Intuitionistic Fuzzy Implications Research Articles

Research on intuitionistic fuzzy implications. Part 3

Continuing the research from [1, 2], here we give the lists of the intuitionistic fuzzy implications, introduced in [2], that satisfy at least one of two forms of Modus Ponens (MP) – a standard and a new one, called “mixed”, forms. We show the relationship between every two of these implications that satisfy the mixed form of MP.

A new intuitionistic fuzzy implication and the negation, conjunctions and disjunctions generated by it

A new – the 207-th – intuitionistic fuzzy implication is introduced over intuitionistic fuzzy sets. Over its basis, new intuitionistic fuzzy negation, conjunctions and disjunctions are constructed. For the first time, it is shown that all three conjunctions generated by the same implication coincide, whence their respective disjunctions are different. Some properties of the newly constructed operations are studied.

Intuitionistic fuzzy interpretations of formula (A → B) → ((¬A → B) → B)

One of the essential formulas of the classical mathematical logic is (A → B) → ((¬A → B) → B). In the present paper, its intuitionistic fuzzy interpretation is introduced, and lists of all defined intuitionistic fuzzy implications that satisfy it as a tautology and an intuitionistic fuzzy tautology are given.

The Hauber's law with intuitionistic fuzzy implications

25 years ago, in [7], it was proved that the Hauber's law is an intuitionistic fuzzy tautology. In this case, the used implication was the standard intuitionistic fuzzy one. In the present paper, we check which intuitionistic fuzzy implications, defined during these 25 years, satisfy the Hauber's law as a tautology and which if them – as an intuitionistic fuzzy tautology.

ON NEW INTUITIONISTIC FUZZY(\ep,η)-NEGATION AND (\ep,η)-IMPLICATION

The concept of an Intuitionistic Fuzzy Set (IFS) was introduced in 1983 in \cite{ITKR} and again there in it, the first definition of operation intuitionistic fuzzy negation over IFS was introduced. Later, in \cite{IFPC}, two forms of the operation intuitionistic fuzzy implication were introduced. As of the 2000s, more than 200 different implications and more than 50 different negations arise and in [5 -- 21, 23, 24], their basic properties were discussed. In \cite{e,ee}, two negations, namely $\ep$-negation and $(\ep,\eta)$-negation, were introduced and over their bases $\ep$- and $(\ep,\eta)$-implications were defined, respectively. It was shown that the $(\ep,\eta)$-negation and $(\ep,\eta)$-implication are extensions of the $\ep$-negation and $\ep$-implication, respectively.

Research on intuitionistic fuzzy implications. Part 2

Continuing the research from [2], here we give the list of the axioms that are \linebreak satisfied by the intuitionistic fuzzy implications, introduced in Part 1 of the present research. Given the large number of implications that have already been defined and the naturally arising question about their comparability and usability, we discuss the criteria of usability of implications and outline 44 ones that satisfy at least 28 from the discussed 38 axioms, that is about 1/4 of all implications that satisfy at least about 3/4 of all axioms. Finally, we show the relationships between each pair of implications in the form of a graph.

On modal forms of the two-parametric weak intuitionistic fuzzy implication

In this paper new modal forms of the two-parametric weak intuitionistic fuzzy implication are introduced. Fulfillment of some properties, together with Modus Ponens and Modus Tollens inference rules, are investigated. Negations generated by implications are presented.

The weak intuitionistic fuzzy implication based on △* operation

In this paper, a new weak intuitionistic fuzzy implication is introduced based on the recently defined △* operation. Fulfillment of some axioms and properties, together with Modus Ponens and Modus Tollens inference rules, are investigated. The negation induced by the studied implication is presented.

Robustness of Interval-Valued Intuitionistic Fuzzy Reasoning Quintuple Implication Method

The interval-valued intuitionistic fuzzy quintuple implication algorithm, as an extension of the fuzzy reasoning algorithm, may better characterize and deal with uncertainty in the reasoning, but how to select distance measure and analyze the algorithm’s robustness is an important and unsolved topic. In this paper, a novel distance measure of interval-valued intuitionistic fuzzy sets is constructed based on interval-valued intuitionistic fuzzy biresiduum similarity. The unified form of the conclusion about the robustness of interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm for interval-valued intuitionistic fuzzy modus ponens(IVIFMP) and interval-valued intuitionistic fuzzy modus tollens(IVIFMT) is obtained. Especially, the robustness of the interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G\ddot {o}del$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Lukasiewicz$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Goguen$ </tex-math></inline-formula> implication operators is presented. An application example and experiment are offered to demonstrate the validity of the obtained conclusion. Furthermore, the new distance metric is compared to traditional distances, and its benefits and limits are discussed. The results show that our approach to research the robustness is simpler and more representative, and the robustness of the algorithm based on other implication operators can be obtained by simple substitution.

Modifications of the Goguen's intuitionistic fuzzy implication

On the basis of the Goguen’s intuitionistic fuzzy implication, seven of its modifications are constructed. Some of their basic properties are studied. The negations, generated by these implications are introduced and some of their properties are also described.

Modal Type of Weak Intuitionistic Fuzzy Implications Generated by the Operation Δ

Abstract In 2020 L. Atanassova has been introduced the new operation Δ over intuitionistic fuzzy sets and over intuitionistic fuzzy pairs. Some of its properties have been studied in 2021 from L. Atanassova and P. Dworniczak. In 2021 L. Atanassova and P. Dworniczak generated an intuitionistic fuzzy implication by the operation Δ, and it has been introduced and some of its basic properties have been described. Here, eight modal type of intuitionistic fuzzy implications are generated by the first one and some of their properties are discussed.

Modifications of Łukasiewicz’s intuitionistic fuzzy implication

In [6], G. Klir and B. Yuan named after J. Łukasiewicz the implication p \rightarrow q = min(1, p+q). In a series of papers, 198 different intuitionistic fuzzy implications have been introduced, and their basic properties have been studied. Here we introduce six new implications which are modifications of Łukasiewicz’s intuitionistic fuzzy implication, and we describe and prove some of their properties.

On the weak intuitionistic fuzzy implication based on △ operation

In this paper, a new weak intuitionistic fuzzy implication is introduced. Fulfillment of some axioms and properties, together with the Modus Ponens and Modus Tollens inference rules are investigated. Negation induced by the newly proposed implication is presented.

On an Intuitionistic Fuzzy Form of the Goguen’s Implication

In the present paper we construct a new intuitionistic fuzzy implication, giving intuitionistic fuzzy form to Goguen’s implication. Some of its basic properties are studied and illustrated with examples. Geometrical interpretations of the different forms of the new implication are given. Other forms of the Goguen’s implication are discussed.

Rule extraction based on linguistic-valued intuitionistic fuzzy layered concept lattice

As one of the research tools for data processing and knowledge discovery, concept lattice can effectively extract information. In daily life, due to the ambiguity and uncertainty of the decision environment, different experts may provide different evaluation information according to individual needs which may be expressed with linguistic values. In order to handle these problems, we study the rule extraction method of linguistic-valued intuitionistic fuzzy layered concept lattice. First, we propose a linguistic-valued intuitionistic fuzzy formal decision context based on the intuitionistic fuzzy lattice implication algebra, which can simultaneously process the obtained comparable and incomparable linguistic information from both positive and negative aspects. Furthermore, by setting different linguistic-valued trust degrees, we put forward the linguistic-valued hierarchical concept construction operator. On this basis, a linguistic-valued intuitionistic fuzzy layered concept lattice can be constructed to meet the requirements of different experts at different levels. And then through the relationship between the conditional concept and decision concept in the obtained concept set, we get a rule set of the formal decision context. Finally, we present a rule extraction method to help people make more reasonable decisions, combining the confidence and support degree of linguistic-valued intuitionistic fuzzy decision rules. And a practical example involving individual financial investment decision-making is used to verify the efficiency and applicability of the proposed approach.

Formula omitted]-implications and distributivity of implications on [formula omitted] over t-representable t-norms: The case of strict and nilpotent t-norms

In this paper, a new class of intuitionistic fuzzy implications (IFIs) known as (fI,ω)-implications is introduced which is a generalized form of Yager’s f-implications in intuitionistic fuzzy environment (IFE). The basic properties of these implications are discussed in detail. It is shown that (fI,ω)-implications are not only the generalizations of Yager’s f-implications, but also the generalizations of R-, (S,N)- and QL-implications in IFE. The distributivity equations II(T(u,v),w)=S(II(u,w),II(v,w)) and II(u,T1(v,w))=T2(II(u,v),II(u,w)) over t-representable t-norms and t-conorms generated from nilpotent and strict t-norms in IF set theory are discussed. Also, we solve the open problems concerning characterize all of the correct solutions of the distributive equation II(u,T1(v,w))=T2(II(u,v),II(u,w)) when t-norms are strict and nilpotent.

Knowledge reasoning approach with linguistic-valued intuitionistic fuzzy credibility

Linguistic term evaluations are always collected from two opposite sides at the same time in an assessment system. To process the linguistic knowledge, we propose an approximate reasoning approach with linguistic-valued intuitionistic fuzzy credibility based on linguistic-valued intuitionistic fuzzy lattice implication algebra and apply it to the assessment system. Firstly, we give a knowledge representation model with linguistic-valued intuitionistic fuzzy credibility. Based on the representation model, the forms and patterns of linguistic intuitionistic fuzzy modus ponens (LI-FMP) and linguistic intuitionistic fuzzy modus tollens (LI-FMT) are defined. Then there are three main phases of the knowledge reasoning with linguistic-valued intuitionistic fuzzy credibility. For a single rule, the similarity-based algorithms for LI-FMP and LI-FMT are given to get the sub-conclusion and the properties of similarity-based algorithms are discussed. For the multi-rule, we propose a rule aggregation operator to get the final conclusion by combining all the sub-conclusions. Some incomparable results are further processed if it is necessary. An intuitionistic linguistic-real valuation function is defined implying a linguistic intuitionistic fuzzy distance which is proved to be a positive valuation function. The ranking method of the incomparable results utilizes the linguistic intuitionistic distance. Lastly, the example about individual credit risk assessment shows how the proposed approach work and the contrast example illustrates that the proposed approach is rational and applied.

SISO Intuitionistic Fuzzy Systems: IF-${t}$ -Norm, IF-${R}$ -Implication, and Universal Approximators

Fuzzy system, as the main method of approximate reasoning, has been widely used to solve practical problems. The existing results on fuzzy systems were based on fuzzy sets, fuzzy if-then rule, and fuzzy inference. In this work, we investigate the component of the intuitionistic fuzzy systems, such as the intuitionistic fuzzy triangular norms (IF-t-norm), the intuitionistic fuzzy residual implications (IF-R-implication), the intuitionistic fuzzy if-then rule, the intuitionistic fuzzy inference, and the intuitionistic defuzzification. Accordingly, the single input single output (SISO) intuitionistic fuzzy systems are built. Some examples are paid to illustrate the methods and their application. Moreover, the approximation properties of the intuitionistic Mamdani, the intuitionistic Larsen, the IF-t-norm, and the intuitionistic triple-I fuzzy systems are obtained, which show that the SISO intuitionistic fuzzy systems are the universal approximators.

Covering-based generalized IF rough sets with applications to multi-attribute decision-making

Multi-attribute decision-making (MADM) can be regarded as a process of selecting the optimal one from all objects. Traditional MADM problems with intuitionistic fuzzy (IF) information are mainly focused on an IF binary relation. However, some complicated problems can not be effectively solved by an IF relation. In order to solve these issues, we set forth two novel decision-making methods that are stated in terms of novel and flexible generalized IF rough set models. For defining these models, four types of IF neighborhoods based on an IF implicator I and an IF triangular norm T are firstly proposed. Secondly, by means of four types of IF neighborhoods, four types of coverings-based generalized IF rough set models are proposed. Furthermore, the relationships among the four types of coverings-based generalized IF rough set models and other types of IF rough set models are also discussed. By means of the principle of the IF-TOPSIS methods, MADM with IF information based on covering-based generalized IF rough sets or based on covering-based generalized fuzzy rough sets is put forward. Finally, we solve MADM problems with the evaluation of IF information based on covering-based generalized IF rough set models. By comparative analysis, we find that the results of this method based on covering-based generalized IF rough set models and based on covering-based generalized fuzzy rough set models are highly consistent. In particular, this method based on covering-based generalized IF rough set models is more effective to deal with MADM than that based on covering-based generalized fuzzy rough set models.

Intuitionistic fuzzy interpretations of Barcan formulas

In a series of papers, 186 different intuitionistic fuzzy implications have been defined. For each of them, the four Barcan formulas are checked in the case of the standard intuitionistic fuzzy quantifiers. The list of the intuitionistic fuzzy implications that satisfy the Barcan formulas is given. Modifications of these formulas for the case of the extended intuitionistic fuzzy quantifiers are introduced and an open problem for them is formulated.