Interval-valued fuzzy reasoning full implication algorithms based on the t-representable t-norm☆
Introduction
Fuzzy reasoning is the mathematical and logical foundation of fuzzy control. It is well known that the most fundamental forms of fuzzy reasoning are fuzzy modus ponens and fuzzy modus tollens, which are given as follows [43]:
Given the input “x is ” and fuzzy rule “if x is A then y is B”, try to deduce a reasonable output “y is ”, fuzzy modus ponens (FMP);
Given the input “y is ” and fuzzy rule “if x is A then y is B”, try to deduce a reasonable output “x is ”, fuzzy modus tollens (FMT).
In the above models, and , where and stand for the sets of all fuzzy subsets of the universes X and Y respectively.
These most basic reasoning models FMP and FMT are also called generalized modus ponens (GMP) and generalized modus tollens (GMT) respectively, which were proposed by Baczyński and Jayaram [2].
Zadeh [43], [44] proposed the compositional rule of inference (CRI for short) to deal with the above inference forms for GMP and GMT. In reasoning processes, the CRI method has some arbitrariness and lacks solid logical basis. As an alternative for the CRI method, Wang [40] proposed a new method, called the full implication triple I method, or simply the triple I method. This method sets fuzzy reasoning within the framework of logical semantic implication [41], and it can be considered as a reasonable complement for the CRI method. Consequently, a considerable number of studies on the triple I method have been reported in recent years. Wang and Fu [42] established the unified triple I method based on regular implications and normal implications. Pei [34] constructed unified triple I algorithms based on left-continuous t-norm. Meanwhile, the sound logical foundation for the triple I algorithms based on the left-continuous t-norm was established by Pei [35]. Liu and Wang [24] discussed continuity problems of triple I methods based on several implications. Luo and Yao [25] investigated triple I algorithms based on Schweizer-Sklar operators. However, the triple I algorithms enable the computed solutions for FMP and FMT to be useless or misleading in some cases. Zhou et al. [45] proposed the quintuple implication principle of fuzzy reasoning and characterized the closeness of and A (or B and ) in the process of fuzzy reasoning, which efficiently improves triple I method. Luo and Zhou [29] gave the predicate formal representation of the solution for the quintuple implication method and the strict logic proof, placing brought the quintuple implication method within the logical framework. Nevertheless, the quintuple implication method does not consider the similarity measures of and A (B and ). This shortcoming can lead the quintuple implication method into unreasonable situations of the computed solutions for FMP (FMT). In order to overcome the defect of the quintuple implication method, Luo and Zhao [30] proposed fuzzy reasoning algorithms based on similarity.
Although fuzzy set theory has been widely used, it has some shortcomings in dealing with uncertain information. Therefore, various extensions of fuzzy sets have been given. In 1975, Zadeh [44] proposed interval-valued fuzzy set (IVFS for short). The definition of IVFS was given in the same year by Sambuc [36] as follows: the membership degree of the element is given by a closed subinterval of the unit interval [0, 1]. The interval-valued fuzzy sets can not only effectively reduce the loss of fuzzy information but also reflect the vagueness and uncertainty in information processing. Due to the advantages of interval-valued fuzzy sets, in recent years, researchers have investigated the properties of interval-valued fuzzy reasoning as well as the numerous applications in various fields such as neural networks, control systems and so on [4], [16], [22], [32]. Deschrijver [6], [7], [8], [9] discussed some classes of interval-valued t-norms. Gasse et al. [11], [12] introduced the triangle algebra, and showed that the triangle algebra is isomorphic to the interval-valued residuated lattice. Gasse et al. [13], [14] introduced the interval-valued monoidal t-norm based logic and some of its extensions, and proved the strong standard completeness of the interval-valued monoidal t-norm based logic. Alcalde et al. [1] gave a method to construct the interval-valued fuzzy implication operator associated with t-representable t-norm. Bedregal et al. [3], [37], [38] discussed the interval-valued fuzzy S-implications, QL-implications and D-implications. Dimuro et al. [10] provided a systematic methodology for the selection of interval-valued t-norms and interval-valued t-conorms in various applications of fuzzy systems. Mendel [31] proposed the set representation theorem for a general interval-valued fuzzy set, and this theorem can be used as a starting point to solve many diverse problems involving the interval-valued fuzzy sets. Li et al. [20] extended the CRI method to the case of the interval-valued fuzzy sets and discussed the robustness of the interval-valued CRI method. Luo et al. [26], [27] studied the interval-valued triple I algorithms and the interval-valued reverse triple I algorithms based on the left-continuous associated t-norm and their robustness. Luo et al. [28] proposed the interval-valued quintuple implication principle based on the left-continuous associated t-norm. Liu and Li [23] investigated the interval-valued fuzzy reasoning with multi-antecedent rules. These studies focus exclusively on the interval-valued fuzzy reasoning algorithms based on a special class of the t-representable t-norm.
In this paper, we study the interval-valued fuzzy reasoning full implication triple I algorithms based on generalized left-continuous t-representable t-norm . The rest of the paper is organized as follows. Section 2 contains some basic concepts related to t-representation t-norm. In Section 3, we establish the interval-valued fuzzy reasoning triple I algorithms based on the left-continuous t-representable t-norm . Section 4 analyzes the robustness of the interval-valued fuzzy reasoning triple I algorithms based on the left-continuous t-representable t-norm. The final section exhibits the proposed results as well as an outlook on future research.
Section snippets
Preliminaries
In this section, we describe some concepts and an important theorem that will be required for our following work. First some notations used throughout this paper are given: , , , .
Definition 2.1 ([18]) An increasing, commutative, associative mapping T: is called a triangular norm (t-norm for short) if it satisfies = x for any .
Definition 2.2 ([18]) A t-norm T is called left-continuous, if for any and for each , there
Interval-valued fuzzy reasoning triple I algorithms based on the t-representable t-norm
Suppose that is the interval-valued residuated implication induced by the left-continuous interval-valued t-representable t-norm .
The interval-valued triple implication model is denoted as where , .
Definition 3.1 Interval-valued Triple Implication Principle for FMP Suppose that , . Let , . If the smallest element in the set exists
Robustness of the interval-valued fuzzy reasoning triple I algorithms based on the t-representable t-norm
In this paper, we only consider the case of finite universes. Sensitivity of fuzzy connectives and perturbation of interval-valued fuzzy sets are introduced as follows. Definition 4.1 ([20]) Let f be an n-tuple mapping from to SI, and . For arbitrary , the ε sensitivity of f at point is defined by where .
Definition 4.2 ([20]) The maximum ε sensitivity of f
Conclusions
In this paper, we have proved that the interval-valued t-representable t-norm is left-continuous if and only if satisfies the residuation principle. Following this result we have presented the interval-valued fuzzy reasoning triple I algorithms based on left-continuous interval-valued t-representable t-norm . We have also settled the -type triple I solutions of the interval-valued fuzzy reasoning triple I algorithms. Moreover, the robustness of the interval-valued fuzzy
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to thank Editor-in-Chief Professor Thierry Denoeux, Area Editor and anonymous reviewers for their valuable comments and suggestions.
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