Generalized Extension Principle Research Articles

A note on a note to the finite T-sum of L-R fuzzy numbers and further considerations

This paper addresses a flaw in the application of the Nguyen type theorem recently published in Fuzzy Sets and Systems by Urbanski et al. (2021). A counterexample is given to prove wrong the suggested application. Appropriate applications to product t-norm arithmetic of fuzzy intervals are provided. It is observed that the Nguyen type theorem introduces equivalent level of complexity in fuzzy arithmetic compared with the direct use of the generalized extension principle.

A Generalized Fuzzy Extension Principle and Its Application to Information Fusion

Zadeh's extension principle (ZEP) is a fundamental concept in fuzzy set (FS) theory that enables crisp mathematical operation on FSs. A well-known shortcoming of ZEP is that the height of the output FS is determined by the lowest height of the input FSs. In this article, we introduce a generalized extension principle (GEP) that eliminates this weakness and provides flexibility and control over how membership values are mapped from input to output. Furthermore, we provide a computationally efficient point-based FS representation. In light of our new definition, we discuss two approaches to perform aggregation of FSs using the Choquet integral. The resultant integrals generalize prior work and lay a foundation for future extensions. Last, we demonstrate the extended integrals via a combination of synthetic and real-world examples.

Generalized extension principle for non-normal fuzzy sets

The conventional extension principle is established on the Euclidean space and defined by considering the minimum or t-norm operator in which the fuzzy sets are usually assumed to be normal. The previous work on generalized extension principle was also based on the normal fuzzy sets. Since the non-normal fuzzy sets occur frequently in practical applications, in this paper, the generalized extension principle based on the non-normal fuzzy sets is established in which the general aggregation operator and Hausforff space are taken into account.

Continuity of Fuzzified Functions Using the Generalized Extension Principle

To fuzzify the crisp functions, the extension principle has been widely used for performing this fuzzification. The purpose of this paper is to investigate the continuity of fuzzified function using the more generalized extension principle. The Hausdorff metric will be invoked to study the continuity of fuzzified function. We also apply the principle of continuity of fuzzified function to the fuzzy topological vector space.

Generalized aggregation of fuzzy truth values

This paper deals with generalized extended aggregation operations in accordance with generalized extension principle and provides a theoretical basis for generalized extended aggregation operations. Generalized extended aggregation operations are studied on the algebra of convex (resp. normal) fuzzy truth values. When an aggregation operation is continuous, the generalized extended aggregation operation is proven to be an aggregation operation on fuzzy truth values (FTV aggregation operation, for short) for arbitrary t-norm on the algebra of convex normal fuzzy truth values.

On Advanced Computing With Words Using the Generalized Extension Principle for Type-1 Fuzzy Sets

In this paper, we propose and demonstrate an effective methodology for implementing the generalized extension principle to solve Advanced Computing with Words (ACWW) problems. Such problems involve implicit assignments of linguistic truth, probability, and possibility. To begin, we establish the vocabularies of the words involved in the problems, and then collect data from subjects about the words after which fuzzy set models for the words are obtained by using the Interval Approach (IA) or the Enhanced Interval Approach (EIA). Next, the solutions of the ACWW problems, which involve the fuzzy set models of the words, are formulated using the Generalized Extension Principle. Because the solutions to those problems involve complicated functional optimization problems that cannot be solved analytically, we then develop a numerical method for their solution. Finally, the resulting fuzzy set solutions are decoded into natural language words using Jaccard's similarity measure. We explain how ACWW problems can solve some potential prototype engineering problems and connect the methodology of this paper with Perceptual Computing.

Generalized extended fuzzy implications

This paper is devoted to investigate generalized extended fuzzy implications in accordance with the generalized extension principle. The algebraic properties of generalized extended fuzzy implications are discussed on different algebras of fuzzy truth values. On this basis, fuzzy-valued fuzzy implications are constructed from arbitrary fuzzy implications and t-norms. Moreover, a fuzzy-valued fuzzy implication is identical with a generalized extended fuzzy implication on the algebra of fuzzy values, when they are induced by exactly the same continuous fuzzy implication and same upper semicontinuous t-norm.

Probabilistic automata for computing with words

Usually, probabilistic automata and probabilistic grammars have crisp symbols as inputs, which can be viewed as the formal models of computing with values. In this paper, we first introduce probabilistic automata and probabilistic grammars for computing with (some special) words, where the words are interpreted as probabilistic distributions or possibility distributions over a set of crisp symbols. By probabilistic conditioning, we then establish a retraction principle from computing with words to computing with values for handling crisp inputs and a generalized extension principle from computing with words to computing with all words for handling arbitrary inputs. These principles show that computing with values and computing with all words can be respectively implemented by computing with some special words. To compare the transition probabilities of two near inputs, we also examine some analytical properties of the transition probability functions of generalized extensions. Moreover, the retractions and the generalized extensions are shown to be equivalence-preserving. Finally, we clarify some relationships among the retractions, the generalized extensions, and the extensions studied by Qiu and Wang.

On fuzzifications of discrete dynamical systems

Let X denote a locally compact metric space and φ : X → X be a continuous map. In the 1970s Zadeh presented an extension principle helping us to fuzzify the dynamical system ( X, φ), i.e., to obtain a map Φ for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets Syst. 61 (1994) 277–283] to generalize Zadeh’s original extension principle. In this paper we study basic properties of so-called g-fuzzifications, such as their continuity properties. We also show that, for any g-fuzzification: (i) a uniformly convergent sequence of uniformly continuous maps on X induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and (ii) a conjugacy (resp., a semi-conjugacy) between two discrete dynamical systems can be extended to a conjugacy (resp., a semi-conjugacy) between fuzzified dynamical systems. Throughout this paper we consider different topological structures in the space of fuzzy sets, namely, the sendograph, endograph and levelwise topologies.

Generalized Extension Principle

The conventional extension principle was established on the Euclidean space and defined by considering the minimum or t-norm operator. The generalized extension principle proposed in this paper is established on the Hausdorff space and defined by considering an operator that is more general than the t-norm operator. On the other hand, based on the topological structure, we also discuss the properties of 0-level sets by considering the closure. Many interesting and useful equalities considering the 0-level sets will be obtained in this paper.

Three new cut sets of fuzzy sets and new theories of fuzzy sets

Three new cut sets are introduced from the view points of neighborhood and Q -neighborhood in fuzzy topology and their properties are discussed. By the use of these cut sets, new decomposition theorems, new representation theorems, new extension principles and new fuzzy linear mappings are obtained. Then inner project of fuzzy relations, generalized extension principle and new composition rule of fuzzy relations are given. In the end, we present axiomatic descriptions for different cut sets and show the three most intrinsic properties for each cut set.

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L 2(ℝ d ) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H s (ℝ d ) and H −s (ℝ d ). In terms of masks for φ,ψ 1,…,ψ L ∈H s (ℝ d ) and $\tilde{\phi},\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}\in H^{-s}({\mathbb{R}}^{d})$ , simple sufficient conditions are given to ensure that (X s (φ;ψ 1,…,ψ L ), $X^{-s}(\tilde{\phi};\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}))$ forms a pair of dual wavelet frames in (H s (ℝ d ),H −s (ℝ d )), where $$\begin{array}{ll}X^{s}\bigl(\phi;\psi^{1},\ldots,\psi^{L}\bigr):=&\bigl\{\phi(\cdot-k):k\in {\mathbb{Z}}^{d}\bigr\}\\[9pt]&{}\cup\bigl\{2^{j(d/2-s)}\psi^{\ell}(2^{j}\cdot-k):j\in {\mathbb{N}}_{0},\ k\in{\mathbb{Z}}^{d},\ \ell=1,\ \ldots,L\bigr\}.\end{array}$$ For s>0, the key of this general mixed extension principle is the regularity of φ, ψ 1,…,ψ L , and the vanishing moments of $\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}$ , while allowing $\tilde{\phi}$ , $\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}$ to be tempered distributions not in L 2(ℝ d ) and ψ 1,…,ψ L to have no vanishing moments. So, the systems X s (φ;ψ 1,…,ψ L ) and $X^{-s}(\tilde{\phi};\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L})$ may not be able to be normalized into a frame of L 2(ℝ d ). As an example, we show that {2 j(1/2−s) B m (2 j ⋅−k):j∈ℕ0,k∈ℤ} is a wavelet frame in H s (ℝ) for any 0<s<m−1/2, where B m is the B-spline of order m. This simple construction is also applied to multivariate box splines to obtain wavelet frames with short supports, noting that it is hard to construct nonseparable multivariate wavelet frames with small supports. Applying this general mixed extension principle, we obtain and characterize dual Riesz bases $(X^{s}(\phi;\psi^{1},\ldots,\psi^{L}),X^{-s}(\tilde{\phi};\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}))$ in Sobolev spaces (H s (ℝ d ),H −s (ℝ d )). For example, all interpolatory wavelet systems in (Donoho, Interpolating wavelet transform. Preprint, 1997) generated by an interpolatory refinable function φ∈H s (ℝ) with s>1/2 are Riesz bases of the Sobolev space H s (ℝ). This general mixed extension principle also naturally leads to a characterization of the Sobolev norm of a function in terms of weighted norm of its wavelet coefficient sequence (decomposition sequence) without requiring that dual wavelet frames should be in L 2(ℝ d ), which is quite different from other approaches in the literature.

INFLUENCE OF PARTIAL SAFETY FACTORS ON DESIGN RELIABILITY OF STEEL STRUCTURES ‐ PROBABILITY AND FUZZY PROBABILITY ASSESSMENTS

The paper is aimed at the analysis of the influence of partial safety factors on the reliability of designing steel members. The reliability assessment of a steel member is based on extensive experimental research on real material and geometric characteristics evaluated by samples obtained from frequently manufactured products. Material and geometric characteristics are random variables represented by histograms during the probabilistic assessment of reliability. The random variability of structural dimensions and of material characteristics does not present the only source of uncertainty present during design according to standards. The reliability of steel structures design is also dependent on the partial safety factor values, which are specified by the design standards. Partial safety factors are basic indicators, which determine structural dimensions in relation to loading. The probabilistic assessment of reliability is performed as a parametric study in the first part of the numerical analysis. The probability of failure is analysed in dependence on values of partial safety factors of material, permanent loading and long time variable loading. Partial safety factors are considered as fuzzy numbers with triangular membership functions in the second part of the numerical analysis. Membership functions of failure probability are evaluated using the general extension principle. The most important results are then summarised and additional applications of the verifications of limit states design criteria are described.

Retraction and Generalized Extension of Computing With Words

Fuzzy automata, whose input alphabet is a set of numbers or symbols, are a formal model of computing with values. Motivated by Zadeh's paradigm of computing with words rather than numbers, Ying proposed a kind of fuzzy automata, whose input alphabet consists of all fuzzy subsets of a set of symbols, as a formal model of computing with all words. In this paper, we introduce a somewhat general formal model of computing with (some special) words. The new features of the model are that the input alphabet only comprises some (not necessarily all) fuzzy subsets of a set of symbols and the fuzzy transition function can be specified arbitrarily. By employing the methodology of fuzzy control, we establish a retraction principle from computing with words to computing with values for handling crisp inputs and a generalized extension principle from computing with words to computing with all words for handling fuzzy inputs. These principles show that computing with values and computing with all words can be respectively implemented by computing with words. Some algebraic properties of retractions and generalized extensions are addressed as well.

Universal Bias Dependence of Excess Current Induced by Self-Heating Effect for a-Si:H TFTs

Fuzzy automata, whose input alphabet is a set of numbers or symbols, are a formal model of computing with values. Motivated by Zadeh's paradigm of computing with words rather than numbers, Ying proposed a kind of fuzzy automata, whose input alphabet consists of all fuzzy subsets of a set of symbols, as a formal model of computing with all words. In this paper, we introduce a somewhat general formal model of computing with (some special) words. The new features of the model are that the input alphabet only comprises some (not necessarily all) fuzzy subsets of a set of symbols and the fuzzy transition function can be specified arbitrarily. By employing the methodology of fuzzy control, we establish a retraction principle from computing with words to computing with values for handling crisp inputs and a generalized extension principle from computing with words to computing with all words for handling fuzzy inputs. These principles show that computing with values and computing with all words can be respectively implemented by computing with words. Some algebraic properties of retractions and generalized extensions are addressed as well.

Measure extension theorems for [formula omitted]-spaces

The theme of this paper is the extension of continuous valuations on the lattice of open sets of a T 0 -space to Borel measures. A general extension principle is derived that provides a unified approach to a variety of extension theorems including valuations that are directed suprema of simple valuations, continuous valuations on locally compact sober spaces, and regular valuations on coherent sober spaces.

Fuzzy reasoning based on generalized fuzzy If-Then rules

This paper focuses on a fuzzy reasoning method based on a generalized If-Then rule. Firstly, the antecedent and the consequent of an If-Then rule are considered and expressed as a component of a kind of binary L-type fuzzy relation on the product of the universes of discourse and the range of definition for a certain fuzzy attribute. Then a generalized extension principle based on this L-type fuzzy relation (FR-GEP) is constructed. Moreover, the paper gives a detailed description of this generalized If-Then rule using 20 well-known common implication operators in the framework of the composition of L-type fuzzy relations. Consequently, an L-type binary fuzzy reasoning method based on this generalized If-Then rule is established according to FR-GEP. © 2002 Wiley Periodicals, Inc.

From Computing with Numbers to Computing with Words

Interest in issues relating to consciousness has grown markedly during the last several years. And yet, nobody can claim that consciousness is a well-understood concept that lends itself to precise analysis. It may be argued that, as a concept, consciousness is much too complex to fit into the conceptual structure of existing theories based on Aristotelian logic and probability theory. An approach suggested in this paper links consciousness to perceptions and perceptions to their descriptors in a natural language. In this way, those aspects of consciousness which relate to reasoning and concept formation are linked to what is referred to as the methodology of computing with words (CW). Computing, in its usual sense, is centered on manipulation of numbers and symbols. In contrast, computing with words, or CW for short, is a methodology in which the objects of computation are words and propositions drawn from a natural language (e.g., small, large, far, heavy, not very likely, the price of gas is low and declining, Berkeley is near San Francisco, it is very unlikely that there will be a significant increase in the price of oil in the near future, etc.). Computing with words is inspired by the remarkable human capability to perform a wide variety of physical and mental tasks without any measurements and any computations. Familiar examples of such tasks are parking a car, driving in heavy traffic, playing golf, riding a bicycle, understanding speech, and summarizing a story. Underlying this remarkable capability is the brain's crucial ability to manipulate perceptions--perceptions of distance, size, weight, color, speed, time, direction, force, number, truth, likelihood, and other characteristics of physical and mental objects. Manipulation of perceptions plays a key role in human recognition, decision and execution processes. As a methodology, computing with words provides a foundation for a computational theory of perceptions: a theory which may have an important bearing on how humans make--and machines might make--perception-based rational decisions in an environment of imprecision, uncertainty, and partial truth. A basic difference between perceptions and measurements is that, in general, measurements are crisp, whereas perceptions are fuzzy. One of the fundamental aims of science has been and continues to be that of progressing from perceptions to measurements. Pursuit of this aim has led to brilliant successes. We have sent men to the moon; we can build computers that are capable of performing billions of computations per second; we have constructed telescopes that can explore the far reaches of the universe; and we can date the age of rocks that are millions of years old. But alongside the brilliant successes stand conspicuous underachievements and outright failures. We cannot build robots that can move with the agility of animals or humans; we cannot automate driving in heavy traffic; we cannot translate from one language to another at the level of a human interpreter; we cannot create programs that can summarize non-trivial stories; our ability to model the behavior of economic systems leaves much to be desired; and we cannot build machines that can compete with children in the performance of a wide variety of physical and cognitive tasks. It may be argued that underlying the underachievements and failures is the unavailability of a methodology for reasoning and computing with perceptions rather than measurements. An outline of such a methodology--referred to as a computational theory of perceptions--is presented in this paper. The computational theory of perceptions (CTP) is based on the methodology of CW. In CTP, words play the role of labels of perceptions, and, more generally, perceptions are expressed as propositions in a natural language. CW-based techniques are employed to translate propositions expressed in a natural language into what is called the Generalized Constraint Language (GCL). In this language, the meaning of a proposition is expressed as a generalized constraint, X isr R, where X is the constrained variable, R is the constraining relation, and isr is a variable copula in which r is an indexing variable whose value defines the way in which R constrains X. Among the basic types of constraints are possibilistic, veristic, probabilistic, random set, Pawlak set, fuzzy graph, and usuality. The wide variety of constraints in GCL makes GCL a much more expressive language than the language of predicate logic. In CW, the initial and terminal data sets, IDS and TDS, are assumed to consist of propositions expressed in a natural language. These propositions are translated, respectively, into antecedent and consequent constraints. Consequent constraints are derived from antecedent constraints through the use of rules of constraint propagation. The principal constraint propagation rule is the generalized extension principle. (ABSTRACT TRUNCATED)

From computing with numbers to computing with words. From manipulation of measurements to manipulation of perceptions

Discusses a methodology for reasoning and computing with perceptions rather than measurements. An outline of such a methodology-referred to as a computational theory of perceptions is presented in this paper. The computational theory of perceptions, or CTP for short, is based on the methodology of CW. In CTP, words play the role of labels of perceptions and, more generally, perceptions are expressed as propositions in a natural language. CW-based techniques are employed to translate propositions expressed in a natural language into what is called the Generalized Constraint Language (GCL). In this language, the meaning of a proposition is expressed as a generalized constraint, N is R, where N is the constrained variable, R is the constraining relation and isr is a variable copula in which r is a variable whose value defines the way in which R constrains S. Among the basic types of constraints are: possibilistic, veristic, probabilistic, random set, Pawlak set, fuzzy graph and usuality. The wide variety of constraints in GCL makes GCL a much more expressive language than the language of predicate logic. In CW, the initial and terminal data sets, IDS and TDS, are assumed to consist of propositions expressed in a natural language. These propositions are translated, respectively, into antecedent and consequent constraints. Consequent constraints are derived from antecedent constraints through the use of rules of constraint propagation. The principal constraint propagation rule is the generalized extension principle. The derived constraints are retranslated into a natural language, yielding the terminal data set (TDS). The rules of constraint propagation in CW coincide with the rules of inference in fuzzy logic. A basic problem in CW is that of explicitation of N, R, and r in a generalized constraint, X is R, which represents the meaning of a proposition, p, in a natural language.

Chaos, entropy and a generalized extension principle

We show a relationship between a very simple criterion, positive topological entropy and Li-Yorke chaos. A general definition of fuzzification and level set, based on t-norms/conorms and their diagonal functions, is introduced. The chaos theorem of Benhabib and Day for set valued mappings is considerably strengthened and generalised.