On Fuzzy Propositional Logic with Different Negations

We explore the basic fuzzy logic BL as well as propositional fuzzy logics with modalities □ and ◊ and a total accessibility relation. Formulations and proofs are given to replacement theorems for BL. A basic calculus of modal fuzzy logic is introduced. For this calculus and its extensions, we prove replacement and deduction theorems.

This article provides a high-level retrospective of type-2 fuzzy sets and fuzzy logic systems. It explains how type-2 fuzzy sets can be used to model membership function uncertainties, and how by doing this smoother performance can be obtained than by using type-1 fuzzy sets. It also summarizes the notation that should be used for type-2 fuzzy sets, describes four important mathematical representations for these fuzzy sets, explains the differences between type-1 and type-2 fuzzy logic systems and which of the four representations is most useful when designing an optimal type-2 fuzzy logic system, provides a very useful strategy for optimal designs of fuzzy logic systems – one that guarantees performance improvement as one goes from a type-1 fuzzy logic system to a type-2 fuzzy logic system design – , and describes four methods for simplifying the designs of type-2 fuzzy logic systems. Finally, it explains why type-2 fuzzy sets can capture two kinds of linguistic uncertainties simultaneously (the uncertainty of an individual and the uncertainties of a group about a word), whereas type-1 fuzzy sets cannot, and that such type-2 fuzzy set word models are what should be used to implement Zadeh’s Computing With Words paradigm.

Fuzzy logic control based on the Takagi–Sugeno inference method has been applied for the yeast alcoholic fermentation running in a continuous-time biochemical reactor in this paper. A type-1 fuzzy PID controller was designed to temperature control in a biochemical reactor. The fuzzy PID controller was also designed using type-2 fuzzy sets. The advantage of the fuzzy control is that it can be used very successfully for control of strongly non-linear processes and processes that are difficult to model because of complicated reaction kinetics. Obtained simulation results confirm this fact. The disadvantage of the fuzzy control design lies in the time-consuming tuning of controllers. The subtractive clustering method was used to identify the rule base. This approach was chosen to minimize the number of rules of the designed fuzzy logic controllers and to simplify the fuzzy controller design. Simulation results confirm that fuzzy PID controllers can assure better performance than conventional PID controllers. Yeast alcoholic fermentation is an exothermic process and the cooling is necessary to maintain the optimal temperature in the reactor. Fuzzy PID control reduces coolant consumption in comparison with conventional PID control and so it assures energy-efficient control of continuous-time alcoholic fermentation. Using type-2 fuzzy sets in the fuzzy PID controller design is very promising as the type-2 fuzzy controller offers even better results than the type-1 fuzzy controller.

The paper presents a rule-based fuzzy logic controller to control the output power of a pulse width modulated (PWM) inverter used in a stand-alone wind energy conversion scheme (SAWECS). The self-excited induction generator used in SAWECS has the inherent problem of fluctuations in the magnitude and frequency of its terminal voltage with changes in wind velocity and load. To overcome this drawback the variable magnitude, variable frequency voltage at the generator terminals is rectified and the DC power is transferred to the load through a PWM inverter. The objective is to track and extract maximum power from the wind energy system and transfer this power to the local isolated load, This is achieved by using the fuzzy logic controller which regulates the modulation index of the PWM inverter based on the input signals: the power error; and its rate of change. These input signals are fuzzified, that is defined by a set of linguistic labels characterized by their membership functions predefined for each class. Using a set of 49 rules which relate the fuzzified input signals to the fuzzy controller output, fuzzy set theory and associated fuzzy logic operations, the fuzzy controller's output is obtained. The fuzzy set describing the controller's output (in terms of linguistic labels) is defuzzified to obtain the actual analog (numerical) output signal which is then used to control the PWM inverter and ensure complete utilization of the available wind energy. The proposed rule-based fuzzy logic controller is simulated and the results are experimentally verified on a scaled down laboratory prototype of the SAWECS.

Consistency degrees of finite theories in Łukasiewicz propositional fuzzy logic

One of the promising and effective approaches of the modern control theory of complex technological systems in the conditions of uncertainty caused by the fuzziness of the initial information is an approach based on the use of expert evaluation methods and the theory of fuzzy sets. This article presents a study of the properties of an automatic control system with the use of a pseudo-linear fuzzy PID controller built on the basis of a correction device with phase advance. One of the alternative methods of constructing control and regulation systems for objects that are indistinctly defined from the point of view of classical theory is the use of so-called fuzzy logic controllers. Therefore, the development of fuzzy control controllers based on existing microcontrollers is a very urgent task, since a pseudo-linear fuzzy controller built on the basis of fuzzy sets and fuzzy logical I/O, under the condition of uncertainty of the disturbing effect, is able to provide higher quality indicators of the transient process than a traditional PID controller. As pseudo-linear correction devices (PKU) we use: CU with amplitude suppression, with phase advance and with separate channels for amplitude and phase, since one of the main disadvantages of the PID controller is the presence of phase lag and high sensitivity to interference in the measuring channel.

Based on the Schweizer-Sklar t-norm, a fuzzy logic system UL* is established, and its soundness theorem and completeness theorem are proved. The following facts are pointed out: the well-known formal system SBL∼ is a semantic extension of UL*; the fuzzy logic system IMTLΔ is a special case of UL* when two negations in UL* coincide. Moreover, the connections between the system UL* and some fuzzy logic formal systems are investigated. Finally, starting from the concepts of “the strength of an ‘AND’ operator” by R.R. Yager and “the strength of fuzzy rule interaction” by T. Whalen, the essential meaning of a parameter p in UL* is explained and the use of fuzzy logic system UL* in approximate reasoning is presented.

Emerging trends in soft set theory and related topics.

Rough set theory for the interval-valued fuzzy information systems

Interval type‐2 fuzzy logic systems have favorable abilities to cope with uncertainties in many applications. While the block type‐reduction under the guidance of inference plays the central role in the systems, Karnik‐Mendel (KM) iterative algorithms are standard algorithms to perform the type‐reduction; however, the high computational cost of type‐reduction process may hinder them from real applications. The comparison between the KM algorithms and other alternative algorithms is still an open problem. This paper introduces the related theory of interval type‐2 fuzzy sets and discusses the blocks of fuzzy reasoning, type‐reduction, and defuzzification of interval type‐2 fuzzy logic systems by combining the Nagar‐Bardini (NB) and Nie‐Tan (NT) noniterative algorithms for solving the centroids of output interval type‐2 fuzzy sets. Moreover, the continuous version of NT (CNT) algorithms is proved to be accurate algorithms for performing the type‐reduction. Four computer simulation examples are provided to illustrate and analyze the performances of two kinds of noniterative algorithms. The NB and NT algorithms are superior to the KM algorithms on both calculation accuracy and time, which afford the potential application value for designer and adopters of type‐2 fuzzy logic systems.

This paper proposes a novel method to estimate the influence of domestic consumers on power quality, namely harmonic pollution, using fuzzy logic. Comparing the conventional measurement approach, the proposed fuzzy logic method is simpler and time saving. The fuzzy logic system used to quantify the impact on harmonic distortion, contains several modules: the fuzzifier, database, base rule and defuzzifier. The fuzzy system uses the Mandami type rule structure and linguistic fuzzy approach. The fuzzy sets were developed considering the domestic consumers' electrical apparatus (electrical lamps and appliances). Thus the linguistic variables were used to implement the behavior of these loads working together.

Soft set theory was first introduced by D. A. Molodtsov [D. Molodtsov, Soft set theory first results, Computers & Mathematics with Applications, 37(4–5) (1999) 19–31] in 1999 as an effective approach to deal with uncertainty. Over time, fuzzy soft set theory has evolved significantly, integrating concepts from both fuzzy set theory and soft set theory to enhance decision-making processes. Acharjee and Medhi [S. Acharjee and S. Medhi, Logical connectives of fuzzy soft set theory, New Mathematics and Natural Computation, 21(1) (2025) 339–352] introduced several advanced concepts related to logical connectives in fuzzy soft set theory. Building on Molodtsov’s foundational work, this paper explores new developments in fuzzy soft logic, extending the findings of Acharjee and Medhi. Specifically, we introduce the concepts of [Formula: see text]-fuzzy soft implication and [Formula: see text]-fuzzy soft bi-implication, along with their related results within the framework of fuzzy soft logic.

Fuzzy normal forms can be generated with an application of "Normal Form Generation Algorithm" on fuzzy truth tables. This takes place at the third level of knowledge representation, i.e., propositional level. It is shown that at least three distinct sets of normal forms can be generated depending on the axioms one is willing to impose on the propositional fuzzy set and logic theories. All are conjunctive-disjunctive and complement based De Morgan logics with the following three classes of axioms that identify each general class of fuzzy normal forms in order of least to most restrictive set of axioms in the following sense: 1) boundary and monotonicity; 2) boundary, monotonicity, associativity and commutativity; and 3) boundary, monotonicity, associativity, commutativity and idempotency.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In cultivation of vegetable plants, plant factory system requires flexible approach and resolution methods with a high degree of accuracy, considering the main requirement quality of plant factory is internal environmental conditions that must be a constant value for a long period, while the value of temperature and humidity environment is fluctuate easily. The purpose of this study is to find out the steps of obtaining fuzzy values and knowing the value of PWM (Pulse Width Modulation) output in the plant factory for a set point temperature of 28 0 C and a set point humidity of 75% using the Tsukamoto fuzzy approach. Fuzzy logic is an approach method and problem solving by modeling linear and non-linear functions, the main advantages of this method are being very flexible, high tolerance and high-value precision as well natural language use in this method, so it can be understood easily. Pak choy plants ( Brassica Chinensis L. ) can grow ideally at temperatures of 19-30 o C and 75% humidity. The steps of research is, first determine the input and output variables which are firm sets. Second, fuzzy compositions, are the steps of changing input values to the value of functions involve. Third, inference fuzzy set of Tsukamoto with the minimum method. Fourth, defuzzication with re-convert data in fuzzy set to crisp data. The last step is to get the output value of the defuzzification result which converted to PWM (Pulse Width Modulation) value. The calculation results obtained a PWM (Pulse Width Modulation) value of 760.2450 for the set point temperature of 28 o C and humidity of 75% in the plant factory of pak choy’s cultivation. Keywords : Fuzzy Logic, Fuzzy Tsukamoto, High Accuracy, Pak Choy, Plant Factory, PWM.

Fuzzy logic (FL) has been regarded as a useful methodology in modelling. The modeling performance of FL heavily relies on the modeling capability of a fuzzy set (FS). However, the MF of a FS cannot be arbitrarily accurate in practice and thus the centroid as a measure of a FS cannot be accurate to any degree. In face of inaccurate MF, the sensitivity of the centroid of a FS to the deviation of its MF from the ideal MF determines the accuracy of the centroid. To evaluate the modeling capability of a type-1 (T1) FS and an interval type-2 (IT2) FS, this paper will present a comparative study of the sensitivity of the centroid of a T1 FS and an IT2 FS to the deviation of their MFs from the ideal MFs. In this paper, equations relating the amount that the centroid of a T1 FS and an IT2 FS vary with the deviation of their MFs from the ideal MFs were established. Based on the derived equations, it was found that an IT2 FS is able to provide an additional design freedom to vary the sensitivity of its centroid compared to a T1 FS. This unique characteristic may enable an IT2 FS to minimize the sensitivity of the centroid and thus to achieve higher modeling accuracy. Furthermore, guidelines for designing an IT2 FS were provided.