In this paper, a presented definition of **type**-**2** **fuzzy** sets and **type**-**2** **fuzzy** set operation on it was given. The aim of this work was to introduce the concept of **general** topological spaces were extended in **type**-**2** **fuzzy** sets with the structural properties such as open sets, closed sets, interior, closure and neighborhoods in topological spaces were extended to **general** **type**-**2** **fuzzy** topological spaces and many related theorems are proved.

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. Other work using an alpha-planes representation has been applied, e.g. as a method for edge-detection [35] and a learning method to forecast Mackey-Glass time-series [41]. The latter showed a better performance of **general** **type**-**2** **fuzzy** logic systems using a simpler model known as “triangle quasi- **type**-**2** **fuzzy** logic system” first presented in [40]. Some other researchers used some neural network concepts or classification algorithms such as: **type** **2** Adaptive Network Based **Fuzzy** Inference System (ANFIS) [28], **general** **type**-**2** **fuzzy** neural network (GT2FNN) [24] and **fuzzy** C-means algorithm with a model known as “efficient tri- angular **type**-**2** **fuzzy** logic system” [43]. To the best of the authors’ knowledge, no attempt to employ a learning method to **general** **type**-**2** **fuzzy** logic systems using the vertical-slices representation has been reported. To achieve this objective, apart from using a practical **type**-reducer, some kinds of parametrization are needed for **general** **type**-**2** sets to allow learning or optimization techniques to deal with these parame- ters easily rather than having all the secondary grades or membership functions chosen manually. The parametrization method should preserve most of the freedom associated with GT2FLS.

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Abstract. Centroid of **general** **type**-**2** **fuzzy** set can be used as a measure of uncertainty in highly uncertain environments. Computing centroid of **general** **type**-**2** **fuzzy** set has received an increasing research attention during recent years. Although computation complexity of such sets is higher than that of interval **type**-**2** **fuzzy** sets, with the advent of new representation techniques, e.g. -planes and z-Slices, computation eorts needed to deal with **general** **type**-**2** **fuzzy** sets have decremented. A very rst method to calculate the centroid of a **general** **type**-**2** **fuzzy** set was to use Karnik-Mendel algorithm on each - plane, independently. Because of the iterative nature of this method, running time in this approach is rather high. To tackle such a drawback, several emerging algorithms such as Sampling method, Centroid-Flow algorithm, and, recently, Monotone Centroid- Flow algorithm have been proposed. The aim of this paper is to present a new method to calculate centroid intervals of each -plane, independently, while reducing convergence time compared with other algorithms like iterative use of Karnik-Mendel algorithm on each -plane. The proposed approach is based on estimating an initial switch point for each -plane. Exhaustive computations demonstrate that the proposed method is considerably faster than independent implementation of existing iterative methods on each -plane. © 2015 Sharif University of Technology. All rights reserved.

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Abstract—Similarity measures provide one of the core tools that enable reasoning about **fuzzy** sets. While many types of similarity measures exist for **type**-1 and interval **type**-**2** **fuzzy** sets, there are very few similarity measures that enable the comparison of **general** **type**-**2** **fuzzy** sets. In this paper, we introduce a **general** method for extending existing interval **type**-**2** similarity measures to similarity measures for **general** **type**-**2** **fuzzy** sets. Specifically, we show how similarity measures for interval **type**-**2** **fuzzy** sets can be employed in conjunction with the zSlices based **general** **type**-**2** representation for **fuzzy** sets to provide measures of similarity which preserve all the common properties (i.e. reflexivity, symmetry, transitivity and overlapping) of the original interval **type**-**2** similarity measure. We demonstrate examples of such extended **fuzzy** measures and provide comparisons between (different types of) interval and **general** **type**-**2** **fuzzy** measures.

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The IAA enables these capabilities while minimising the number and scope of assumptions during the model-creation process. In concrete terms, it generates non-parametric FS models of the interval-based data without preselecting a spe- cific FS **type** (such as triangular or Gaussian) and minimises the loss of information during the model creation process (from the original, potentially uncertain intervals). Finally, the IAA leverages the different degrees of freedom of zGT2 FSs to represent and to distinguish the different types of uncer- tainty. Specifically, agreement/variation over multiple interval samples (e.g., survey responses) from the same source (e.g., expert) is commonly captured using the primary memberships (y ∈ [0, 1]), with lower and upper bounds of primary member- ship modelling potential uncertainty about interval endpoints. Uncertainty originating from the agreement/variation between multiple sources (e.g., experts) of information is modelled through the secondary memberships of the zGT2 FSs (z ∈ [0, 1]). Note that the choice of in which dimension to model which **type** of uncertainty is arbitrary and either choice is valid - as long as data collection and the interpretation of the re- sulting sets is modified accordingly. To illustrate, consider the case where only a single sample/survey was administered to a large number of subjects; here, it is preferable to model inter- expert uncertainty using the primary membership, resulting in simpler and easier interpretable (**type**-1 FS) models.

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Jonathan M. Garibaldi received the B.Sc (Hons) degree in Physics from Bristol University, UK in 1984, and the M.Sc. degree in Intelligent Systems and the Ph.D. degree in Uncertainty Handling in Immediate Neonatal Assessment from the University of Plymouth, UK in 1990 and 1997, respectively. He is Professor of Computer Science, and leads the Intelligent Modelling and Analysis (IMA) Research Group, in the School of Computer Science at the University of Nottingham, UK. The IMA research group undertakes research into intelligent modelling, utilising data analysis and transformation techniques to enable deeper and clearer understanding of complex problems. His main research interests are modelling uncertainty and variation in human reasoning, and in modelling and interpreting complex data to enable better decision making, particularly in medical domains. He has made many theoretical and practical contributions in **fuzzy** sets and systems, and in a wide range of generic machine learning techniques in real-world applications. Prof. Garibaldi has published over 200 papers on **fuzzy** systems and intelligent data analysis, is an Associate Editor of Soft Computing, and three other journals. He has served regularly in the organising committees and programme committees of a range of leading international conferences and workshops, such as FUZZ-IEEE, WCCI, EURO and PPSN. He is a member of the IEEE.

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is the source of extra parameters, is in itself the origin of extra computational cost. The quest for a representation that allow practical systems to be implemented is a fertile field of research. There are four main representation theorems for T2FSs, in which practical applications and theoretical defi- nition have been investigated. The vertical slice, wavy slice [22], alpha-plane (or zSlices) [17], [27] and geometric [5] representations. Zadeh [33] was the first to define operations for T2FSs, utilising α-cuts of each **fuzzy** membership grade. Recently, Chen and Kawase [4], Tahayori et al. [26], Liu et al. [17], [19], and Wagner and Hagras [27], [28] focused their attention towards decomposing T2FSs into several IVFSs. In particular, Liu [17] defined α-planes and Wagner and Hagras [27] defined zSlices as part of their effort to calculate the Centroid of T2FSs. In his work, Liu concluded that the union, intersection and centroid of T2FSs is equal to their respective operations of its constituent α-planes. Wagner and Hagras independently concluded the same. Hamrawi and Coupland [9], [10] derived arithmetic operations and defined non-specificity for T2FSs using the same concept and stated a generalised formula in [11], [12]. In this paper we investigate the use of the concept of α-cuts and its extension principle for T2FSs. We explain, step by step, the development phases of the theory and definitions. We believe it is a significant step forward in the theory and application of T2FSs. The novel ideas provided in this paper, are themselves built upon existing theories and definitions well accepted in the literature and is an extension to available and definitions. We show how operations on **general** **type**-**2** **fuzzy** sets can be broken down into a collection of interval **type**-**2** or crisp interval operations. The paper is organised as follows: Section **2** provides the notations and necessary back ground for the following work; Section 3 revisits the α- plane representation and defines the α-plane extension prin- ciple; Section 4 discusses the α-cut representation of IVFSs; Section 5 defines the α-cut representation for T2FSs and the extension principle associated with this representation; Section 6 provides a conclusion.

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Abstract—Facial expressions of a person representing similar emotion is not always unique. Naturally, the facial features of a subject taken from different instances of the same emotion have wide variations. In presence of two or more facial features, the variation of the attributes together makes the emotion recognition problem more complicated. This variation is the main source of uncertainty in the emotion recognition problem, which has been addressed here in two steps using **type**-**2** **fuzzy** sets. First a **type**-**2** **fuzzy** face-space is constructed with the background knowledge of facial features of different subjects for different emotions. Second, the emotion of an unknown facial expression is determined based on the consensus of the measured facial features with the **fuzzy** face-space. Both Interval and **General** **Type**-**2** **Fuzzy** Sets have been used separately to model the **fuzzy** face space. The interval **type**-**2** **fuzzy** set involves primary membership functions for m facial features obtained from n-subjects, each having l–instances of facial expressions for a given emotion. The **general** **type**-**2** **fuzzy** set besides employing the primary membership functions mentioned above, also involves the secondary memberships for individual primary membership curve, which has been obtained here by formulating and solving an optimization problem. The optimization problem here attempts to minimize the difference between two decoded signals: the first one being the **type**-1 defuzzification of the average primary membership functions obtained from the n- subjects, while the second one refers to the **type**-**2** defuzzified signal for a given primary membership function with secondary memberships as unknown. The uncertainty management policy adopted using **general** **type**-**2** **fuzzy** set has resulted in a classification accuracy of 98.333% in comparison to 91.667% obtained by its interval **type**-**2** counterpart. A small improvement (approximately 2.5%) in classification accuracy by IT2FS has been attained by pre-processing measurements using the well- known Interval Approach.

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Then in mobile robot control by **fuzzy** logic, all the cited forms of uncertainties will be multiplied over fuzzi- fication, inference and defuzzification. Those errors can degrade the performance of the whole robot controller. **Type**-**2** **fuzzy** logic has been used by researchers to try and overcome some of these difficulties [6-8]. And since many researchers have explored the use of **type**-**2** **fuzzy** logic controllers in various applications [9-16]. In fact, **type**-**2** **fuzzy** sets were initially introduced by Zadeh [17]. Firstly a **general** **type**-**2** **fuzzy** set was defined, where it represents a 3D set in which each membership grade is a **type**-1 **fuzzy** set bounded in [0,1]. Due to the complexity of the join (OR) and meet (AND) operations performing the inference part [18] and **type**-reduction in the defuzzi- fication part [19], the application of **general** **type**-**2** **fuzzy** sets has been limited. Hence, a simplified version of **general** **type**-**2** set called interval **type**-**2** **fuzzy** set is used more widely [6,20]. This kind of set has membership grades that are crisp interval sets bounded in [0,1]. The uncertainty here is represented as a 2D bounded region that is called the Footprint of Uncertainty. Various re- searchers have explored the advantages of interval **type**-**2** **fuzzy** sets [21,22]. Moreover, a geometric approach has been introduced by Coupland and John [23] distinguish- ing between **fuzzy** logic over discrete and continuous do- main. But this approach is not fast enough in control ap- plications [24]. In the rest of this paper we treat only in- terval **type**-**2** **fuzzy** sets over discrete domains.

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In **general** frequency divider categorize into two groups, analog and digital (Ching et al,. 2009). Regenerative frequency divider and the injection locking frequency divider (ILFD) belong to the analog frequency dividers while the static and dynamic logic dividers are included in digital frequency dividers. Since the dynamic logic has benefits of high speed and programmability, it is more common in the application of communication systems (Ting et al,. 2010).

In this chapter, we will present the novel applications of the IT2-FLCs into the research area of computer games. In this context, we will handle two well-known computer games, namely Flappy Bird [11], [12] and Lunar Lander [13], [14], to show the abilities of the IT2-FLCs. From a control engineering point of view, as the game player can be seen as the main controller in a feedback loop, we will transform the game logic of flappy bird into a reference tracking problem while handling the moon landing problem as a position control problem. Thus, we will construct an intelligent control system composed of three main subsystems: reference generator, the main controller, and game dynamics. In this chapter, we will design and then employ an IT2-FLC as the main controller in a feedback loop such that to have a satisfactory performance and to be able to handle the various uncertainties of the games. In this context, we will briefly present the **general** structure and the design methods of two IT2-FLCs which are the SIT2-FLC and DIT2-FLC. In this chapter, we will design a SIT2-FLC for the game Flappy Bird while a DIT2-FLC structure for the game moon lander. The IT2-FLCs have been designed and implemented by using the Interval **Type**-**2** **Fuzzy** Logic Toolbox [28] for Matlab/Simulink. We will examine the performance of both IT2 **fuzzy** control systems with respect to their control system and game performances, in comparison with its T1 and conventional counterparts, to show that the presented structure can handle the uncertainties caused by the nature of the games much better.

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The UK 2001 Output Area Classification (OAC) is a free geodemographic classification which at the highest taxonomic level is comprised of seven classes named ‘Supergroups’. This classification employed a variant of hard c-means and critically in addition to the assigned class, distances 𝑑 𝑖𝑗 are also available for all n elements and c classes. Fisher and Tate (in press) employed [Equation 4] to create possibilistic memberships for each of the seven classes for each census reporting zone (Output Area - OA) for the City of Leicester. They compared PCM outcomes with **fuzzy** memberships from the equivalent FCM calculation favouring the PCM approach because of the constraint change in Equations 1 and **2**. Following **general** practice (Bezdek 1981 among others) Fisher and Tate (in press) selected a crisp value of m = **2**. However, m may have any value greater than 1, and following the method of Fisher (2010; see also Hwang and Rhee, 2007) by allowing m to vary [1.1 to 3.5 in this instance] we can generate **type**-**2** **fuzzy** sets for each Output Area.

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Zadeh [**2**] initiated another important extension of the concept of **type**-1 **fuzzy** sets in the form of **type**-**2** **fuzzy** sets. These sets are **fuzzy** sets whose membership grades themselves are **type**-1 **fuzzy** sets. Mendel and John [1] gave a simple representation for **Type**-**2** **fuzzy** sets. Due to the dependence of the membership functions on available linguistic information and numerical data. Linguistic information (e.g., rules from experts), in **general**, does not give any information about the shapes of the membership functions when membership functions are determined or tuned based on numerical data, the uncertainty in the numerical data e.g., noise, translates into uncertainty in the membership functions. In all such cases, **type**-**2** framework of **fuzzy** sets can be used to model information about the linguistic, numerical uncertainty very well [3].

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Decision making is broadly defined as include any choice or selection of alternatives and is therefore of importance in many fields in both “soft” social sciences and the “hard” disciplines of natural sciences. Soft set theory [6], firstly proposed by Molodtsov, is a **general** mathematical tool for dealing with uncertainty. The advantage of soft set theory is that it is free from the inadequacy of the parametrization tools of existing theories. It has been demonstrated that soft set theory brings about a rich potential for applications in many fields. It was growing rapidly over the years.

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In [5] the authors claim that the **general** definition of a T2 or IT2 FS, is too generic in certain contexts since there are no conditions on the MFs. As a consequence, the resulting FOUs and ESs that can be completely unrelated to the modelling of the concepts that they should represent. To overcome these limitations, they introduce a new kind of set, named Constrained **Type**-**2** (CT2) **fuzzy** sets, by imposing the concepts of shape coherency and explicit variability on the FOU and the ES.