Type-2 Fuzzy Logic: Circumventing the Defuzzification Bottleneck

Type-2 fuzzy inferencing for generalised, discretised type-2 fuzzy sets has been impeded by the computational complexity of the defuzzification stage of the fuzzy inferencing system. Indeed this stage is so complex computationally that it has come to be known as the defuzzification bottleneck. The computational complexity derives from the enormous number of embedded sets that have to be individually processed in order to effect defuzzification. Two new approaches to type-2 defuzzification are presented, the sampling method and the Greenfield-Chiclana Collapsing Defuzzifier. The sampling method and its variant, elite sampling, are techniques for the defuzzification of generalised type-2 fuzzy sets. In these methods a relatively small sample of the totality of embedded sets is randomly selected and processed. The small sample size drastically reduces the computational complexity of the defuzzification process, so that it may be speedily accomplished. The Greenfield-Chiclana Collapsing Defuzzifier relies upon the concept of the representative embedded set, which is an embedded set having the same defuzzified value as the type-2 fuzzy set that is to be defuzzified. By a process termed collapsing the type-2 fuzzy set is converted into a type-1 fuzzy set which, as an approximation to the representative embedded set, is known as the representative embedded set approximation. This type-1 fuzzy set is easily defuzzified to give the defuzzified value of the original type-2 fuzzy set. By this method the computational complexity of type-2 defuzzification is reduced enormously, since the representative embedded set approximation replaces the entire collection of embedded sets. The strategy was conceived as a generalised method, but so far only the interval version has been derived mathematically. The grid method of discretisation for type-2 fuzzy sets is also introduced in this thesis. Work on the defuzzification of type-2 fuzzy sets began around the turn of the millennium. Since that time a number of investigators have contributed methods in this area. These different approaches are surveyed, and the major methods implemented in code prior to their experimental evaluation. In these comparative experiments the grid method of defuzzification is employed. The experimental results show beyond doubt that the collapsing method performs the best of the interval alternatives. However, though the sampling method performs well experimentally, the results do not demonstrate it to be the best performing generalised technique

Fuzzy in 3-D: Contrasting Complex Fuzzy Sets with Type-2 Fuzzy Sets

CCIComplex fuzzy sets come in two forms, the standard form, postulated in 2002 by Ramot et al., and the 2011 innovation of pure complex fuzzy sets, proposed by Tamir et al.. In this paper we compare and contrast both forms of complex fuzzy set with type-2 fuzzy sets, as regards their rationales, applications, definitions, and structures. In addition, pure complex fuzzy sets are compared with type-2 fuzzy sets in relation to their inferencing operations. Complex fuzzy sets and type-2 fuzzy sets differ in their roles and applications. Their definitions differ also, though there is equivalence between those of a pure complex fuzzy set and a type-2 fuzzy set. Structural similarity is evident between these three-dimensional sets. Complex fuzzy sets are represented by a line, and type-2 fuzzy sets by a surface, but a surface is simply a generalisation of a line. This similarity is particularly evident between pure complex fuzzy sets and type-2 fuzzy sets, which are both mappings from the domain onto the unit square. Type-2 fuzzy sets were found not to be isomorphic to pure complex fuzzy sets

Defuzzification of the Discretised Generalised Type-2 Fuzzy Set: Experimental Evaluation

CCI - Centre for Computational Intelligence NOTICE: this is the author’s version of a work that was accepted for publication in Information Science. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version can be found by following the DOIThe work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised generalised type-2 fuzzy sets as created by the inference stage of a Mamdani Fuzzy Inferencing System. The exhaustive method of defuzzification for type-2 fuzzy sets is extremely slow, owing to its enormous computational complexity. Several approximate methods have been devised in response to this defuzzification bottleneck. In this paper we begin by surveying the main alternative strategies for defuzzifying a generalised type-2 fuzzy set: (1) Vertical Slice Centroid Type-Reduction; (2) the sampling method; (3) the elite sampling method; and (4) the $\alpha$-planes method. We then evaluate the different methods experimentally for accuracy and efficiency. For accuracy the exhaustive method is used as the standard. The test results are analysed statistically by means of the Wilcoxon Nonparametric Test and the elite sampling method shown to be the most accurate. In regards to efficiency, Vertical Slice Centroid Type-Reduction is demonstrated to be the fastest technique

Type-Reduction of the Discretised Interval Type-2 Fuzzy Set: Approaching the Continuous Case through Progressively Finer Discretisation.

The defuzzification of a type-2 fuzzy set is a two stage process consisting of firstly type-reduction, and secondly defuzzification of the resultant type-1 set. This paper considers three approaches to discrete interval type-reduction: 1. The exhaustive method which produces the Type-Reduced Set, 2. the Greenfield-Chiclana Collapsing Defuzzifier which gives rise to the Representative Embedded Set Approximation, and 3. the Nie-Tan Method from which the Nie-Tan Set is derived. In the discrete case these three type-1 sets are distinct. The behavior of the three sets under fine discretisation is investigated experimentally, in order to shed light on the relationships between the continuous versions of these type-1 sets

Secure Trapdoor Hash Functions Based on Public-Key Cryptosystems

In this paper we systematically consider examples representative of the various families of public-key cryptosystems to see if it would be possible to incorporate them into trapdoor hash functions, and we attempt to evaluate the resulting strengths and weaknesses of the functions we are able to construct. We are motivated by the following question: Question 1.2 How likely is it that the discoverer of a heretofore unknown public-key cryptosystem could subvert it for use in a plausible secure trapdoor hash algorithm? In subsequent sections, our investigations will lead to a variety of constructions and bring to light the non-adaptability of public-key cryptosystems that are of a \low density. More importantly, we will be led to consider from a new point of view the effects of the unsigned addition, shift, exclusive-or and other logical bit string operators that are presently used in constructing secure hash algorithms: We will show how the use of publickey cryptosystems leads to \fragile secure hash algorithms, and we will argue that circular shift operators are largely responsible for the security of modern high-speed secure hash algorithms

Join and Meet Operations for Interval-Valued Complex Fuzzy Logic

DMU Interdisciplinary Group in Intelligent Transport SystemsInterval-valued complex fuzzy logic is able to handle scenarios where both seasonality and uncertainty feature. The interval-valued complex fuzzy set is defined, and the interval valued complex fuzzy inferencing system outlined. Highly pertinent to complex fuzzy logic operations is the concept of rotational invariance, which is an intuitive and desirable characteristic. Interval-valued complex fuzzy logic is driven by interval-valued join and meet operations. Four pairs of alternative algorithms for these operations are specified; three pairs possesses the attribute of rotational invariance, whereas the other pair lacks this characteristic

Open Source Natural Language Processing

Our MQP aimed to introduce finite state machine based techniques for natural language processing into Hunspell, the world\u27s premiere Open Source spell checker used in several prominent projects such as Firefox and Open Office. We created compact machine-readable finite state transducer representations of 26 of the most commonly used languages on Wikipedia. We then created an automata based spell checker. In addition, we implemented an transducer based stemmer, which will be used in the future of transducer based morphological analysis

Geometric Defuzzification Revisited

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.In this paper the Geometric Defuzzification strategy for type-2 fuzzy sets is reappraised. For both discretised and geometric fuzzy sets the techniques for type-1, interval type-2, and generalised type-2 defuzzification are presented in turn. In the type-2 case the accuracy of Geometric Defuzzification is assessed through a series of test runs on interval type-2 fuzzy sets, using Exhaustive Defuzzification as the benchmark method. These experiments demonstrate the Geometric Defuzzifier to be wildly inaccurate. The test sets take many shapes; they are not confined to those type-2 sets with rotational symmetry that have previously been acknowledged by the technique’s developers to be problematic as regards accuracy. Type-2 Geometric Defuzzification is then examined theoretically. The defuzzification strategy is demonstrated to be built upon a fallacious application of the concept of centroid. This explains the markedly inaccurate experimental results. Thus the accuracy issues of type-2 Geometric Defuzzification are revealed to be inevitable, fundamental and significant

The Grid Method of Discretisation for Type-2 Fuzzy Sets

In order to perform fuzzy inferencing, it is normal practice to discretise fuzzy sets. For type-1 fuzzy sets there is only one method of discretisation, but type-2 fuzzy sets may be discretised in more than one way. This paper introduces the grid method of type-2 discretisation — a simpler, more convenient alternative to the established technique