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2 pages, 155 KiB

Open AccessEditorial

Special Issue for the 10th Anniversary of Axioms: Logic

by Oscar Castillo

Axioms 2023, 12(5), 455; https://doi.org/10.3390/axioms12050455 - 06 May 2023

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Abstract

Published for the first time in 2012, Axioms is celebrating its 10th anniversary [...] Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Logic)

Research

Jump to: Editorial

22 pages, 565 KiB

Open AccessArticle

Granular Computing Approach to Evaluate Spatio-Temporal Events in Intuitionistic Fuzzy Sets Data through Formal Concept Analysis

by Imran Ali, Yongming Li and Witold Pedrycz

Axioms 2023, 12(5), 407; https://doi.org/10.3390/axioms12050407 - 22 Apr 2023

Cited by 1 | Viewed by 1290

Abstract

Knowledge discovery through spatial and temporal aspects of data related to occurrences of events has many applications in digital forensics. Specifically, in electronic surveillance, it is helpful to construct a timeline to analyze information. The existing techniques only analyze the occurrence and co-occurrence [...] Read more.

Knowledge discovery through spatial and temporal aspects of data related to occurrences of events has many applications in digital forensics. Specifically, in electronic surveillance, it is helpful to construct a timeline to analyze information. The existing techniques only analyze the occurrence and co-occurrence of events; however, in general, there are three aspects of events: occurrences (and co-occurrences), nonoccurrences, and uncertainty of occurrences/non-occurrences with respect to spatial and temporal aspects of data. These three aspects of events have to be considered to better analyze periodicity and predict future events. This study focuses on the spatial and temporal aspects given in intuitionistic fuzzy (IF) datasets using the granular computing (GrC) paradigm; formal concept analysis (FCA) was used to understand the granularity of data. The originality of the proposed approach is to discover the periodicity of events data given in IF sets through FCA and the GrC paradigm that helps to predict future events. An experimental evaluation was also performed to understand the applicability of the proposed methodology. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Logic)

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31 pages, 8574 KiB

Open AccessArticle

Application of the Methodology of Multi-Valued Logic Trees with Weighting Factors in the Optimization of a Proportional Valve

by Adam Deptuła, Michał Stosiak, Rafał Cieślicki, Mykola Karpenko, Kamil Urbanowicz, Paulius Skačkauskas and Anna Małgorzata Deptuła

Axioms 2023, 12(1), 8; https://doi.org/10.3390/axioms12010008 - 22 Dec 2022

Viewed by 1190

Abstract

Hydraulic valves are used to determine the set values of hydraulic quantities (flow rate, pressure, or pressure difference) in a hydraulic system or its part. This is achieved through the appropriate throttling of the stream flowing through the valve, which is automatically set [...] Read more.

Hydraulic valves are used to determine the set values of hydraulic quantities (flow rate, pressure, or pressure difference) in a hydraulic system or its part. This is achieved through the appropriate throttling of the stream flowing through the valve, which is automatically set by the operator (e.g., opening the throttle valve). The procedures for determining its static and dynamic properties were described using the example of modeling a two-stage proportional relief valve. Subsequently, the importance of the design and operational parameters was determined using multi-valued logic trees. Modeling began with the determination of equations describing the flow and movement of moving parts in a valve. Based on the equations, a numerical model was then created, e.g., in the Matlab/Simulink environment (R2020b). The static characteristics were obtained as the result of a model analysis of slow changes in the flow rate through the valve. Various coefficients of logical products have not been taken into account in the separable and common minimization processes of multi-valued logic equation systems in any available literature. The results of the model tests can be used to optimize several types of hydraulic valve constructions. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Logic)

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18 pages, 366 KiB

Open AccessArticle

Linear Diophantine Fuzzy Rough Sets on Paired Universes with Multi Stage Decision Analysis

by Saba Ayub, Muhammad Shabir, Muhammad Riaz, Faruk Karaaslan, Dragan Marinkovic and Djordje Vranjes

Axioms 2022, 11(12), 686; https://doi.org/10.3390/axioms11120686 - 30 Nov 2022

Cited by 5 | Viewed by 1244

Abstract

Rough set (RS) and fuzzy set (FS) theories were developed to account for ambiguity in the data processing. The most persuasive and modernist abstraction of an FS is the linear Diophantine FS (LD-FS). This paper introduces a resilient hybrid linear Diophantine fuzzy RS [...] Read more.

Rough set (RS) and fuzzy set (FS) theories were developed to account for ambiguity in the data processing. The most persuasive and modernist abstraction of an FS is the linear Diophantine FS (LD-FS). This paper introduces a resilient hybrid linear Diophantine fuzzy RS model (LDF-RS) on paired universes based on a linear Diophantine fuzzy relation (LDF-R). This is a typical method of fuzzy RS (F-RS) and bipolar FRS (BF-RS) on two universes that are more appropriate and customizable. By using an LDF-level cut relation, the notions of lower approximation (L-A) and upper approximation (U-A) are defined. While this is going on, certain fundamental structural aspects of LD-FAs are thoroughly investigated, with some instances to back them up. This cutting-edge LDF-RS technique is crucial from both a theoretical and practical perspective in the field of medical assessment. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Logic)

10 pages, 250 KiB

Open AccessArticle

Proving, Refuting, Improving—Looking for a Theorem

by Branislav Boričić

Axioms 2022, 11(10), 559; https://doi.org/10.3390/axioms11100559 - 15 Oct 2022

Viewed by 1010

Abstract

Exploring the proofs and refutations of an abstract statement, conjecture with the aim to give a formal syntactic treatment of its proving–refuting process, we introduce the notion of extrapolation of a possibly unprovable statement having the form if A, then B, and propose [...] Read more.

Exploring the proofs and refutations of an abstract statement, conjecture with the aim to give a formal syntactic treatment of its proving–refuting process, we introduce the notion of extrapolation of a possibly unprovable statement having the form if A, then B, and propose a procedure that should result in the new statement if $A^{'}$ , then $B^{'}$ , which is similar to the starting one, but provable. We think that this procedure, based on the extrapolation method, can be considered a basic methodological tool applicable to prove–refute–improve any conjecture. This new notion, extrapolation, presents a dual counterpart of the well-known interpolation introduced in traditional logic sixty-five years ago. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Logic)

14 pages, 3864 KiB

Open AccessArticle

Spatial Fuzzy C-Means Clustering Analysis of U.S. Presidential Election and COVID-19 Related Factors in the Rustbelt States in 2020

by Shianghau Wu

Axioms 2022, 11(8), 401; https://doi.org/10.3390/axioms11080401 - 15 Aug 2022

Viewed by 1489

Abstract

The rustbelt states play a key role in determining the vote turnout in the U.S. elections. The current study attempts to utilize the spatial fuzzy C-means method to analyze the U.S. presidential election in the rustbelt states in 2020. We intend to explore [...] Read more.

The rustbelt states play a key role in determining the vote turnout in the U.S. elections. The current study attempts to utilize the spatial fuzzy C-means method to analyze the U.S. presidential election in the rustbelt states in 2020. We intend to explore that the U.S. presidential election had related factors, including COVID-19-related factors, such as the mask-wearing percentage and the COVID-19 death tolls in each county of the rust belt states. Contrary to the related literature, the study uses education level, number of house units, unemployment rate, household income, COVID-19-related factors and the share of Republican’s votes in the presidential election. The results indicate that spatial generalized fuzzy C-means analysis has better clustering results than the C-means clustering method. Moreover, the COVID-19 death toll in each county did not affect the Republican’s vote share in the rustbelt states, while the mask-wearing behavior in some regions had a negative impact on the Republican’s vote share. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Logic)

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13 pages, 295 KiB

Open AccessArticle

Does Set Theory Really Ground Arithmetic Truth?

by Alfredo Roque Freire

Axioms 2022, 11(7), 351; https://doi.org/10.3390/axioms11070351 - 21 Jul 2022

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Abstract

We consider the foundational relation between arithmetic and set theory. Our goal is to criticize the construction of standard arithmetic models as providing grounds for arithmetic truth. Our method is to emphasize the incomplete picture of both theories and to treat models as [...] Read more.

We consider the foundational relation between arithmetic and set theory. Our goal is to criticize the construction of standard arithmetic models as providing grounds for arithmetic truth. Our method is to emphasize the incomplete picture of both theories and to treat models as their syntactical counterparts. Insisting on the incomplete picture will allow us to argue in favor of the revisability of the standard-model interpretation. We start briefly characterizing the expansion of arithmetic ‘truth’ provided by the interpretation in a set theory. Interpreted versions of an arithmetic theory into set theories generally have more theorems than the original. This theorem expansion is not complete however. Using this, the set theoretic multiversalist concludes that there are multiple legitimate standard models of arithmetic. We suggest a different multiversalist conclusion: while there is a single arithmetic structure, its interpretation in each universe may vary or even not be possible. We continue by defining the coordination problem. We consider two independent communities of mathematicians responsible for deciding over new axioms for ZF and PA. How likely are they to be coordinated regarding PA’s interpretation in ZF? We prove that it is possible to have extensions of PA not interpretable in a given set theory ST. We further show that the number of extensions of arithmetic is uncountable, while interpretable extensions in ST are countable. We finally argue that this fact suggests that coordination can only work if it is assumed from the start. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Logic)

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14 pages, 588 KiB

Open AccessArticle

A Story of Computational Science: Colonel Titus’ Problem from the 17th Century

by Trond Steihaug

Axioms 2022, 11(6), 287; https://doi.org/10.3390/axioms11060287 - 14 Jun 2022

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Abstract

Experimentation and the evaluation of algorithms have a long history in algebra. In this paper we follow a single test example over more than 250 years. In 1685, John Wallis published A treatise of algebra, both historical and practical, containing a solution [...] Read more.

Experimentation and the evaluation of algorithms have a long history in algebra. In this paper we follow a single test example over more than 250 years. In 1685, John Wallis published A treatise of algebra, both historical and practical, containing a solution of Colonel Titus’ problem that was proposed to him around 1650. The Colonel Titus problem consists of three algebraic quadratic equations in three unknowns, which Wallis transformed into the problem of finding the roots of a fourth-order (quartic) polynomial. When Joseph Raphson published his method in 1690, he demonstrated the method on 32 algebraic equations and one of the examples was this quartic equation. Edmund Halley later used the same polynomial as an example for his new methods in 1694. Although Wallis used the method of Vietè, which is a digit–by–digit method, the more efficient methods of Halley and Raphson are clearly demonstrated in the works by Raphson and Halley. For more than 250 years the quartic equation has been used as an example in a wide range of solution methods for nonlinear equations. This paper provides an overview of the Colonel Titus problem and the equation first derived by Wallis. The quartic equation has four positive roots and the equation has been found to be very useful for analyzing the number of roots and finding intervals for the individual roots, in the Cardan–Ferrari direct approach for solving quartic equations, and in Sturm’s method of determining the number of real roots of an algebraic equation. The quartic equation, together with two other algebraic equations, have likely been the first set of test examples used to compare different iteration methods of solving algebraic equations. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Logic)

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19 pages, 11264 KiB

Open AccessArticle

Interval Type-3 Fuzzy Control for Automated Tuning of Image Quality in Televisions

by Oscar Castillo, Juan R. Castro and Patricia Melin

Axioms 2022, 11(6), 276; https://doi.org/10.3390/axioms11060276 - 09 Jun 2022

Cited by 15 | Viewed by 2288

Abstract

In this article, an intelligent system utilizing type-3 fuzzy logic for automated image quality tuning in televisions is presented. The tuning problem can be formulated as controlling the television imaging system to achieve the requirements of production quality. Previously, the tuning process has [...] Read more.

In this article, an intelligent system utilizing type-3 fuzzy logic for automated image quality tuning in televisions is presented. The tuning problem can be formulated as controlling the television imaging system to achieve the requirements of production quality. Previously, the tuning process has been carried out by experts, by manually adjusting the television imaging system on production lines to meet the quality control standards. In this approach, interval type-3 fuzzy logic is utilized with the goal of automating the tuning of televisions manufactured on production lines. An interval type-3 fuzzy approach for image tuning is proposed, so that the best image quality is obtained and, in this way, meet quality requirements. A system based on type-3 fuzzy control is implemented with good simulation results. The validation of the type-3 fuzzy approach is made by comparing the results with human experts on the process of electrical tuning of televisions. The key contribution is the utilization of type-3 fuzzy in the image tuning application, which has not been reported previously in the literature. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Logic)

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22 pages, 609 KiB

Open AccessArticle

Analysis of Interval-Valued Intuitionistic Fuzzy Aczel–Alsina Geometric Aggregation Operators and Their Application to Multiple Attribute Decision-Making

by Tapan Senapati, Radko Mesiar, Vladimir Simic, Aiyared Iampan, Ronnason Chinram and Rifaqat Ali

Axioms 2022, 11(6), 258; https://doi.org/10.3390/axioms11060258 - 29 May 2022

Cited by 33 | Viewed by 2311

Abstract

When dealing with the haziness that is intrinsic in decision analysis-driven decision making procedures, interval-valued intuitionistic fuzzy sets (IVIFSs) can be quite effective. Our approach to solving the multiple attribute decision making (MADM) difficulties, where all of the evidence provided by the decision-makers [...] Read more.

When dealing with the haziness that is intrinsic in decision analysis-driven decision making procedures, interval-valued intuitionistic fuzzy sets (IVIFSs) can be quite effective. Our approach to solving the multiple attribute decision making (MADM) difficulties, where all of the evidence provided by the decision-makers is demonstrated as interval-valued intuitionistic fuzzy (IVIF) decision matrices, in which all of the components are distinguished by an IVIF number (IVIFN), is based on Aczel–Alsina operational processes. We begin by introducing novel IVIFN operations including the Aczel–Alsina sum, product, scalar multiplication, and exponential. We may then create IVIF aggregation operators, such as the IVIF Aczel–Alsina weighted geometric operator, the IVIF Aczel–Alsina ordered weighted geometric operator, and the IVIF Aczel–Alsina hybrid geometric operator, among others. We present a MADM approach that relies on the IVIF aggregation operators that have been developed. A case study is used to demonstrate the practical applicability of the strategies proposed in this paper. By contrasting the newly developed technique with existing techniques, the method is capable of demonstrating the advantages of the newly developed approach. A key result of this work is the discovery that some of the current IVIF aggregation operators are subsets of the operators reported in this article. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Logic)

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18 pages, 654 KiB

Open AccessArticle

New Fuzzy Extensions on Binomial Distribution

by Gia Sirbiladze, Janusz Kacprzyk, Teimuraz Manjafarashvili, Bidzina Midodashvili and Bidzina Matsaberidze

Axioms 2022, 11(5), 220; https://doi.org/10.3390/axioms11050220 - 09 May 2022

Cited by 3 | Viewed by 1969

Abstract

The use of discrete probabilistic distributions is relevant to many practical tasks, especially in present-day situations where the data on distribution are insufficient and expert knowledge and evaluations are the only instruments for the restoration of probability distributions. However, in such cases, uncertainty [...] Read more.

The use of discrete probabilistic distributions is relevant to many practical tasks, especially in present-day situations where the data on distribution are insufficient and expert knowledge and evaluations are the only instruments for the restoration of probability distributions. However, in such cases, uncertainty arises, and it becomes necessary to build suitable approaches to overcome it. In this direction, this paper discusses a new approach of fuzzy binomial distributions (BDs) and their extensions. Four cases are considered: (1) When the elementary events are fuzzy. Based on this information, the probabilistic distribution of the corresponding fuzzy-random binomial variable is calculated. The conditions of restrictions on this distribution are obtained, and it is shown that these conditions depend on the ratio of success and failure of membership levels. The formulas for the generating function (GF) of the constructed distribution and the first and second order moments are also obtained. The Poisson distribution is calculated as the limit case of a fuzzy-random binomial experiment. (2) When the number of successes is of a fuzzy nature and is represented as a fuzzy subset of the set of possible success numbers. The formula for calculating the probability of convolution of binomial dependent fuzzy events is obtained, and the corresponding GF is built. As a result, the scheme for calculating the mathematical expectation of the number of fuzzy successes is defined. (3) When the spectrum of the extended distribution is fuzzy. The discussion is based on the concepts of a fuzzy-random event and its probability, as well as the notion of fuzzy random events independence. The fuzzy binomial upper distribution is specifically considered. In this case the fuzziness is represented by the membership levels of the binomial and non-binomial events of the complete failure complex. The GF of the constructed distribution and the first-order moment of the distribution are also calculated. Sufficient conditions for the existence of a limit distribution and a Poisson distribution are also obtained. (4) As is known, based on the analysis of lexical material, the linguistic spectrum of the statistical process of word-formation becomes two-component when switching to vocabulary. For this, two variants of the hybrid fuzzy-probabilistic process are constructed, which can be used in the analysis of the linguistic spectrum of the statistical process of word-formation. A fuzzy extension of standard Fuchs distribution is also presented, where the fuzziness is reflected in the growing numbers of failures. For better representation of the results, the examples of fuzzy BD are illustrated in each section. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Logic)

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