Transactions on Fuzzy Sets and Systems
Volume:2 Issue: 2, Fall - Winter 2023

The United Nation's Sustainable Development Goals encourage countries to solve many social problems‎. ‎One of these problems is homelessness‎. ‎We consider those goals which are most pertinent to homelessness according to \cite{13}‎. ‎We rank countries with respect to the‎  achievement of these goals‎. ‎We use fuzzy similarity measures to determine‎ ‎ the degree of similarity between these rankings‎. ‎We use three methods to‎ ‎ rank the counties‎, ‎namely‎, ‎the Analytic Hierarchy Process‎, ‎the Guiasu‎ ‎ method,‎ ‎and the Yen method‎. ‎Overall scores of categories in some basic‎ ‎ research papers pertaining to Sustainable Development Goals were obtained by‎ ‎ using multiplication of the scores of the category's targets‎. ‎Multiplication‎ ‎ was used to agree with the philosophy that in order for a high score to be‎ ‎ obtained,‎ ‎all targets must have a high score‎. ‎To support this philosophy in‎ ‎ the decision process‎, ‎we use the $t$-norms bounded difference‎, ‎algebraic‎ ‎ product,‎ ‎and standard intersection as experts‎. ‎We also suggest a way the‎ ‎ techniques used here can be extended to nonstandard analysis‎.

Here we research the univariate fuzzy ordinary and fractional quantitative‎ ‎approximation of fuzzy real valued functions on a compact interval by‎ ‎quasi-interpolation general sigmoid activation function relied on fuzzy neural‎ ‎network operators‎. ‎These approximations are derived by establishing fuzzy‎ ‎Jackson type inequalities involving the fuzzy moduli of continuity of the‎ ‎function‎, ‎or of the right and left Caputo fuzzy fractional derivatives of‎ ‎the involved function‎. ‎The approximations are fuzzy pointwise and fuzzy‎ ‎uniform‎. ‎The related feed-forward fuzzy neural networks are with one hidden‎ ‎layer‎. ‎We study in particular the fuzzy integer derivative and just fuzzy‎  ‎continuous cases‎. ‎Our fuzzy fractional approximation result using higher‎ ‎order fuzzy differentiation converges better than in the fuzzy just‎ ‎continuous case‎.

In real life‎, ‎structural problems can be described in linear and nonlinear forms‎. ‎This nonlinear structural problem is very challenging to solve when its all parameters are imprecise in nature‎. ‎Intuitionistic fuzzy sets were proposed to manage circumstances in which experts have some membership and non-membership value to judge an option‎. ‎Hesitant fuzzy sets were used to manage scenarios in which experts pause between many possible membership values while evaluating an alternative‎. ‎A new growing area of a generalized fuzzy set theory called intuitionistic hesitant fuzzy set (IHFS) provides useful tools for dealing with uncertainty in structural design problem that is observed in the actual world‎. ‎In this article‎, ‎we have developed a procedure to solve non-linear structural problem in an intuitionistic hesitant fuzzy (IHF) environment‎. ‎The concept of an intuitionistic hesitant fuzzy set is introduced to provide a computational basis to manage the situations in which experts assess an alternative in possible membership values and non-membership values‎. ‎This important feature is not available in the intuitionistic fuzzy optimization technique‎. ‎Here we have discussed the solution procedure of intuitionistic hesitant fuzzy optimization technique dedicatedly for linear‎, ‎exponential‎, ‎and hyperbolic types of membership and non-membership functions‎. ‎Some theoretical development based on these functions has been discussed‎. ‎A numerical illustration is given to justify the effectiveness and efficiency of the proposed method in comparison with fuzzy multi-objective nonlinear programming method and intuitionistic fuzzy multi-objective nonlinear programming method‎. ‎Finally‎, ‎based on the proposed work‎, ‎conclusions and future research directions are addressed‎.

This paper has a twofold goal‎: ‎The first is to study how the inferential zigzag can be activated‎, ‎even computationally‎, ‎trying to analyse what kind of reasoning consists of‎, ‎where its 'mechanism' is rooted‎, ‎how it can be activated since without all this it can just seem a metaphysical idea‎. ‎The second‎, ‎not so deeply different‎ - ‎as it can be presumed at a first view‎ - ‎but complementary‎, ‎is to explore the subject's link with the old thought on conjectures of the 15th Century Theologist and Philosopher Nicolaus Cusanus‎ ‎who was the first thinker consciously and extensively using conjectures‎.

‎Recently‎, ‎Gautama algebras were defined and investigated as a common generalization of the variety $\mathbb{RDBLS}\rm t$ of regular double Stone algebras and the variety $\mathbb{RKLS}\rm t$ of regular Kleene Stone algebras‎, ‎both of which are‎, ‎in turn‎, ‎generalizations of Boolean algebras‎. ‎Those algebras were named in honor and memory of the two founders of Indian Logic--{\bf Akshapada Gautama} and {\bf Medhatithi Gautama}‎. ‎The purpose of this paper is to define and investigate a generalization of Gautama algebras‎, ‎called ``Almost Gautama algebras ($\mathbb{AG}$‎, ‎for short).''‎ ‎More precisely‎, ‎we give an explicit description of subdirectly irreducible Almost Gautama algebras‎. ‎As consequences‎, ‎explicit description of the lattice of subvarieties of $\mathbb{AG}$ and the equational bases for all its subvarieties are given‎. ‎It is also shown that the variety $\mathbb{AG}$ is a discriminator variety‎. ‎Next‎, ‎we consider logicizing $\mathbb{AG}$; but the variety $\mathbb{AG}$ lacks an implication operation‎. ‎We‎, ‎therefore‎, ‎introduce another variety of algebras called ``Almost Gautama Heyting algebras'' ($\mathbb{AGH}$‎, ‎for short) and show that the variety $\mathbb{AGH}$ %of Almost Heyting algebras‎ ‎is term-equivalent to that of $\mathbb{AG}$‎. ‎Next‎, ‎a propositional logic‎, ‎called $\mathcal{AG}$ (or $\mathcal{AGH}$)‎, ‎is defined and shown to be algebraizable (in the sense of Blok and Pigozzi) with the variety $\mathbb{AG}$‎, ‎via $\mathbb{AGH},$ as its equivalent algebraic semantics (up to term  equivalence)‎. ‎All axiomatic extensions of the logic $\mathcal{AG}$‎, ‎corresponding to all the subvarieties of $\mathbb{AG}$ are given‎. ‎They include the axiomatic extensions $\mathcal{RDBLS}t$‎, ‎$\mathcal{RKLS}t$ and $\mathcal{G}$ of the logic $\mathcal{AG}$ corresponding to the varieties $\mathbb{RDBLS}\rm t$‎, ‎$\mathbb{RKLS}\rm t$‎, ‎and $\mathbb{G}$ (of Gautama algebras)‎, ‎respectively‎. ‎It is also deduced that none of the axiomatic extensions of‎ ‎$\mathcal{AG}$ has the Disjunction Property‎. ‎Finally‎, ‎We revisit the classical logic with strong negation $\mathcal{CN}$ and classical Nelson algebras $\mathbb{CN}$ introduced by Vakarelov in 1977 and improve his results by showing that $\mathcal{CN}$ is algebraizable with $\mathbb{CN}$ as its algebraic semantics and that the logics $\mathcal{RKLS}\rm t$‎, ‎$\mathcal{RKLS}\rm t\mathcal{H}$‎, ‎3-valued \L ukasivicz logic and the classical logic with strong negation are all equivalent‎.

‎After the introduction of the concept of fuzzy sets‎ ‎by Zadeh‎, ‎several researches were conducted on‎ ‎the generalizations of the notion of fuzzy sets‎. ‎There are many viewpoints on the notion of metric space in fuzzy topology‎. ‎One of the most important problems in fuzzy topology is obtaining an appropriate concept of fuzzy metric space‎. ‎This problem has been investigated by many authors from different points of view‎. ‎Atanassov gives the concept of the intuitionistic fuzzy set as a generalization of the fuzzy set‎. ‎Park introduced the notion of intuitionistic fuzzy metric space as a natural generalization of fuzzy metric spaces due to George and Veeramani‎. ‎This paper introduces the concept of intuitionistic fuzzy modular space‎. ‎Afterward‎, ‎a Hausdorff topology induced by a $\delta$-homogeneous intuitionistic fuzzy modular is defined and some related topological properties are also examined‎. ‎After giving the fundamental definitions and the necessary examples‎, ‎we introduce the definitions of intuitionistic fuzzy boundedness‎, ‎intuitionistic fuzzy compactness‎, ‎and intuitionistic fuzzy convergence‎, ‎and obtain several preservation properties and some characterizations concerning them‎. ‎Also‎, ‎we investigate the relationship between an intuitionistic fuzzy modular and an intuitionistic fuzzy metric‎. ‎Finally‎, ‎we prove some known results of metric spaces including Baire’s theorem and the Uniform limit theorem for intuitionistic fuzzy modular spaces‎.

‎We define and study two completeness notions for saturated $\mathsf{L}$-quasi-uniform limit spaces‎. ‎The one‎, ‎that we term Lawvere completeness‎, ‎is defined using the concept of promodule and lends a lax algebraic interpretation of completeness also for saturated $\mathsf{L}$-quasi-uniform limit spaces‎. ‎The other‎, ‎termed Cauchy completeness‎, ‎is defined using saturated Cauchy pair prefilters‎. ‎We show that both concepts coincide with related notions in the case of saturated $\mathsf{L}$-quasi-uniform spaces and that also for saturated $\mathsf{L}$-quasi-uniform limit spaces‎, ‎both completeness notions are equivalent‎.

This paper develops the notion of fuzzy ideal and generalized fuzzy ideal on double Boolean algebra (dBa)‎. ‎According to Rudolf Wille‎, ‎a double Boolean algebra $\underline{D}:=(D‎, ‎\sqcap‎, ‎\sqcup‎, ‎\neg‎, ‎\lrcorner‎, ‎\bot‎, ‎\top)$ is an algebra of type $(2‎, ‎2‎, ‎1‎, ‎1‎, ‎0‎, ‎0),$ which satisfies a set of properties‎. ‎This algebraic structure aimed to capture the equational theory of the algebra of protoconcepts‎. ‎We show that collections of fuzzy ideals and generalized fuzzy ideals are endowed with lattice structures‎. ‎We further prove that (by isomorphism) lattice structures obtained from fuzzy ideals and generalized fuzzy ideals of a double Boolean algebra D can entirely be determined by sets of fuzzy ideals and generalized fuzzy ideals of the Boolean algebra $D_{\sqcup}.$

Cloud-based machine learning tools for enhanced Big Data applications}‎, ‎where the main idea is that of predicting the ``\emph{next}'' \emph{workload} occurring against the target Cloud infrastructure via an innovative \emph{ensemble-based approach} that combines the effectiveness of different well-known \emph{classifiers} in order to enhance the whole accuracy of the final classification‎, ‎which is very relevant at now in the specific context of \emph{Big Data}‎. ‎The so-called \emph{workload categorization problem} plays a critical role in improving the efficiency and reliability of Cloud-based big data applications‎. ‎Implementation-wise‎, ‎our method proposes deploying Cloud entities that participate in the distributed classification approach on top of \emph{virtual machines}‎, ‎which represent classical ``commodity'' settings for Cloud-based big data applications‎. ‎Given a number of known reference workloads‎, ‎and an unknown workload‎, ‎in this paper we deal with the problem of finding the reference workload which is most similar to the unknown one‎. ‎The depicted scenario turns out to be useful in a plethora of modern information system applications‎. ‎We name this problem as \emph{coarse-grained workload classification}‎, ‎because‎, ‎instead of characterizing the unknown workload in terms of finer behaviors‎, ‎such as CPU‎, ‎memory‎, ‎disk‎, ‎or network intensive patterns‎, ‎we classify the whole unknown workload as one of the (possible) reference workloads‎. ‎Reference workloads represent a category of workloads that are relevant in a given applicative environment‎. ‎In particular‎, ‎we focus our attention on the classification problem described above in the special case represented by \emph{virtualized environments}‎. ‎Today‎, ‎\emph{Virtual Machines} (VMs) have become very popular because they offer important advantages to modern computing environments such as cloud computing or server farms‎. ‎In virtualization frameworks‎, ‎workload classification is very useful for accounting‎, ‎security reasons‎, ‎or user profiling‎. ‎Hence‎, ‎our research makes more sense in such environments‎, ‎and it turns out to be very useful in a special context like Cloud Computing‎, ‎which is emerging now‎. ‎In this respect‎, ‎our approach consists of running several machine learning-based classifiers of different workload models‎, ‎and then deriving the best classifier produced by the \emph{Dempster-Shafer Fusion}‎, ‎in order to magnify the accuracy of the final classification‎. ‎Experimental assessment and analysis clearly confirm the benefits derived from our classification framework‎. ‎The running programs which produce unknown workloads to be classified are treated in a similar way‎. ‎A fundamental aspect of this paper concerns the successful use of data fusion in workload classification‎. ‎Different types of metrics are in fact fused together using the Dempster-Shafer theory of evidence combination‎, ‎giving a classification accuracy of slightly less than $80\%$‎. ‎The acquisition of data from the running process‎, ‎the pre-processing algorithms‎, ‎and the workload classification are described in detail‎. ‎Various classical algorithms have been used for classification to classify the workloads‎, ‎and the results are compared‎.

‎In this paper we consider MV-algebras and their prime spectrum‎. ‎We show that there is an uncountable MV-algebra that has the same spectrum as the free MV-algebra over one element‎, ‎that is‎, ‎the MV-algebra $Free_1$ of McNaughton functions from $[0,1]$ to $[0,1]$‎, ‎the continuous‎, ‎piecewise linear functions with integer coefficients‎. ‎The construction is heavily based on Mundici equivalence between MV-algebras and lattice ordered abelian groups with the strong unit‎. ‎Also‎, ‎we heavily use the fact that two MV-algebras have the same spectrum if and only if their lattice of principal ideals is isomorphic‎.‎As an intermediate step we consider the MV-algebra $A_1$ of continuous‎, ‎piecewise linear functions with rational coefficients‎. ‎It is known that $A_1$ contains $Free_1$‎, ‎and that $A_1$ and $Free_1$ are equispectral‎. ‎However‎, ‎$A_1$ is in some sense easy to work with than $Free_1$‎. Now‎, ‎$A_1$ is still countable‎. ‎To build an equispectral uncountable MV-algebra $A_2$‎, ‎we consider certain ``almost rational'' functions on $[0,1]$‎, ‎which are rational in every initial segment of $[0,1]$‎, ‎but which can have an irrational limit in $1$‎.‎We exploit heavily‎, ‎via Mundici equivalence‎, ‎the properties of divisible lattice ordered abelian groups‎, ‎which have an additional structure of vector spaces over the rational field‎.

‎As users in an online social network are overwhelmed by the abundant amount of information‎, ‎it is very hard to retrieve the preferred or required content‎. ‎In this context‎, ‎an online recommender system helps to filter and recommend content such as people,items or services‎. ‎But‎, ‎in a real scenario‎, ‎people rely more on recommendations‎ ‎from trusted sources than distrusting sources‎. ‎Though‎, ‎there are many trust based recommender systems that exist‎, ‎it lag in prediction error‎. ‎In order to improve the accuracy of the prediction‎, ‎this paper proposes a Trust-Boosted Recommender System (TBRS)‎. ‎Since‎, ‎the provenance derives the trust in a better way than other approaches‎, ‎TBRS is built‎ ‎from the provenance concept‎. ‎The proposed recommender system takes the provenance based fuzzy rules which were derived from the Fuzzy Decision Tree‎. ‎TBRS then computes the multi-attribute vector similarity score and boosts the score with trust weight‎. ‎This system is tested on the book-review dataset to recommend the top-k trustworthy reviewers.The performance of the proposed method is evaluated in terms of MAE and RMSE‎. ‎The result shows that the error value of boosted similarity is lesser than without boost‎. ‎The reduced error rates of the Jaccard‎, ‎Dice and Cosine similarity measures are 18\%‎, ‎15\% and 7\% respectively‎. ‎Also‎, ‎when the model is subjected to failure analysis‎, ‎it gives better performance for unskewed data than slewed data‎. ‎The models fbest‎, ‎average and worst case predictions are 90\%‎, ‎50\% and $<$23\% respectively‎.

‎In fuzzy set theory‎, ‎the concept of a non-membership function and the hesitation margin were not considered while these two concepts have been included along with the membership function for intuitionistic fuzzy sets‎. ‎It is also to be noted that the intuitionistic fuzzy set is reflected as an extension of the fuzzy set accommodating both membership and non-membership functions together with a hesitation margin‎. ‎In the intuitionistic fuzzy set theory‎, ‎the sum of the membership function and the non-membership function is a value between 0 and 1‎. ‎In recent times‎, ‎intuitionistic fuzzy rough set theory has emerged as a powerful tool for dealing with imprecision and uncertain information in relational database theory‎. ‎Measures of similarity between fuzzy rough sets as well as intuitionistic fuzzy rough sets provide wide applications in real-life problems and that is why many researchers paid more attention to this concept‎. ‎Intuitionistic fuzzy rough set theory behaves like an excellent tool to tackle impreciseness or uncertainties‎. ‎In this paper‎, ‎we propose a new approach of similarity measure on an intuitionistic fuzzy rough set based on a set-theoretic approach‎. ‎The proposed measure is able to give an exact result‎. ‎In the application part‎, ‎we consider a real-life problem for selecting a fair play award-winning team in a cricket tournament and describe the algorithm‎.