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As the theory of picture fuzzy sets has been developed, more information in life can be expressed in mathematical terms. Similarity measure is a special tool for quantifying the similarity between two sets, so studying similarity measure on picture fuzzy sets has become a trending topic. This new research direction has drawn a great deal of attention from experts and has led to a number of important results which have led to significant results in a number of practical applications. By examining these new findings, we discovered that there are many studies on similarity measure of picture fuzzy sets, some of them are deficient in solving certain problems, and such similarity measures can lead to the calculation of unreasonable data in practical applications, affecting the final results. Secondly, there is still room for research similarity measures on exponential functions. Considering these two aspects, we propose two new similarity measures based on exponential function, which not only satisfy the axiomatic definition of similarity measures, but also show reasonable computational results in practical applications. [ABSTRACT FROM AUTHOR]

Similarity measure is an important tool to evaluate the degree of similarity between different objects. Therefore, a large number of similarity measures on picture fuzzy sets have been studied. However, some existing similarity measures have not been unified into a framework. The aim of this paper is to study a construction method of similarity measure on picture fuzzy sets, based on this, we can unify some existing similarity measures into a framework, and then propose some new similarity measures. Firstly, the concept of picture fuzzy equivalence is introduced. Some construction methods for picture fuzzy equivalence are given. Secondly, a construction method of similarity measure on picture fuzzy sets is proposed. Some new similarity measures on picture fuzzy sets based on picture fuzzy equivalences are given and some existing similarity measures are unified into a framework. Finally, the proposed similarity measures are applied to pattern recognition. By comparing the degrees of confidence, we find that our degrees of confidence are higher than some similarity measures. The experimental results show that the proposed similarity measures can effectively distinguish some similar picture fuzzy sets. [ABSTRACT FROM AUTHOR]

In previous studies, based on picture fuzzy sets, full implication algorithm and quintuple implication principle have been studied. The problem is, full implication algorithm did not consider the relationship between the input and antecedent of rule, while quintuple implication principle only considered the inclusion relationship between the rule's antecedent and input. Both algorithms did not consider the similarity of rule's antecedent and input. This leads to unreasonable calculation results in some cases. Therefore, we study fuzzy reasoning methods based on picture similarity measure for generalized modus ponens and generalized modus tollens in this paper. First of all, based on picture biresiduation, a new picture similarity measure between picture fuzzy sets is constructed. Then, representations of solutions of fuzzy reasoning methods based on picture similarity measure are given. Moreover, reducibility and robustness of fuzzy reasoning methods based on picture similarity measure are studied. Finally, two issues related to medical diagnosis are addressed by the proposed methods, demonstrate the rationality and effectiveness of our suggested methods. [ABSTRACT FROM AUTHOR]

In previous studies, based on picture fuzzy sets, full implication algorithm and quintuple implication principle have been studied. The problem is, full implication algorithm did not consider the relationship between the input and antecedent of rule, while quintuple implication principle only considered the inclusion relationship between the rule's antecedent and input. Both algorithms did not consider the similarity of rule's antecedent and input. This leads to unreasonable calculation results in some cases. Therefore, we study fuzzy reasoning methods based on picture similarity measure for generalized modus ponens and generalized modus tollens in this paper. First of all, based on picture biresiduation, a new picture similarity measure between picture fuzzy sets is constructed. Then, representations of solutions of fuzzy reasoning methods based on picture similarity measure are given. Moreover, reducibility and robustness of fuzzy reasoning methods based on picture similarity measure are studied. Finally, two issues related to medical diagnosis are addressed by the proposed methods, demonstrate the rationality and effectiveness of our suggested methods. [ABSTRACT FROM AUTHOR]

In this paper, we focus on studying picture fuzzy reasoning full implication methods. First, picture associated t -norm and its residuated implication operator are given. Then, picture fuzzy reasoning full implication principles and the solutions of picture fuzzy reasoning full implication methods based on picture associated t -norm are given. Moreover, the reducibility and robustness of picture fuzzy reasoning full implication methods based on picture associated t -norm are proved. Finally, pattern recognition problems can be successfully solved by the proposed method. [ABSTRACT FROM AUTHOR]

Picture fuzzy sets, which are the extension of intuitionistic fuzzy sets, can deal with inconsistent information better in practical applications. A distance measure is an important mathematical tool to calculate the difference degree between picture fuzzy sets. Although some distance measures of picture fuzzy sets have been constructed, there are some unreasonable and counterintuitive cases. The main reason is that the existing distance measures do not or seldom consider the refusal degree of picture fuzzy sets. In order to solve these unreasonable and counterintuitive cases, in this paper, we propose a dynamic distance measure of picture fuzzy sets based on a picture fuzzy point operator. Through a numerical comparison and multi-criteria decision-making problems, we show that the proposed distance measure is reasonable and effective. [ABSTRACT FROM AUTHOR]