzzy formulas by using finite algebras.