DOI: http://dx.doi.org/10.26483/ijarcs.v9i2.5712 Volume 9, No. 2, March-April 2018 International Journal of Advanced Research in Computer Science RESEARCH PAPER Available Online at www.ijarcs.info © 2015-19, IJARCS All Rights Reserved 314 ISSN No. 0976-5697 FUZZY THEORY – A SURVEY ON ITS FOUNDATIONS AND ADVANCEMENTS Saniya Tasleem Zafar M. Tech. Scholar, Department of CSE School of Engg. Sciences and Technology, Jamia Hamdard New Delhi, India Dr. Safdar Tanweer Assistant Professor, Department of CSE School of Engg. Sciences and Technology, Jamia Hamdard New Delhi, India Nafisur Rahman Assistant Professor, Department of CSE School of Engg. Sciences and Technology, Jamia Hamdard New Delhi, India Abstract: The term ‘Fuzzy’ means vague, unclear, or imprecise. Fuzzy Logic is a many-valued logic that is based on the theory of Fuzzy Sets and it facilitates the representation of approximate reasoning. It finds its use in various areas where binary representations do not suffice. In this paper, we have confined our discussions on the theoretical foundations and advancements of modern Fuzzy Logic. We start with a brief account of Fuzzy Sets followed by the operations they support. Then we discuss how the Linguistic Variables allow more realistic reasoning as opposed to traditional binary reasoning. Then we introduce the theoretical aspects of the calculus of Fuzzy Restrictions. Finally, we discuss the theory of possibility as an alternative to the theory of probability. For the sake of simplicity and intelligibility, we have tried to avoid incommodious mathematical equations throughout this paper. Keywords: Fuzzy Sets; Approximate reasoning; Linguistic Variables; Fuzzy Restriction; Possibility theory I. INTRODUCTION The theoretical foundations of Fuzzy Logic are based on Fuzzy Sets that deal with the information that is uncertain, generic, imprecise, or vague. The theory of Fuzzy Sets is associated with the idea of membership function by the course of human thinking and understanding. It also gives an effective means for understanding better assessment options. Fuzzy Sets are being used extensively in the development of intelligent systems. Fuzzy Sets are distinguished from crisp sets in that they cater to the requirements of approximate reasoning. Very often, Linguistic Variables are used for this purpose. A Linguistic Variable is a value which is given in the form of word or sentences rather than in a numeric form. Fuzzy Restrictions allow the expression of Fuzzy propositions in the form of relational assignment equations. Treating the incomplete information as a vague concept, the theory of Possibility provides an alternative to Probability theory by dealing with certain types of ambiguity. II. FUZZY SETS Before discussing Fuzzy Sets [1, 2], it is imperative to define Fuzzy Logic. Fuzzy Logic is a contrast of Boolean logic. Boolean logic consists of value either true or false on the other hand, Fuzzy Logic consists of all the values which come in between completely true to completely false. The fundamental difference between Fuzzy Logic and Boolean logic is that it gives a sliding measure rather than a discrete value. Any value between 0 and 1, it may be 0.45, 0.8, 0.978, and so on and could be represented in a Fuzzy system whereas, the Boolean system will only accept the value 0 or 1, or true or false. Zadeh introduced Fuzzy Sets in his seminal paper on the subject in 1965. Fuzzy Sets contain elements with have different orders of membership. They are characterized by a membership function which is valued in the interval [0, 1]. Fuzzy Set is, in essence, a class of objects with a continuum of grades of membership. While Fuzzy Set is an augmentation of the crisp set - a set in which the elements are present or not present. A crisp set is often described as an ordinary or classical set. The membership function is defined as the mapping of each point in the input value to a degree of membership or membership value which is marked in between the values of 0 and 1. III. FUZZY SET OPERATIONS The operations on Fuzzy Set are a generalization of the crisp set. The most extensively used operations are Intersection, Union, and Complement [2, 3]. Let M and N be the two Fuzzy Sets where both belong to the universal set U. Fig. 1 Intersection Operation: The intersection between two Fuzzy Sets M and N is determined by the degree of membership using the function operation: μ(M ∩ N)(y) = max [μM(y) ∩ μN(y)], y ∈ U