Methods of knowledge representation using type-1 fuzzy sets

Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes.

Fuzzy logic (FL) has been regarded as a useful methodology in modelling. The modeling performance of FL heavily relies on the modeling capability of a fuzzy set (FS). However, the MF of a FS cannot be arbitrarily accurate in practice and thus the centroid as a measure of a FS cannot be accurate to any degree. In face of inaccurate MF, the sensitivity of the centroid of a FS to the deviation of its MF from the ideal MF determines the accuracy of the centroid. To evaluate the modeling capability of a type-1 (T1) FS and an interval type-2 (IT2) FS, this paper will present a comparative study of the sensitivity of the centroid of a T1 FS and an IT2 FS to the deviation of their MFs from the ideal MFs. In this paper, equations relating the amount that the centroid of a T1 FS and an IT2 FS vary with the deviation of their MFs from the ideal MFs were established. Based on the derived equations, it was found that an IT2 FS is able to provide an additional design freedom to vary the sensitivity of its centroid compared to a T1 FS. This unique characteristic may enable an IT2 FS to minimize the sensitivity of the centroid and thus to achieve higher modeling accuracy. Furthermore, guidelines for designing an IT2 FS were provided.

The Wang-Mendel Approach (WMA) focuses on combining the numerical as well as linguistic information for achieving greater explainability for inference models. The standard WMA models the linguistic information using type-1 (T1) fuzzy sets (FSs), which have a reduced capability to model the semantics of linguistic information. Therefore, we propose a novel Enhanced WMA, which models the linguistic information using the type-2 (T2) FSs. Further, our Enhanced T2 FS-based WMA can be modified to reflect the use of interval type-2 (IT2) FSs, for modelling linguistic uncertainty. IT2 FSs are suitable when better uncertainty handling capabilities are required compared to T1 FSs, however, at a computational cost lesser than the T2 FSs. Performance of Enhanced WMA is demonstrated through a real-world crop-yield prediction problem in smart agriculture and an additional exemplar application on users’ satisfaction ratings. Further, we have compared our approach with the performance obtained from the T1 FS-based WMA and the original estimations given in the original data. We found that our Enhanced WMA provides more precise estimates than the other two with 95% confidence level. To the best of our knowledge, this is the first proposal of a T2 FSs method for enhancing the modeling of linguistic uncertainty in the WMA.

This chapter formally introduces type-1 fuzzy sets and fuzzy logic. It is the backbone for Chap. 3 and provides the foundation upon which type-2 fuzzy sets and systems are built in later chapters. Its coverage includes: crisp sets, type-1 fuzzy sets and associated concepts [including a short biography of Prof. Zadeh (the father of fuzzy sets and fuzzy logic)], type-1 fuzzy set defined, linguistic variables, returning to linguistic variables from a numerical value of a membership function, set theoretic operations for crisp and type-1 fuzzy sets, crisp and fuzzy relations and compositionson the same or different product spaces , compositions of a type-1 fuzzy set with a type-1 fuzzy relation, hedges, the Extension Principle (which is about functions of fuzzy sets), α-cuts (which are a powerful way to represent a type-1 fuzzy set in terms of intervals), functions of type-1 fuzzy sets computed by using α-cuts, multivariable membership functions and Cartesian products, crisp logic, going from crisp logic to fuzzy logic, Mamdani (engineering) implications, some final remarks, and an appendix about properties/laws of type-1 fuzzy sets. 35 examples are used to illustrate this chapter’s important concepts.

Knowledge representation is the core of artificial intelligence research. Knowledge representation methods include predicate logic, semantic network, computer programming language, database, mathematical model, graphics language, natural language, etc. To establish the intrinsic link between various knowledge representation methods, a unified knowledge representation model is necessary. According to ontology, system theory, and control theory, a standard model of knowledge representation that reflects the change of the objective world is proposed. The model is composed of input, processing, and output. This knowledge representation method is not a contradiction to the traditional knowledge representation method. It can express knowledge in terms of multivariate and multidimensional. It can also express process knowledge, and at the same time, it has a strong ability to solve problems. In addition, the standard model of knowledge representation provides a way to solve problems of non-precision and inconsistent knowledge.

Finite-interval-valued Type-2 Gaussian fuzzy numbers applied to fuzzy TODIM in a healthcare problem

In this paper, the Type-Reduced Set (TRS) of the continuous type-2 fuzzy set is considered as an object in its own right. The structures of the TRSs of both the interval and generalised forms of the type-2 fuzzy set are investigated. In each case the respective TRS structure is approached by first examining the TRS of the discretised set. The TRS of a continuous interval type-2 fuzzy set is demonstrated to be a continuous horizontal straight line, and that of a generalised type-2 fuzzy set, a continuous, convex curve. This analysis leads on to the concept of truncation, and the definition of the truncation grade. The truncated type-2 fuzzy set is then defined, whose TRS (and hence defuzzified value) is identical to that of the non-truncated type-2 fuzzy set. This result is termed the Type-2 Truncation Theorem, an immediate corollary of which is the Type-2 Equivalence Theorem which states that the defuzzified values of type-2 fuzzy sets that are equivalent under truncation are equal. Experimental corroboration of the equivalence of the non-truncated and truncated generalised type-2 fuzzy set is provided. The implications of these theorems for uncertainty quantification are explored. The theorem's repercussions for type-2 defuzzification employing the α-Planes Representation are examined; it is shown that the known inaccuracies of the α-Planes Method are deeply entrenched.

Assessment method sends messages to students to define and priorities what is important to learn and ultimately how they spend their time leaning it. Traditional grading methods are largely based on human judgments, which tend to be subjective. In addition, it is based on sharp criteria instead of fuzzy criteria and suffers from erroneous scores assigned by indifferent or inexperienced examiners, which represent a rich source of uncertainties, which might impair the credibility of the system. In an attempt to reduce uncertainties and provide more objective, reliable, and precise grading, a sophisticated assessment approach based on type-2 fuzzy set theory is developed. In this paper, interval type-2 (IT2) fuzzy sets, which are a special case of the general T2 fuzzy sets, are used. The transparency and capabilities of type-2 fuzzy sets in handling uncertainties is expected to provide an evaluation system able to justify and raise the quality and consistency of assessment judgments. A simplified implementation of interval type-2 fuzzy system using the basic knowledge of type-1 fuzzy is presented. A comparison between the use of type-1, interval type-2 fuzzy systems and the simplified IT2 fuzzy systems in reducing uncertainties and providing more transparent and fair assessment that can reflect needs of individual students and foster development is presented.

In many contexts, type-2 fuzzy sets (T2 FS) are obtained from a type-1 fuzzy set to which we wish to add uncertainty. However, in the current type-2 representation, there is no restriction on the shape of the footprint of uncertainty and the embedded sets (ESs) that can be considered acceptable. This leads, usually, to the loss of the semantic relationship between the T2 FS and the concept it models. As a consequence, the interpretability of some of the ESs and the explainability of the uncertainty measures obtained from them can decrease. To overcome these issues, constrained type-2 (CT2) fuzzy sets have been proposed. However, no formal definitions for some of their key components [e.g., acceptable ESs (AESs)] and constrained operations have been given. In this article, we provide some theoretical underpinning for the definition of CT2 sets, their inferencing and defuzzification method. To conclude, the constrained inference framework is presented, applied to two real-world cases and briefly compared to the standard interval type-2 inference and defuzzification method.

Modeling, design and control of a 4-arm delta parallel manipulator employing type-1 and interval type-2 fuzzy logic-based techniques for precision applications

Type-2 fuzzy set and type-2 intuitionistic fuzzy set are important tools for dealing with problems involving uncertainty and linguistic variables. However, it is sometimes difficult to model problems involving inconsistent data by using these approaches. In this paper, the concept of type-2 single-valued neutrosophic sets is defined in order to cope with this difficulty. Also, some distance measure methods for type-2 single-valued neutrosophic sets based on Hausdorff, Hamming and Euclidean distances are introduced and some properties of them are investigated. Furthermore, a multi-criteria group decision-making method is developed based on TOPSIS approach under the type-2 single-valued neutrosophic environment. Finally, an illustrative example is given in order to show the validity and process of the proposed method.

Application of Type-2 Interval Fuzzy Sets to Contractor Qualification Process

Multivariate modeling and type-2 fuzzy sets

This article provides a high-level retrospective of type-2 fuzzy sets and fuzzy logic systems. It explains how type-2 fuzzy sets can be used to model membership function uncertainties, and how by doing this smoother performance can be obtained than by using type-1 fuzzy sets. It also summarizes the notation that should be used for type-2 fuzzy sets, describes four important mathematical representations for these fuzzy sets, explains the differences between type-1 and type-2 fuzzy logic systems and which of the four representations is most useful when designing an optimal type-2 fuzzy logic system, provides a very useful strategy for optimal designs of fuzzy logic systems – one that guarantees performance improvement as one goes from a type-1 fuzzy logic system to a type-2 fuzzy logic system design – , and describes four methods for simplifying the designs of type-2 fuzzy logic systems. Finally, it explains why type-2 fuzzy sets can capture two kinds of linguistic uncertainties simultaneously (the uncertainty of an individual and the uncertainties of a group about a word), whereas type-1 fuzzy sets cannot, and that such type-2 fuzzy set word models are what should be used to implement Zadeh’s Computing With Words paradigm.

This paper employs the concept of fuzzify upgrade operator to describe the relationship between crisp sets, fuzzy sets and higher order fuzzy sets, such as type-2 fuzzy sets. Theory analysis and case studies demonstrate that a crisp set can be represented by a type-1 fuzzy set, a type-1 fuzzy set can be represented by a type-2 fuzzy set, and a type-n (n>=2) fuzzy set can be represented by a type-(n+1) (n>=2) fuzzy set. The fuzzify upgrade operator makes crisp sets, type-1 fuzzy sets, type-2 fuzzy sets and any fuzzy sets much more accessible to all readers. The relationship between crisp sets and fuzzy sets are explored.