Diffuse logic

fuzzy logic (also called fuzzy logic) is a multivalued paraconsistent logic in which the truth values of the variables can be any real number between 0 and 1. It was formulated by the mathematician and engineer Lotfi A. Zadeh.

Operation

Fuzzy logic (fuzzy logic, in English) allows more or less intense decisions to be made based on intermediate degrees of compliance with a premise; it is better adapted to the real world we live in, and can even understand and function with our expressions, such as "it's very hot", "it's not very high", "heart rate is a little fast", etc.

Takes two random values, but contextualized and referred to each other. For example, a person who is two meters tall is clearly a tall person, if the short person value has previously been taken and set to one meter. Both values are contextualized to people and refer to a linear metric measure.

The key to this adaptation to language is based on understanding the quality quantifiers for our inferences (in the examples above, “a lot”, “very” and “a little”).

In fuzzy set theory, the union, intersection, difference, negation or complement operations, and other operations on sets (see also fuzzy subset), on which this logic is based, are also defined.

For each fuzzy set, there is associated a membership function for its elements, which indicates to what extent the element is part of that fuzzy set. The most typical membership function shapes are trapezoidal, linear, and curved.

It is based on heuristic rules of the form IF (antecedent) THEN (consequent), where the antecedent and the consequent are also fuzzy sets, either pure or the result of operating with them. Serve as heuristic rule examples for this logic (note the importance of the words "very much", "drastically", "a little" and "slightly" for fuzzy logic):

• It's really cold. Then the temperature dropped drastically.
• I'm gonna be a little late. Then slightly increase the speed.

The inference methods for this rule base must be simple, versatile, and efficient. The results of these methods are a final area, the result of a set of overlapping areas (each area is the result of an inference rule). To choose a specific exit from so many fuzzy premises, the most used method is the centroid method, in which the final exit will be the center of gravity of the resulting total area.

The rules available to the inference engine of a fuzzy system can be formulated by experts or else learned by the system itself, making use of neural networks in this case to strengthen future decision-making.

Input data is usually collected by sensors that measure the input variables of a system. The inference engine is based on fuzzy chips, which are exponentially increasing their rule processing capacity year by year.

A typical scheme of operation for a fuzzy system could be as follows:

Operation of a diffuse control system.

In the figure, the control system makes the calculations based on its heuristic rules, commented previously. The final output would act on the physical environment, and the values on the physical environment of the new inputs (modified by the output of the control system) would be taken by system sensors.

For example, imagining that our fuzzy system were the air conditioning of a car that self-regulates according to needs: The fuzzy chips of the air conditioning collect the input data, which in this case could well be simply the temperature and humidity. These data are submitted to the rules of the inference engine (as mentioned before, in the form IF... THEN...), resulting in a results area. The center of gravity will be chosen from that area, providing it as an exit. Depending on the result, the air conditioner could increase the temperature or decrease it depending on the degree of the output.

Fuzzy Compensatory Logic (FLC)

The LDC is a multivalent logic model that allows simultaneous modeling of deductive and decision-making processes. The use of the LDC in mathematical models allows the use of concepts related to reality following behavior patterns similar to human thought. The most important characteristics of these models are: flexibility, tolerance for imprecision, the ability to model non-linear problems and its foundation in common sense language. Under this foundation, we specifically study how to condition the model without conditioning reality.

The LDC uses the DL scale, which can vary from 0 to 1 to measure the degree of truth or falsity of its propositions, where propositions can be expressed by predicates. A predicate is a function of the universe X in the interval [0, 1], and the operations of conjunction, disjunction, negation and implication are defined in such a way that when restricted to the domain [0, 1] Boolean Logic is obtained.

The different ways of defining operations and their properties determine different multivalent logics that are part of the DL paradigm. Multivalent logics are generally defined as those that allow intermediate values between the absolute truth and the total falsehood of an expression. Then 0 and 1 are both associated with the certainty and accuracy of what is affirmed or denied and 0.5 with maximum vagueness and uncertainty. In processes that require decision-making, the exchange with experts leads to complex and subtle formulations that require compound predicates. The truth values obtained on these compound predicates must be sensitive to changes in the truth values of the basic predicates.

This need is satisfied with the use of the LDC, which renounces compliance with the classical properties of conjunction and disjunction, contrasting them with the idea that the increase or decrease in the truth value of conjunction or disjunction caused by the change in the truth value of one of its components, can be "compensated" with the corresponding decrease or increase in the other. These properties naturally make possible the work of translation from natural language to Logic, including long predicates if they arise from the modeling process.

In the LDC, the conjunction operator, expressed as c (and) is the geometric mean. [1]

The modeling of vagueness in Compensatory Fuzzy Logic

In LDC, vagueness modeling is achieved through linguistic variables, which makes it possible to take advantage of the knowledge of experts, contrary to what happens in other methods closer to black boxes and exclusively based on data, such as neural networks.

There are authors such as Jesús Cejas Montero in his article La Lógica Diffusa Compensatoria published in 2011 by the Industrial Engineering Magazine of the José Antonio Echeverría Higher Polytechnic Institute, which marked a milestone in the dissemination of the LDC, which recommends the use of functions sigmoidal membership for increasing or decreasing functions. The parameters of these functions are determined by setting two values. The first one is the value from which the statement contained in the predicate is considered to be more true than false, for example it could be established from 0.5. The second is the value for which the data makes the corresponding statement almost unacceptable, for example it could be established from 0.1.

Currently there is a Decision Support System Based on Trees with Fuzzy Logic Operators whose name is Fuzzy Tree Studio 1.0, developed jointly between Universidad CAECE and the Universidad Nacional de Mar del Plata (Argentina), which owns a module that works with the LDC. This allows the decision agent to stop worrying about the mathematical background and focus on the verbal formulation of the model that allows him to make a decision.

In general, LDC-based models combine experience and knowledge with numerical data, so it can be seen as a “grey box”. LD-based models can be seen as “white boxes”, since they allow you to see their structure explicitly. In contrast to models based exclusively on data, such as Neural Networks, which would correspond to "black boxes".

These models can be optimized when real numerical data is available. The optimization method can come from Computational Intelligence. In this context, Genetic Algorithms present an interesting alternative. This approach forms the foundation of hybrid systems.

The trend of research on business management, through LDC techniques, is oriented towards the creation of hybrid systems that integrate it with the skills of Neural Networks and the possibilities of Genetic Algorithms and Set Logic. The creation and implementation of these mixed systems allows solving complex and difficult-to-solve problems; in which subjective estimates based on experience and available information are used, such as: decision models used with optimization criteria, location of shopping centers, market entry strategy, selection of product and service portfolios, development of computer applications, methods for knowledge discovery problems, methods for evaluating the efficiency of different types of institutions, among others.

Compensatory Fuzzy Logic is a multivalent logic model that renounces several classical axioms to achieve an idempotent and “sensible” system, by allowing the “compensation” of the predicates. In LD the truth value of the conjunction is less than or equal to all the components, while the truth value of the disjunction is greater than or equal to all the components. The waiver of these restrictions constitutes the basic idea of the LDC.

In conclusion, the LDC is a new approach to multivalent systems based on the Geometric Mean that, in addition to providing a formal system with logical properties of notable interest, constitutes a bridge between Logic and Decision Making. The LDC becomes part of the arsenal of methods for multicriteria evaluation, adapting especially to those situations in which the decision-maker can verbally describe, often ambiguously, the heuristics used when executing multicriteria evaluation/classification actions. However, the consistency of the logical platform endows this proposal with a capacity to formalize reasoning that goes beyond descriptive approaches to decision processes. It is an opportunity to use language as a key element of communication in the construction of semantic models that facilitate evaluation, decision making and knowledge discovery.

Applications

General applications

Fuzzy logic is used when the complexity of the process in question is very high and there are no precise mathematical models, for highly non-linear processes and when definitions and knowledge that are not strictly defined (imprecise or subjective) are involved.

On the other hand, it is not a good idea to use it when some mathematical model already solves the problem efficiently, when the problems are linear or when they have no solution.

This technique has been used quite successfully in the industry, mainly in Japan, extending its applications to many fields. The first time it was used in a major way was in the Japanese subway, with excellent results. Later it was generalized according to the uncertainty theory developed by the Spanish mathematician and economist Jaume Gil Aluja.

Here are some examples of its application:

• Air conditioner control systems
• Automatic focus systems in cameras
• Family appliances (refrigerants, washing machines...)
• Control and optimization of industrial processes and systems
• Writing systems
• Improved engine fuel efficiency
• Improvement of the environment
• Expert knowledge systems (emulating the behavior of a human expert)
• Robotics
• Vehicles and autonomous driving
• Computer technology
• Diffuse databases: Store and consult imprecise information. For this point, for example, there is FSQL language.
• ...and, in general, in most of the control systems that do not depend on an Yes/No.

Fuzzy logic in artificial intelligence

Fuzzy logic is a branch of artificial intelligence that allows a computer to analyze information from the real world on a scale between true and false, manipulate vague concepts such as "hot" or "wet", and allows engineers to build devices that judge hard-to-define information.

In artificial intelligence, fuzzy logic, or fuzzy logic is used to solve a variety of problems, mainly those related to the control of complex industrial processes and systems. decision making, resolution, and data compression. Fuzzy logic systems are also widespread in everyday technology, for example in digital cameras, air conditioning systems, washing machines, etc. Fuzzy logic-based systems mimic the way humans make decisions, with the advantage of being much faster. These systems are generally robust and tolerant of inaccuracies and noise in the input data. Some logic programming languages that have incorporated fuzzy logic would be, for example, the various implementations of Fuzzy PROLOG or the Fril language.

It consists of the application of fuzzy logic with the intention of imitating human reasoning in computer programming. With conventional logic, computers can manipulate strictly dual values, such as true/false, yes/no, or bound/unbound. In fuzzy logic, mathematical models are used to represent subjective notions, such as hot/warm/cold, for concrete values that can be manipulated by The computers.

In this paradigm, the time variable also has a special value, since control systems may need to provide feedback in a specific space of time, previous data may be needed to make an average evaluation of the situation in an earlier period.

Advantages and disadvantages

As the main advantage, it is worth noting the excellent results provided by a control system based on fuzzy logic: it offers outputs quickly and precisely, thus reducing the transitions of fundamental states in the physical environment that it controls. For example, if the air conditioning turned on when it reached a temperature of 30º, and the current temperature fluctuated between 29º-30º, our air conditioning system would be turning on and off continuously, with the energy expense that this would entail. If it were regulated by fuzzy logic, those 30º would not be a threshold, and the control system would learn to maintain a stable temperature without continuous switching off and on.

There is also the indecision to opt for either experts or technology (mainly through neural networks) to reinforce the initial heuristic rules of any control system based on this type of logic.