# Compact Fuzzy Systems Based on Boolean Relations
## 1 Introduction
However, a deficiency detected on the M-CFIS is that the base of rules may become redundant to a specific set of rules. In addition, reference [23] presents the analysis of a collection of complex data using properties from the fuzzy concept network and the complex soft set. On the other hand, Mamdani Complex Fuzzy Inference System (M-CFIS) allows managing events not restricted only to values of a point in a specific time but including all the values inside some time intervals (phase term). In such scenarios, fuzzy complex theory permits one to see the range and phase values of an event, which means better performance; nevertheless, the set of rules may be redundant to a specific dataset [2]. Meanwhile, the fuzzy sets theory has been applied to manage uncertainty in numerous areas; even so, it shows limitations to face situations of inaccuracy when sources of vagueness appear [4]. Thus, the literary sources include different extensions of fuzzy sets to overcome such limitations:  
* Intuitionistic Fuzzy Sets (IFS) proposed by [PERSON] [30]. These consider the membership and no membership grade for each element simultaneously. * Type 2 Fuzzy Sets (T2 FS), incorporate uncertainty in defining the membership function through a fuzzy set to model the unitary interval. Type 2 Fuzzy Sets (T2 FS) include uncertainties when defining the membership function using a fuzzy set to model it [31]. * Interval Valued Fuzzy Sets (IVFS). The membership grade of an element is given by a closed subinterval of the unitary interval; thus, the length of that interval is understood as a measurement of lack of certainty to build an element's accurate membership grade [32]. * Multiset-Based Systems, where the membership grade of an element is given by a subset of \([0,1]\)[33]. * The theory of Hesitant Fuzzy Sets (HFS) is about managing situations where a set of values is possible to use in the process of defining the membership grade of an element [34]. Experts use intuitionist fuzzy sets to manage the evaluation of an alternative with specific values of membership and nonmembership. Meanwhile, hesitant fuzzy sets handle situations where experts doubt several possible membership values to evaluate an option [35]. The theory of fuzzy sets of [PERSON] [7] has been applied in several fields. The idea of a fuzzy set is welcome since it manages uncertainty and vagueness that the Cantorian set cannot address. In fuzzy sets theory, the membership of an element is a unique value between zero and one. However, it may not always be true that the degree of nonmembership of an element in a fuzzy set is equal to one minus the degree of membership because there may be some degree of doubt; therefore, reference [30] introduced a generalization called intuitionistic fuzzy sets. This includes a margin of hesitation defined as one minus the sum of the degrees of membership and no membership [36]. On the other hand, hesitant fuzzy sets are useful to deal with group decision-making processes when the experts doubt possible memberships for an element. These possible memberships may include crisp values \([0,1]\) and interval values [37]. [PERSON] [34] uses the same approach introducing an extension for sets called hesitant fuzzy sets, given the standard difficulty that generally appears when it is necessary to define the grade of membership. Such a problem is not given by error range (as in IFS) or any distribution of possibilities (as in T2 FS) but because there are possible values that make it difficult to decide which one is accurate. This is a common situation in the decision-making process for an expert regarding several grades of membership \(\{0.43,0.58,0.79\}\) of element \(x\) in the set \(A\). Thus, hesitant fuzzy sets become relevant since they make it easier to manage hesitation in typical real-life situations. Consequently, many developments have been carried out from different quantitative [37; 38] and qualitative [39] perspectives, given that many doubts arise, which model the uncertainty in both directions.