This work introduces an innovative approach from modeling to simulations, integrating quantum fuzzy interpretations, focusing on the expression of membership degrees in quantum circuits, interpreted as a unitary quantum transformation. Our research contrasts Quantum Fuzzy Computing with Classical Computing, particularly in the context of the prisoner's dilemma and parent-child relationships. Through the Qiskit framework, we conduct simulations, demonstrating the differences in results obtained by both approaches. Well-known fuzzy connectives, such as “exclusive or” and “arithmetic means”, provide an algebraic description for the corresponding composition of quantum operators and circuit representations. Thus, such algorithms can easily extend to model multiple agents' relationships, described by multidimensional quantum registers. The Qiskit simulations provide the structure for the algorithms' computation on the actual quantum platforms, indicating a promising direction for future research in this hybrid research area.

This study introduces an innovative quantum fuzzy approach for modeling and simulating complex decision-making processes, utilizing quantum circuits to express membership degrees as unitary transformations. We contrast Quantum Fuzzy Computing with Classical Computing, highlighting advancements in modeling the prisoner's dilemma. Using the Qiskit framework, we demonstrate how quantum fuzzy algorithms featuring connectives such as ‘exclusive or’ and ‘arithmetic mean’ enable detailed modeling of multi-agent relationships via multidimensional quantum registers, showcasing potential advancements in this hybrid research area.

This study is given at the intersection of three important areas: Modal Logic (ML), Fuzzy Logic (FL), and Quantum Computing (QC), leveraging from their main features. On the one hand, we have the QC ability to handle complex data more efficiently, taking advantage of the quantum mechanics concepts. On the other hand, ML and FL allow us to express uncertainties by mathematically modeling the imprecision of the natural language. Therefore, we first provide an algebraic model to interpret modal operators on fuzzy logic, and then we represent them in a quantum computing environment. Besides, we present some case studies simulating the fuzzy modal connectives in the Qiskit platform, aiming to better understand the evolution of quantum circuits.

Firstly, this work discusses the main conditions guarantying that general overlap (grouping) functions can be obtained from n-dimensional overlap (grouping) functions. Focusing on QL-implications, which are usually generated by strong negations together with t-norms and t-conorms, we consider a non-restrictive construction, by relaxing not only the associativity and the corresponding neutral elements (NE) but also the reverse construction of other properties. Thus, the main properties of the QL-implication class are studied, considering a tuple (G,N,O) generated from grouping and overlap functions together with the greatest fuzzy negation. In addition, in order to provide more flexibility, we define a subclass of QL-implications generated from general overlap and general grouping functions. Some examples are introduced, illustrating the constructive methods to generate such operators.

This paper contributes to classifying video streaming traffic by exploring concepts of Interval-Valued Fuzzy Logic. This approach extends related works by considering the uncertainties generated by variations in network conditions and the parameter imprecision affecting the behavior of network flow, which increases the complexity to reach high accuracy in identifying the network traffic. Some evaluations using the interval-valued logic approach for video streaming traffic classification are presented using application tools and datasets to validate the proposal.

This paper considers the consensual analysis in decision-making (CDM) processes based on fuzzy logic (FL) and interval-valued fuzzy logic (IVFL), providing a CDM-strategy, by exploring the axiomatic properties of fuzzy consensus measures (FCM) via penalty functions. Thus, two models are formalized, FS-FCM and IVFS-FCM. In the former, the fuzzy-valued lattice enables the analysis of fuzzy information for linguistic variables (LV), which is obtained by the aggregation of penalty functions. And, in the latter, the consensus measures of fuzzy sets are aggregated to build a new consensual analysis modeling. Thus, e.g., the cohesion of several terms related to the same LV can be analyzed, and also the coherence between fuzzy sets referring to the lowest and highest projections. Such models decide based on relevance criteria and qualitative assessments, via the selection of alternatives, supporting the corresponding algorithmic strategies: FS-FCM strategy, applied to fuzzy values, and IVFS-FCM strategy, covering fuzzy sets. The Intf-HybridMem approach explores the access patterns to volatile and non-volatile memories related to decision-making in two steps: (i) the FS-FCM strategy explores consensus measures of fuzzy values from membership functions; and (ii) the IVFS-FCM strategy, modeling inaccuracy inherent in input variables, as read/write frequency and access recency, also including the migration recommendation as output, which is validated by evaluations carried out in both proposed strategies.

In this paper, we discussconsensus measures for typical hesitant fuzzy elements (THFE), which are the finite and nonempty fuzzy membership degrees under the scope of typical hesitant fuzzy sets (THFS). In our approach, we present a model that formally constructs consensus measures by means of aggregations functions, fuzzy implication-like functions and fuzzy negations, using admissible orders to compare the THFE, and also providing an analysis of consistency on them. Our theoretical results are applied into a problem of decision making with multicriteria illustrating our methodology to achieve consensus in a group of experts working with THFS.

This paper explores fuzzy logic-based consensus analysis as support to the decision making problem related to page migrations in hybrid memories. The proposed interval fuzzy approach considers the uncertainties on data access profiles in hybrid memories and its main characteristics. The fuzzy set consensus analysis, in its turn, is proposed using arithmetic and exponential extended aggregations. Aiming to evaluate the proposal, a case study was developed in the Intf-HybridMem memory management architecture.

Multi Expert-Multi Criteria Decision Making (ME-MCDM) problems have been well explored on hesitant fuzzy environments dealing with membership degrees as subsets which do not necessarily have the same cardinality. Admissible (total) orders collaborate by reducing the collapse in the ranking of alternatives related to preference relations. In such context, three classes of admissible orders are presented in a non-restrictive way, integrating the concepts of linearity and cardinality and providing support to the comparison between two typical hesitant fuzzy elements. In the sense of hesitant fuzzy logic, the study of fuzzy operators and their main properties is extended considering the admissible linear orders. Namely, the typical hesitant fuzzy aggregation functions, negations and implication functions are discussed mainly related to their (iso/anti)tonicity properties w.r.t. these admissible orders. An algorithmic procedure is introduced illustrating our strategy to solve an ME-MCDM problem, by selecting a Computer Integrating Manufacturing software and making use of the achieved theoretical results.

In the Hybrid Memory research approach, the development of new storing data strategies combines the traditional volatile memories (DRAM) with the emerging non-volatile memory (NVM) to take advantage of their potential and best features. One question in the hybrid memory research is the decision of which memory technology to use in order to store data of distinct applications. These decision processes have to examine the behavior of memory operations, such as read/write frequency, and also their memory characteristics. Also, considering the large volume of parameters and the uncertainties inherent in the data migration, the decision to store data in hybrid memories is not a trivial task. This paper presents the f-HybridMem component, a fuzzy-based system to support the uncertainty in data management for hybrid memory architectures. Thus, this proposal contributes to determine a correct selection between memory modules by improving the data management, in a page level organization. The memory management takes into account the behavior of memory operations and the memory characteristics. Such architecture is conceived based on the following modules: (i) Access Updater, a hardware module to identify the pages access patterns, and (ii) f-HybridMem component, a fuzzy-based software module supporting the migration decision processes. In addition, tests are conducted attempt evaluating the accuracy of the fuzzy-based system. Moreover, the proposed evaluations aim to estimate the influence of the following parameters: Buffer Size, Counters Size, Frequency of Migration, Promote Value, Demote Value, providing a correct recommend for data migrations.

The increasing offer of Internet-connected resources, providing more than one service, confers complexity to the classification procedures and selection of adequate resources to meet customer demands. In this context, this article aims to present a proposal for the classification of resources in IoT, considering non-functional parameters of QoS. The proposal explores two complementary approaches: (i) use of type-2 fuzzy logic; and (ii) dynamic adequacy of the fuzzy rules employed. The results obtained in the classification of the QWS dataset are presented and are satisfactory, adjusting the selection of the most relevant resources according to the preferences specified by the client.

This work extends the study of properties related to the interval extension of the generalized Atanassov’s intuitionistic fuzzy index (A-GIvIFIx) which is defined as a composition between an interval-valued aggregation and fuzzy implication w.r.t partial/total order on $\mathbb{U}$. The relationship between Atanassov’s intuitionistic fuzzy index and its interval extension is discussed. Aggregation and conjugate operators provide two methods to obtain new classes of A-IvIFIx.

Dynamic consolidation of virtual machines (VMs) is an effective way to improve resource utilization and power efficiency in cloud computing, directly affecting Quality of Service aspects. This paper presents Int-FLBCC, a new proposal with exploring a Type-2 Fuzzy Logic approach to address the uncertainties and inaccuracies in determining resource usage, aiming at energy savings with minimal performance degradation. Validation results in a simulated cloud computing environment showed improvements in energy efficiency of 8.83% with IQR_XY, 22.43% with MAD_XY. For fulfillment of Service Level Agreements (SLAs), the best values achieved were 9.06% with MAD_XY and 25% of THR_Lex1.

Computer systems based on intuitionistic fuzzy logic are capable of generating a reliable output even when handling inaccurate input data by applying a rule based system, even with rules that are generated with imprecision. The main contribution of this paper is to show that quantum computing can be used to extend the class of intuitionistic fuzzy sets with respect to representing intuitionistic fuzzy Xor operators. This paper describes a multi-dimensional quantum register using aggregations operators such as t-(co)norms based on quantum gates allowing the modeling and interpretation of intuitionistic fuzzy Xor operations.

This paper studies the correlation coefficient (A-CC) related to the Atanassov’s intuitionistic fuzzy sets (A-IFS) which are obtained as image of modal operators. Extended results from the action of A-CC over necessity and possibility modal operators are considered, determining the A-CC of A-IFS obtained as image of the !A and ?A modal level operators and discussing the main conditions under which the main properties related to such fuzzy sets are preserved by conjugate and complement operations. In addition, a simulation based on the proposal methodology using modal level operators is applied to a medical diagnosis analysis.

Currently, billions of resources are connected to the Internet, many providing services. There is forecast to be a significant increase in the number of resources in the coming years. The adequate selection of resources that best meet the demands of users among a large number of options has been a relevant and current research challenge in the autonomic IoT management. This paper specifies and evaluates the pre-classification of new resources of the EXEHDA middleware based on the non-functional parameters of QoS. Fuzzy logic is used in the treatment of uncertainties in defining the importance weights of QoS attributes. The results obtained in the evaluation of the accuracy of the pre-classification through fuzzy logic and machine learning are presented.

This paper aims to study the correlation between Atanassov's intuitionistic fuzzy sets obtained as image of modal-level operators. We consider the action of modal operators the necessity and possibility in order to verify the conditions under which the correlation coefficient preserve main properties related to such fuzzy sets.

Among the research challenges, in Grid Computing we need to make the system robust as much as the uncertainties of the different measures extracted from the computational infrastructure, as for the imprecision of the computations related to the decision making. In particular, the appropriation of information relevant to task scheduling from a set of heterogeneous sources and dynamic behavior has been the focus of a significant number of studies. Scheduling tasks is a known NP-Hard problem. This paper provides an approach with Fuzzy Type-2 logics to treat uncertainties and dynamic behavior for scheduling tasks in grid environments, named Int-fGrid. The scheduler was validated through simulations in the SimGrid framework with a model of the GridRS architecture. Our results show that the Fuzzy Type-2 approach provides makespans up to 17.60% better than the best alternative tested named scheduler XSufferage.

As a relevant extension of fuzzy sets, n-dimensional fuzzy sets consider membership degrees as ordered n-tuples on U = [0, 1], providing n-dimensional intervals on Ln(U). As our main contribution, n-dimensional fuzzy implications are studied considering duality and conjugation operators. Additionally, the conditions over that main properties are preserved based on their representability from Unto Ln(U) are also presented. Some results in the class of S-implications obtained from t-representable conorms and involutive n-dimensional fuzzy negations on Ln(U) are discussed.

n-dimensional fuzzy sets theory is a broad research area extending fuzzy set theory where the membership values are n-tuples of real numbers in the unit interval [0, 1] orderly increased, called n-dimensional intervals. The set of n-dimensional intervals is denoted by Ln([0, 1]). This paper investigates a special extension from [0, 1]-n-representable fuzzy negations on Ln([0, 1]), summarizing the class of such functions that are continuous and monotone by part. The main properties of (strong) fuzzy negations on [0, 1] are preserved by representable (strong) fuzzy negation on Ln([0, 1]), mainly related to the analysis of degenerate elements and equilibrium points. The conjugate obtained by the action of an n-dimensional automorphism on an n-dimensional fuzzy negation provides a method to obtain another n-dimensional fuzzy negation, in which properties such as representability, continuity, and monotonicity on Ln([0, 1]) are preserved. Finally, we provide a method for multiexpert decisionmaking problems based on N-reciprocal n-dimensional fuzzy preference relations over a set of alternatives preserving the N reciprocity for an n-dimensional fuzzy negation N.

N-dimensional fuzzy sets are an extension of fuzzy sets where the membership values are n-tuples of real numbers in the unit interval [0,1] ordered in increasing order, called n-dimensional intervals. The set of n-dimensional intervals is denoted by Ln([0,1]). In the present paper, we consider the definitions and results obtained for n-dimensional fuzzy negations, applying these studies mainly on natural n-dimensional fuzzy negations for n-dimensional triangular norms and triangular conorms. Additionally, the conjugate obtained by action of an n-dimensional automorphism on an n-dimensional natural fuzzy negations for n-dimensional triangular norms and triangular conorms, provides a method to obtain other n-dimensional strong fuzzy negations, in which its properties on Ln([0,1]) are preserved.

The aim of this work is to study the class II, I1, I2(U) of fuzzy implications obtained by a triple (I, I1, I2) of fuzzy implications. Thus, this paper discusses under which conditions such functions preserve the main properties of fuzzy implications. In addition, by conjugate fuzzy implications it is shown that an II, I1, I2(U) fuzzy implication can be preserved by action of an order automorphism. Finally, we introduce the family I(k)I, I1, I2(U) of fuzzy implications obtained by taking the extended classes of I(k)-implications verifying both generalized properties, exchange principle and distributivity in addition to, their dual construction is also considered.

Summary Because of the expansion of transformations and read/write memory states by tensor products in multidimensional quantum applications, the exponential increase in temporal and spatial complexities constitutes one of the main challenges for quantum computing simulations. Simulation of these systems is very relevant to develop and test new quantum algorithms. In order to overcome the increase in simulation complexity, this work presents reduction and decomposition optimizations for the Distributed Geometric Machine environment. By exploring properties as the sparsity of the Identity operator and partiality of dense unitary transformations, better storage and distribution of quantum information are achieved. The main improvements are reached by decreasing replication and void elements inherited from quantum operators. In the evaluation of this proposal, Hadamard transformations from 21 to 28 qubits and Quantum Fourier Transforms from 26 to 28 qubits were simulated over CPU, sequentially and in parallel, and in graphics processing unit, showing reduced temporal complexity and, consequently, shorter simulation time. Moreover, evaluations of the Shor's algorithm considering 2n + 3 qubits in the order‐finding quantum algorithm were simulated up to 25 qubits. When comparing our implementations running on the same hardware with language‐integrated quantum operation, academic release version, our new simulator was faster and allowed for the simulation of more qubits. Copyright © 2016 John Wiley & Sons, Ltd.

This paper studies the conjugate functions related to main connectives of the Intervalvalued Atanassov’s Intuitionistic Fuzzy Logic. The relationships among automorphism classes are formalized by the ϕ-representability theorem, passing from automorphisms to interval-valued intuitionistic automorphisms, also visiting other two ones, intuitionistic automorphisms and interval-valued automorphisms. Additionally, the ϕ-conjugate of an interval-valued Atanassov’s intuitionistic fuzzy negation can be obtained either from an interval-valued fuzzy negation or from an Atanassov’s intuitionistic fuzzy negation, including a discussion presenting such reverse constructions. The ϕ-conjugate of an interval-valued Atanassov’s intuitionistic fuzzy negation not only preserves the main properties of its corresponding fuzzy negation but also of two other ones, the intuitionistic fuzzy negation and interval-valued fuzzy negation. Moreover, an extension of the intuitionistic fuzzy index as well as the correlation coefficient is discussed in terms of fuzzy negations, by considering the Atanassov’s Intuitionistic Fuzzy Logic.

The exponential increase in the temporal and spatial complexities is one of the main challenges in the widespread use of quantum algorithm simulation, especially in dense quantum transformations (QTs) such as the Hadamard transformation (H), which has found wide applications in computer and communication science and also comprising the simplest quantum universal set of QTs. The main reason for these costs is the expansion of QTs by using tensor product in multi-dimension quantum applications. In this work, new optimizations for the execution of reduction and decomposition based on the Identity operator are introduced in the Distributed Geometric Machine framework (D-GM). Instead of executing the quantum transformation in a single step, they are divided in sub-quantum transformations and only the values different from Identity transformations are stored. Mixed Partial Processes provide control over the increase in the size of read/write memory states in the calculation of a QT, thus contributing to increase the scalability of applications regarding hardware-GPUs memory limit. In the evaluation of this D-GM extension, Hadamard Transformations were simulated up to 28 qubits applications over a single GPU. Our new simulator is 10, 829χ faster and allows for the simulation of more qubits when compared to our previous implementation running on the same GPU.

This paper aims to study the correlation between Atanassov's intuitionistic fuzzy sets (A-IFSs) obtained as image of strong intuitionistic fuzzy negations. We consider the action of strong fuzzy negations in order to verify the conditions under which the correlation coefficient related to such A-IFSs and their corresponding conjugate constructions are obtained. We attempt to present algebraic expressions of correlation relationship by considering representable intuitionistic automorphisms.

n-dimensional fuzzy sets are an extension of fuzzy sets where the membership values are n-truples of real numbers in the unit interval [0, 1] ordered in increasing order, called n-dimensional intervals. The set of n-dimensional intervals is denoted by Ln([0, 1]). This paper aims to investigate the class of functions on Ln([0, 1]) which are continuous and strictly decreasing, called n-dimensional strict fuzzy negations. In particular, investigate the class of representable n-dimensional strict fuzzy negations, i.e., n-dimensional strict fuzzy negation which are determined by strict fuzzy negation. The main properties of strict fuzzy negations on [0, 1] are preserved by representable strict fuzzy negations on Ln([0, 1]). In addition, the conjugate obtained by action of an n-dimensional automorphism on an n-dimensional strict fuzzy negation provides a method to obtain other n-dimensional strict fuzzy negations, in which the properties of the original one are preserved, as well as the Fodor's characterization theorem.

The necessity of making a system robust to the uncertainties of the different measures taken from the computing infrastructure is one of the Grid Computing research challenges. In particular, taking relevant information to tasks scheduling from a set of heterogeneous sources and dynamic behavior has been the focus of a significant number of studies. Considering this background, this paper aims to approach a module for task scheduling in Grid Computing, using a taken decision system under uncertainty regime. The proposed module uses the fuzzy logic, called fGrid, to treat uncertainties. Simulations were carried out using the SimGrid framework and modeling GridRS architecture. In addition to the presented results, the a standard scale use with linguistic variation makes both the manipulation of the priority list as the scheduling easier, which contributes to maintainability.

This paper studies the robustness of intuitionistic fuzzy connectives in fuzzy reasoning. Starting with an evaluation of the sensitivity in representable fuzzy negations, we apply the results in the intuitionistic (S, N)-implication class and its dual construction. As the main contribution, the paper formally states that the robustness preserves the projection functions in this class and corresponding dual operators.

This paper aims to study the robustness of f-Xor connectives, in particular the ETP, SP, NSoperator also including its main properties, NS-dual construction and corresponding fuzzy f-Xor implication and bi-implication. Additionally, we obtained some basic results on the pointwise sensitivity of the f-Xor connective ETP, SP, NS, mainly connected with the endpoints of unit interval U. Our major interest is the robustness analysis related to f-X(N)or (co)implications and f-X(N)or bi-(co)implications.

This paper presents the fuzzy (S, N)- QL- and D-subimplication classes, which are obtained by OWA operators performed over the families of triangular subnorms and sub-conorms along with fuzzy negations. Since these classes of subim-plications are explicitly represented by such connectives, the corresponding (S, N)- QL- and D-subimplicatios are characterized by the generalized associativity and distributive properties together with extensions of the exchange and neutrality principles. As the main results, these families of subimplications extend related implications by preserving their corresponding properties.

Since the seminal paper of fuzzy set theory by Zadeh in 1965, many extensions have been proposed to overcome the difficulty for assigning the membership degrees. In recent years, a new extension, the hesitant fuzzy sets, has attracted a lot of interest due to its usefulness to handle those problems in which it is difficult to provide accurately a single membership value; since for hesitant sets, membership values are given by a whole set of values. On the other hand, since fuzzy negations have an important role in applications as well as in the theoretical approach to of fuzzy logics, it is important to study an extension of the concept of fuzzy negation for hesitant fuzzy degrees (elements). In this paper, we propose such a definition and we study some of the main properties of this new concept.

Due to the unavailability of quantum computers, simulation of quantum algorithms using classical computers is still the most affordable option for research of algorithms and models for quantum computing. As such simulation requires high computing power and memory resources, and computations are regular and data-intensive, GPUs become a suitable solution for accelerating the simulation of quantum algorithms. This work introduces an extension of the execution library for the VPE-qGM environment to support GPU acceleration. Hadamard gates with up to 20 quantum bits were simulated, and speedups of up to approximately 540× over a sequential simulation and of approximately 85× over a 8-core distributed simulation in the VirD-GM environment were achieved.

In fuzzy logic, there are two main approaches to eliciting membership degrees: an approach based on polling experts, and a Likert-scale approach, in which we ask experts to indicate their degree of certainty on a scale - e.g., on a scale form 0 to 10. Both approaches are reasonable, but they often lead to different membership degrees. In this paper, we analyze the relation between these two approaches, and we show that this relation can be made much clearer if we use models from quantum computing.

Computer systems based on fuzzy logic are capable of generating a reliable output even when handling inaccurate input data by applying a rule based system. The main contribution of this paper is to show that quantum computing can be used to extend the class of fuzzy sets. The central idea associates the states of a quantum register to membership functions (mFs) of fuzzy subsets, and the rules for the processes of fuzzyfication are performed by unitary quantum transformations. Thus, this paper describes multi-dimensional quantum registers, associated to mFs on the unitary interval U, in order to introduce a novel interpretation of aggregations found in fuzzy set theory. In particular, t-norms and t-conorms based on quantum gates allows the modeling and interpretation of union, intersection, difference and implication among fuzzy sets, also including an expression for the class of fuzzy S-implications. Furthermore, an interpretation of the symmetric sum was achieved by considering the quantum register sum operator.

The main contribution of this paper is the introduction of an intrinsic definition of the connective “fuzzy exclusive or” E (f-Xor E), based only on the properties of boundary conditions, commutativity and partial isotonicity-antitonicity on the the end-points of the unit interval U = [0,1], in a way that the classical definition of the boolean Xor is preserved. We show three classes of the f-Xor E that can be also obtained from the composition of fuzzy connectives, namely, triangular norms, triangular conorms and fuzzy negations. A discussion about extra properties satisfied by the f-Xor E is presented. Additionally, the paper introduces a class of fuzzy equivalences that generalizes the Fodor and Roubens's fuzzy equivalence, and four classes of fuzzy implications induced by the f-Xor E, discussing their main properties. The relationships between those classes of fuzzy implications and automorphisms are explored. The action of automorphisms on f-Xor E is analyzed.

The simulation of quantum algorithms in classic computers is a task which requires high processing and storing capabilities, limiting the size of quantum systems supported by the simulators. However, optimizations for reduction of temporal and spatial complexities are promising, expanding the capabilities of some simulators. The main contribution of this work consists in designing optimizations resulting from the description of quantum transformations using Quantum Processes and Partial Quantum Processes conceived in the qGM theoretical model. These processes, when computed on the VPE-qGM execution environment, require low execution time and result in the improvement of the performance, allowing the simulation of more complex quantum algorithms. The performance evaluation of this proposal was performed by benchmarks used in similar works and included the sequential simulation of quantum algorithms up to 24 qubits. The results are promising when compared to the state-of-art, indicating the possibilities of advances in this research.

The paper presents a general expression of an interval-valued intuitionistic fuzzy coimplication generated by interval-valued aggregation function acting on a pair of mutual dual functions, named interval-valued fuzzy implications and coimplications. We analyse under which conditions such connectives preserve the main properties of intuitionistic fuzzy implications. In addition, interpretations of conditional fuzzy rules based on interval-valued fuzzy logic and intuitionistic fuzzy logic are considered.

The aim of this work is to analyze interval fuzzy S-implications and interval automorphisms. Starting from any fuzzy S-implication, it is shown how to obtain an interval fuzzy S-implication canonically. We proved that such interval fuzzy S-implications meet the optimality property and preserve the same properties satisfied by fuzzy S-implications. In addition, commutative diagrams are used in order to relate fuzzy S-implications to interval fuzzy S-implications, and to understand how interval automorphisms act on interval S-implications, generating other interval fuzzy S-implications.