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36 pages, 558 KiB

Open AccessEditor’s ChoiceArticle

Fuzzy Property Grammars for Gradience in Natural Language

by Adrià Torrens-Urrutia, Vilém Novák and María Dolores Jiménez-López

Mathematics 2023, 11(3), 735; https://doi.org/10.3390/math11030735 - 1 Feb 2023

Cited by 2 | Viewed by 1706

This paper introduces a new grammatical framework, Fuzzy Property Grammars (FPGr). This is a model based on Property Grammars and Fuzzy Natural Logic. Such grammatical framework is constraint-based and provides a new way to formally characterize gradience by representing grammaticality degrees regarding linguistic [...] Read more.

This paper introduces a new grammatical framework, Fuzzy Property Grammars (FPGr). This is a model based on Property Grammars and Fuzzy Natural Logic. Such grammatical framework is constraint-based and provides a new way to formally characterize gradience by representing grammaticality degrees regarding linguistic competence (without involving speakers judgments). The paper provides a formal-logical characterization of FPGr. A test of the framework is presented by implementing an FPGr for Spanish. FPGr is a formal theory that may serve linguists, computing scientists, and mathematicians since it can capture infinite grammatical structures within the variability of a language. Full article

(This article belongs to the Special Issue FSTA: Fuzzy Set Theory and Applications)

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13 pages, 300 KiB

Open AccessArticle

Coherent Upper Conditional Previsions Defined through Conditional Aggregation Operators

by Serena Doria

Mathematics 2022, 10(24), 4728; https://doi.org/10.3390/math10244728 - 13 Dec 2022

Cited by 1 | Viewed by 1152

Abstract

Conditional aggregation operators are introduced, and coherent upper conditional previsions are constructed by sub-additive, positively homogenous, and shift-invariant conditional aggregation operators. Composed operators defined by positively homogenous conditional aggregation operators are proven to be conditional aggregation operators and they are involved in the [...] Read more.

Conditional aggregation operators are introduced, and coherent upper conditional previsions are constructed by sub-additive, positively homogenous, and shift-invariant conditional aggregation operators. Composed operators defined by positively homogenous conditional aggregation operators are proven to be conditional aggregation operators and they are involved in the construction of coherent upper conditional previsions. The given results show that the composed conditional aggregation operator obtained as the supremum of the class of the Choquet integrals with respect to Hausdorff outer measures defined by bi-Lipschitz equivalent metrics is a coherent upper conditional prevision. Full article

(This article belongs to the Special Issue FSTA: Fuzzy Set Theory and Applications)

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15 pages, 325 KiB

Open AccessArticle

Fuzzy Extension of Crisp Metric by Means of Fuzzy Equivalence Relation

by Olga Grigorenko and Alexander Šostak

Mathematics 2022, 10(24), 4648; https://doi.org/10.3390/math10244648 - 8 Dec 2022

Cited by 1 | Viewed by 1410

Abstract

We develop an alternative approach to the fuzzy metric concept, which we obtain by fuzzy extension of a crisp metric d on a set X by means of a fuzzy equivalence relation E on the set

{I R}^{+}

. We call it [...] Read more.

We develop an alternative approach to the fuzzy metric concept, which we obtain by fuzzy extension of a crisp metric d on a set X by means of a fuzzy equivalence relation E on the set

{I R}^{+}

. We call it an E-d metric and study its properties and relations with “classical” fuzzy metrics. Our special interest is in the topologies and fuzzy topologies induced by E-d metrics. Full article

(This article belongs to the Special Issue FSTA: Fuzzy Set Theory and Applications)

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22 pages, 659 KiB

Open AccessArticle

From Fuzzy Information to Community Detection: An Approach to Social Networks Analysis with Soft Information

by Inmaculada Gutiérrez, Daniel Gómez, Javier Castro and Rosa Espínola

Mathematics 2022, 10(22), 4348; https://doi.org/10.3390/math10224348 - 19 Nov 2022

Cited by 1 | Viewed by 1388

Abstract

On the basis of network analysis, and within the context of modeling imprecision or vague information with fuzzy sets, we propose an innovative way to analyze, aggregate and apply this uncertain knowledge into community detection of real-life problems. This work is set on [...] Read more.

On the basis of network analysis, and within the context of modeling imprecision or vague information with fuzzy sets, we propose an innovative way to analyze, aggregate and apply this uncertain knowledge into community detection of real-life problems. This work is set on the existence of one (or multiple) soft information sources, independent of the network considered, assuming this extra knowledge is modeled by a vector of fuzzy sets (or a family of vectors). This information may represent, for example, how much some people agree with a specific law, or their position against several politicians. We emphasize the importance of being able to manage the vagueness which usually appears in real life because of the common use of linguistic terms. Then, we propose a constructive method to build fuzzy measures from fuzzy sets. These measures are the basis of a new representation model which combines the information of a network with that of fuzzy sets, specifically when it comes to linguistic terms. We propose a specific application of that model in terms of finding communities in a network with additional soft information. To do so, we propose an efficient algorithm and measure its performance by means of a benchmarking process, obtaining high-quality results. Full article

(This article belongs to the Special Issue FSTA: Fuzzy Set Theory and Applications)

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17 pages, 5175 KiB

Open AccessArticle

A Combined Approach of Fuzzy Cognitive Maps and Fuzzy Rule-Based Inference Supporting Freeway Traffic Control Strategies

by Mehran Amini, Miklos F. Hatwagner and Laszlo T. Koczy

Mathematics 2022, 10(21), 4139; https://doi.org/10.3390/math10214139 - 5 Nov 2022

Cited by 5 | Viewed by 1914

Abstract

Freeway networks, despite being built to handle the transportation needs of large traffic volumes, have suffered in recent years from an increase in demand that is rarely resolvable through infrastructure improvements. Therefore, the implementation of particular control methods constitutes, in many instances, the [...] Read more.

Freeway networks, despite being built to handle the transportation needs of large traffic volumes, have suffered in recent years from an increase in demand that is rarely resolvable through infrastructure improvements. Therefore, the implementation of particular control methods constitutes, in many instances, the only viable solution for enhancing the performance of freeway traffic systems. The topic is fraught with ambiguity, and there is no tool for understanding the entire system mathematically; hence, a fuzzy suggested algorithm seems not just appropriate but essential. In this study, a fuzzy cognitive map-based model and a fuzzy rule-based system are proposed as tools to analyze freeway traffic data with the objective of traffic flow modeling at a macroscopic level in order to address congestion-related issues as the primary goal of the traffic control strategies. In addition to presenting a framework of fuzzy system-based controllers in freeway traffic, the results of this study demonstrated that a fuzzy inference system and fuzzy cognitive maps are capable of congestion level prediction, traffic flow simulation, and scenario analysis, thereby enhancing the performance of the traffic control strategies involving the implementation of ramp management policies, controlling vehicle movement within the freeway by mainstream control, and routing control. Full article

(This article belongs to the Special Issue FSTA: Fuzzy Set Theory and Applications)

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30 pages, 2073 KiB

Open AccessArticle

On an Application of Lattice Integral Transforms in Image Processing

by Michal Holčapek, Viec Bui Quoc and Petr Ferbas

Mathematics 2022, 10(21), 4077; https://doi.org/10.3390/math10214077 - 2 Nov 2022

Viewed by 1672

Abstract

The lattice integral transforms have been introduced to generalize lower and upper fuzzy transforms for lattice-valued functions that are used to approximate original functions from below and above. They are defined in complete analogy with classical integral transforms, particularly, the product of a [...] Read more.

The lattice integral transforms have been introduced to generalize lower and upper fuzzy transforms for lattice-valued functions that are used to approximate original functions from below and above. They are defined in complete analogy with classical integral transforms, particularly, the product of a lattice-valued function and a fuzzy relation called the integral kernel is integrated by a Sugeno-like fuzzy integral. In the article, we first investigate the conditions under which lattice integral transforms preserve (reverse) constant functions, which appears to be a fundamental presumption for a successful approximation of lattice-valued functions. Further, we show how the lattice integral transforms can be applied in image processing, more specifically, in non-linear filtering, compression/decompression, and opening/closing of images. We demonstrate that the filters based on integral transforms generalize the popular median filter as well as minimum and maximum filters, and also opening and closing defined using fuzzy morphological erosion and dilation. We illustrate the proposed methods in various selected images. Full article

(This article belongs to the Special Issue FSTA: Fuzzy Set Theory and Applications)

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21 pages, 391 KiB

Open AccessArticle

Weak Inflationary BL-Algebras and Filters of Inflationary (Pseudo) General Residuated Lattices

by Xiaohong Zhang, Rong Liang and Benjamín Bedregal

Mathematics 2022, 10(18), 3394; https://doi.org/10.3390/math10183394 - 19 Sep 2022

Cited by 10 | Viewed by 1514

Abstract

After the research on naBL-algebras gained by the non-associative t-norms and overlap functions, inflationary BL-algebras were also studied as a recent kind of non-associative generalization of BL-algebras, which can be obtained by general overlap functions. In this paper, we show that not every [...] Read more.

After the research on naBL-algebras gained by the non-associative t-norms and overlap functions, inflationary BL-algebras were also studied as a recent kind of non-associative generalization of BL-algebras, which can be obtained by general overlap functions. In this paper, we show that not every inflationary general overlap function can induce an inflationary BL-algebra by a counterexample and thus propose the new concept of weak inflationary BL-algebras. We prove that each inflationary general overlap function corresponds to a weak inflationary BL-algebra; therefore, two mistaken results in the previous paper are revised. In addition, some properties satisfied by weak inflationary BL-algebras are discussed, and the relationships among some non-classical logic algebras are analyzed. Finally, we establish the theory of filters and quotient algebras of inflationary general residuated lattice (IGRL) and inflationary pseudo-general residuated lattice (IPGRL), and characterize the properties of some kinds of IGRLs and IPGRLs by naBL-filters, (weak) inflationary BL-filters, and weak inflationary pseudo-BL-filters. Full article

(This article belongs to the Special Issue FSTA: Fuzzy Set Theory and Applications)

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9 pages, 263 KiB

Open AccessFeature PaperArticle

Remarks on Sugeno Integrals on Bounded Lattices

by Radomír Halaš, Jozef Pócs and Jana Pócsová

Mathematics 2022, 10(17), 3078; https://doi.org/10.3390/math10173078 - 26 Aug 2022

Cited by 2 | Viewed by 1633

Abstract

A discrete Sugeno integral on a bounded distributive lattice L is defined as an idempotent weighted lattice polynomial. Another possibility for axiomatization of Sugeno integrals is to consider compatible aggregation functions, uniquely extending the L-valued fuzzy measures. This paper aims to study [...] Read more.

A discrete Sugeno integral on a bounded distributive lattice L is defined as an idempotent weighted lattice polynomial. Another possibility for axiomatization of Sugeno integrals is to consider compatible aggregation functions, uniquely extending the L-valued fuzzy measures. This paper aims to study the mentioned unique extension property concerning the possible extension of a Sugeno integral to non-distributive lattices. We show that this property is equivalent to the distributivity of the underlying bounded lattice. As a byproduct, an alternative proof of Iseki’s result, stating that a lattice having prime ideal separation property for every pair of distinct elements is distributive, is provided. Full article

(This article belongs to the Special Issue FSTA: Fuzzy Set Theory and Applications)

15 pages, 319 KiB

Open AccessArticle

Pseudo General Overlap Functions and Weak Inflationary Pseudo BL-Algebras

by Rong Liang and Xiaohong Zhang

Mathematics 2022, 10(16), 3007; https://doi.org/10.3390/math10163007 - 20 Aug 2022

Cited by 13 | Viewed by 1759

Abstract

General overlap functions are generalized on the basis of overlap functions, which have better application effects in classification problems, and the (weak) inflationary BL-algebras as the related algebraic structure were also studied. However, general overlap functions are a class of aggregation operators, and [...] Read more.

General overlap functions are generalized on the basis of overlap functions, which have better application effects in classification problems, and the (weak) inflationary BL-algebras as the related algebraic structure were also studied. However, general overlap functions are a class of aggregation operators, and their commutativity puts certain restrictions on them. In this article, we first propose the notion of pseudo general overlap functions as a non-commutative generalization of general overlap functions, so as to extend their application range, then illustrate their relationship with several other commonly used aggregation functions, and characterize some construction methods. Secondly, the residuated implications induced by inflationary pseudo general overlap functions are discussed, and some examples are given. Then, on this basis, we show the definitions of inflationary pseudo general residuated lattices (IPGRLs) and weak inflationary pseudo BL-algebras, and explain that the weak inflationary pseudo BL-algebras can be gained by the inflationary pseudo general overlap functions. Moreover, they are more extensive algebraic structures, thus enriching the content of existing non-classical logical algebra. Finally, their related properties and their relations with some algebraic structures such as non-commutative residuated lattice-ordered groupoids are investigated. The legend reveals IPGRLs include all non-commutative algebraic structures involved in the article. Full article

(This article belongs to the Special Issue FSTA: Fuzzy Set Theory and Applications)

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31 pages, 1528 KiB

Open AccessArticle

Rough Semiring-Valued Fuzzy Sets with Application

by Jiří Močkoř, Petr Hurtik and David Hýnar

Mathematics 2022, 10(13), 2274; https://doi.org/10.3390/math10132274 - 29 Jun 2022

Cited by 6 | Viewed by 1472

Abstract

Many of the new fuzzy structures with complete

M V

-algebras as value sets, such as hesitant, intuitionistic, neutrosophic, or fuzzy soft sets, can be transformed into one type of fuzzy set with values in special complete algebras, called

A M V

-algebras. [...] Read more.

Many of the new fuzzy structures with complete

M V

-algebras as value sets, such as hesitant, intuitionistic, neutrosophic, or fuzzy soft sets, can be transformed into one type of fuzzy set with values in special complete algebras, called

A M V

-algebras. The category of complete

A M V

-algebras is isomorphic to the category of special pairs

(R, R^{*})

of complete commutative semirings and the corresponding fuzzy sets are called

(R, R^{*})

-fuzzy sets. We use this theory to define

(R, R^{*})

-fuzzy relations, lower and upper approximations of

(R, R^{*})

-fuzzy sets by

(R, R^{*})

-relations, and rough

(R, R^{*})

-fuzzy sets, and we show that these notions can be universally applied to any fuzzy type structure that is transformable to

(R, R^{*})

-fuzzy sets. As an example, we also show how this general theory can be used to determine the upper and lower approximations of a color segment corresponding to a particular color. Full article

(This article belongs to the Special Issue FSTA: Fuzzy Set Theory and Applications)

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