# Compact Fuzzy Systems Based on Boolean Relations
## 1 Introduction
Besides, many HFS operators have been introduced to deal with this issue in various applications where decision-making is the most remarkable aspect [4; 40]. Meanwhile, inference fuzzy systems based on Boolean relations are suitable when using [PERSON]'s algebra with sets whose membership values are given by \(\{0,1\}\), then a true table defines a model [41; 42; 43]. Nevertheless, these systems display low performance due to abrupt transitions. Consequently, a way to improve the performance is by replacing the Booleans for fuzzy sets [44]. When the concepts clearly define the membership in the range \([0,1]\) which means that the limits of the concepts are \"softened\" (fuzzy) with a structure equivalent to \(\{0,u,1\}\) makes it possible to employ [PERSON]'s algebra [45]. About related works, document [41] suggests the use of the Boolean algebra system design features. Conversely, paper [46] presents a practical approach of a methodology based on the design of automations employing fuzzy sets for implementing the process associated with defuzzification. Simultaneously, in [44] it is indicated that this proposal can be interpreted as a fuzzy inference system. This technique considers the interaction of sensors, actuators, and Boolean relations involved in control strategies of any industrial process [44; 46]. As part of the design process, [47] employs [PERSON]'s algebra to transform binary logic functions into fuzzy logic. Besides, paper [48] describes a method to simplify formulas using finite algebras. In addition, the work presented in [49] offers the vagueness modeling, ambiguity, and contrariness with trivalent and tetravalent fuzzy logic, employing [PERSON] and [PERSON] algebraic structures, focusing on the tables and graphs. Thus, the design methodology based on these elements is presented in [50]. The proposed methodology considers a design using [PERSON]'s algebra, later extended to a fuzzy system, employing [PERSON]'s algebra. Nevertheless, it is necessary to keep in mind that in [PERSON]'s algebra, the variable simplification is rooted in the law of excluded middle, which is not accomplished using [PERSON]'s. However, a similar simplification with Kleene algebra may be achieved employing the absorption law. _Proposal Approach and Document Organization_  
This document offers a design proposal to fuzzy logic systems based on Boolean relations with compact structure. Previously in [50], a methodological design is proposed to implement fuzzy inference systems based on Boolean relations. This work is currently continued considering specific design cases of fuzzy systems with simplifications applied to equations in the inference process, providing compact systems useful in different applications as in identification and control of dynamic systems. Related applications of these systems can be observed in [51] to predict chaotic time series while reference [52] identifies a variable charge in a distribution system; besides, document [53] displays a control of a synchronous generator of permanent magnets. Finally, the voltage control in a electrical grid is shown in [54]. Although in these works compact fuzzy structures are employed, these do not use the design methodology presented in [50] via [PERSON]'s algebra that allow the formalization of these systems which is the main topic of this work. The document is organized as follows: Section 2 describes the definitions employed for the design process of fuzzy inference systems based on Boolean relations and the simplification procedure with Kleene algebra used to determine the structures of compact systems. Section 3 shows the design process for specific cases of fuzzy inference systems based on Boolean relations called \"compact systems\", where Kleene algebra is employed for the simplification analysis. Section 4 introduces compact systems structures that are obtained considering two cases where the simplification of configurations that are represented in a truth table can be performed.