# Compact Fuzzy Systems Based on Boolean Relations
## 1 Introduction  
Modeling, controlling, and performing any system is an essential task for the designer/user to apply the necessary corrections to make it more effective. Frequently, evaluating a system is done under fuzzy rules, for instance, by employing qualitative instead of numeric marks or incomplete numeric functions, etc. [1]. Thus, fuzzy systems theory is built on concrete and practical formation for many relevant applications, especially in decision-making processes [2, 3]. Various real-life problems are problematic since they involve human beings, mechanical elements, and other issues. Consequently, cases like these or even simpler ones may include diverse uncertainty conditions of high relevance when achieving satisfactory outcomes. Uncertainty may include randomness, fuzziness, indistinguishability, and incompleteness [4, 5]. The theory on fuzzy sets and fuzzy logic permits manipulating incomplete and imprecise information of objects belonging to a concept [6]. [PERSON] proposed a Fuzzy Set (FS) as a focus to process and represent vagueness in the real world [7]. Fuzzy Inference Systems (FIS) are utilized to generate sets of fuzzy rules in problem solving in applications like detection [8, 9], prediction [10, 11], classification [12, 13, 14], and other tasks [15, 16, 17]. In [18], a review of the main theories involved in hybrid models based on neuro-fuzzy systems is presented. The basic concepts regarding the history of hybrid models, the features of the leading models, and their applications also are presented; some neuro-fuzzy structures observed are: fuzzy regression, adaptive neuro-based fuzzy inference systems, autoassociative neuro-fuzzy systems, and fuzzy cognitive maps. It also reviews fuzzification and defuzzification techniques and training algorithms. The authors exposed that fuzzy neural network models and their derivations are useful for the design of systemwith high degree of accuracy and a suitable level of interpretability allowing applications in different areas of science. Using fuzzy sets provides a foundation for systematically handling vague and inaccurate concepts; mainly, fuzzy sets can be used when representing linguistic variables, which are values defined in a linguistic way [19]. Fuzzy logic sets are a suitable solutions alternative in cases with uncertainty, inaccuracy, and ambiguity in the presence of phenomena to be controlled or modeled [20; 21; 22]. Regarding fuzzy inference systems, the most widely known are Mamdani, Takagi-Sugeno, and Tsukamoto. Mamdani systems use fuzzy sets in inputs and outputs and include and aggregation of fuzzy sets acquired from each rule's implication. The result is obtained by defuzzification. Takagi-Sugeno systems employ functions that depend on the input value; the inference process result is accepted as a weighted addition whose value is given by each rule's inference process. On the other hand, Tsukamoto systems employ fuzzy sets for input and output. In the output, it uses injective membership functions to get the preimage of the value from the implication process; later, the results are weighted with the values obtained in implication process. Concerning additional fuzzy inference developments, complex fuzzy sets are proposed to improve the design and modeling of real-life applications [23]. A Complex Fuzzy Set (CFS) [24] is an extension of a fuzzy set where the membership function consists of an amplitude term and one phase. Accordingly, paper [25] suggested a framework for logic reasoning called Complex Fuzzy Logic (CFL), employing the CFL theory since this is useful to make systems capable of managing different problems. As a variation, the Mamdani Complex Inference System (M-CFIS) is proposed in [26]. Others are the CFIS like Neuro-complex, Adaptive Neuro-Complex Fuzzy Inferential System (ANCFIS) with higher-order TSK models [27], the Randomized Adaptive Neuro-Complex Fuzzy Inference System (RANCFIS) [28], and the Fast Adaptive Neuro-Complex Fuzzy Inference System (FANCFIS) [29].