Fuzzy Logic

TEOR? A SOCIAL CHAOS / CAP? CHAPTER 3: Fuzzy logic / ISBN 9789801241312

1 .- By way of introduction topic:

It is inevitable to cross the desert of fundamental epistemology associated with the topic. Nevertheless, we will give a first conceptual approach, which if not addressed in depth the semantic scope of the challenge, at least serves to establish the knowledge base on the subject, as it provides a holistic and inclusive:

Blurred or fuzzy logic is based on the relativity of what is observed.

In this relativity, this kind of logic for comparative analysis takes two or more random values ​​but contextualized and self referrals. For example, a person that measures 2 meters is clearly a tall person, if and only if you have previously assumed that the value of a person standard height is 1.75 meters and height is low if it is, say, 1.60 m or less. We say, then, that the three values ​​are contextualised people and dealing with a linear metric measure. The trial that produced such reasoning, based on relative terms, may not be accurate, because it represents a subjective impression, perhaps likely, but not exact. Therefore, the theory of fuzzy sets is more appropriate than classical logic to represent and analyze most of human knowledge, as it enables observations of phenomena and more than two logical stages.

One way to address the 'fuzzy logic' would be this: John is 2 meters and is right handed. Mary measures 1.65 meters and has blue eyes. Peter is left-handed brown eyes and is it high or low? This paradox becomes paradigm of fuzzy logic as linear removed by introducing an unexpected uncertainty, or at least inconsistent with the initial assumptions.

Uncertainty validates the Fuzzy Logic

But not surprising that uncertainty is 'rife' in our lives Have you ever stopped to think what were your expectations of yesterday? Yes, yesterday. How much security and certainty were you yesterday morning about what would happen during the 24 hours spent to date? Uncertainty can arise from a lack of information or even that there is disagreement on what is known or what could be known. You can have multiple types of sources, from quantifiable errors in the data to ambiguously defined terminology or uncertain projections of human behavior. Uncertainty can therefore be represented by quantitative measures (for example, a range of values ​​calculated by various models) or by qualitative statements (eg, reflecting the trial of a group of experts).

Social reality, in all areas and levels of uncertainty is saturated. The concept of uncertainty, so important to highlight the validity of fuzzy logic, has several meanings, which introduces an external variable and linguistic semantics 'slow' conceptual clearance operations. Let's see: For scientists, the ISO 3534-1 [ISO 1993] defines uncertainty as "an estimate attached to a test result that characterizes the range of values ​​within which it says is the true value." This definition has little practical application because the "true value" can not be known. This has made the International Vocabulary of Metrology, VIM [BIPM, 1993] avoids the term "true value" in its latest issue and redefine the uncertainty as "a parameter associated with the result of a measurement, that characterizes the range of values ​​that can be reasonably attributed to the measurand ".

In this definition, the measurand states: "the subject property as" [BIPM 1993]. The content of zinc in a steel or octane in gasoline are examples of measurands in chemical analysis. The concept of uncertainty, therefore, reflects doubts about the veracity of the result once you have evaluated all possible sources of error and have applied the necessary corrections. Therefore, the uncertainty gives us an idea of ​​the quality of the result as we show an interval around the estimated value within which the value is considered true.

In the sphere of social insecurity can be a feature of contemporary societies, because it delimits the states (individual and / or social) invaded by the perplexity and mistrust. That is, we speak of a reality in which their agents are emotionally confused to make sense when they have their ideas, values ​​and actions with respect to the things that happen in reality. The perplexity and distrust we feel we have every day when going outside are closely related to custom, increasingly developed, to perceive reality without contrasts. We are not referring to the organic incapacity to distinguish between heat and cold, night or day, sweet or bitter, but the cultural inability to find a coherent meaning to the inability to see clearly the contrasts of the realities whose main characteristic is precisely the high contrasts, for example, do not perceive a contradiction between the extreme poverty of some countries and the fact that obesity appears as a significant health problem in developed countries.

The situations of uncertainty are increasingly recurrent, ie, the frequency not only responds to a statistical repetition (frequency) but rather a cultural repetition makes it, paradoxically, more common. However, the single greatest possible security that can be given in such circumstances is concentrated on the sensation of increased risk to suffer a change or a destructive disorder, ie reactive insecurity itself. Uncertainty is not a condition (disease, destruction or exchange by the occurrence of something tangible), but rather pure vulnerability to the disorders that, despite not having completed, are perceived very close and appeared imminent.

Consequently, uncertainty is a state of susceptibility to a disease, destruction or unwanted change, manifested in highly prone to insecurity to surface. Finally the uncertainty is a situation in person, regretfully experienced by those involved under its influence, not a conclusion in itself, but always points to a future binding. If at the end are consolidated and embodied the worst forecasts, uncertainty gives way to a reality (and upset) that is revealed with certainty very raw, and renewed assurances necessarily a new and different been quiet compared to that given origin. But, in the case of a consensus or not projected forecasts, the uncertainty does not disappear completely, remains in the background, both in spirit and in memory of people and societies.

Theories of Fuzzy Logic

Have recently emerged from different fields and disciplines, a number of theories that allow approaches to social reality, approaching it in all its complexity, all with a clear emphasis epistemological. One such theory is the complexity of fuzzy sets or fuzzy (Zadeh) [1] as a mathematical formalization of a logical model of the vagueness of the indeterminate, the diffuse and fuzzy.

Studies on fuzzy logic we have a much more credible social nature of human groups and understanding of events, because conventional systems are used daily in the social sciences, in some cases do not provide a real decision; happens contrary in fuzzy logic, which has become a fundamental part in the development of intelligent systems. This unconventional logic allows us to generate a decision-making from a wealth of information vague, because it is a type of logic that recognizes the values ​​that are in the gray zone of simple true and false estimates, and because the logic fuzzy propositions can be represented with varying degrees of truth or falsehood, such as those expressed semantically from a subjective perspective, "but ..."," almost like ... "

The professor of electrical engineering at the University of California (Berkeley), the Iranian Zadeh in 1965 published an article in the journal "Information and Control" [2], in which multivalent logic applied to Lukasiewicz sets or groups of objects. He begins to refer to some sort of undefined and diffuse groups as "fuzzy sets" ("fuzzy sets" was the title of that article). Thus, with respect to the concept of "fuzzy logic", we will consider the broadest sense, understanding it as ...

"... A logical system that is dedicated to the formalization of modes of reasoning that are approximate and not accurate."

Based on fuzzy set theory, a theory of classes "with no clear boundaries" (Zadeh, 1996: Information and Control, pp. 422), fuzzy logic is reasoning with vague sets. But what is a fuzzy set? How can we analyze recent events in Venezuela from fuzzy logic?

Reality does not always manifest itself in black-and-white

Talking and talking about our perceptions, in everyday life we ​​mention plenty of fuzzy sets. That is, "concepts that have no sharply defined boundaries, such as 'high', 'fat', 'many', 'most', 'slow', 'old', 'family' names ... colors as' red ',' green ',' blue ',' purple'..." (Zadeh, 1996:: Information and Control, pp. 425). Zadeh called "granulation" to the process of forming fuzzy concepts classes grouped by similarity essentially subjective, a process that is determined by "the limited ability of humans to resolve and / or store details" (Zadeh, 1996:: Information and Control p. 426).

If you stop reading this essay and the question asks you to, say, the first 10 people he meets: "How is today's political climate in the country?" inevitably discover that there is a conceptual imprecision occurs diffusely difficulty for the quantitative measurement of responses.

For this circumstance, the blurring of the concepts can be plotted in a "fuzzy curve" or blurring curve and in doing so we get a more accurate correspondence with reality, a new way of knowing or at least conceptually build, with no logical transactions probabilistic . Then the probability overcomes the concept of possibility, a long road that opens qualitatively epistemology. (Munné, [3] 1995).

Although fuzzy logic has been applied primarily to the civil engineering control systems and industrial processes (also called 'expert systems'), fuzzy logic has begun to regard as a key element in the study of social reality, a context permeated permanently blur like most things in the human sciences, in which particular concepts such as emotion, group, cognitive or social class are essentially vague.

An example is a fuzzy category Likert scale [4] as well as all applications are made directly or indirectly the same, as the grid technique Kelly [5] for the exploration of the constructs personal.

Smithson presents another application of fuzzy logic in the construction of a questionnaire about drug use. Fuzzy variables are used to collect the responses (qualitative) questionnaire. Specifically, Smithson discusses the possibility distribution that is "hidden" behind the pre-categorized answers "sometimes" or "a few times" to the question "How many times have you used drugs?".

Gil Quesada [6] (1990) performed an empirical application of the theory of fuzzy sets for school evaluation. This produces a test with a series of items rated as continuous assessment (O to 1) with a sharp cutoff and a "fuzzy cutting zone" of acceptance of the responses. With this, Gil Quesada adequacy promoting greater academic achievement assessments in group tests. With this application blurred, Gil Quesada enters the calculation of a series of new indices, such as: (1) the clear sufficiency index:% of students in the group who score above the cutoff point clear, (2) the index adequacy fuzzy probability that a student passes the test if we consider the membership function as a function of probability, (3) the index of fuzziness:% of students in the group who are clearly labeled. It is noteworthy that fuzzy theory in computational logic or artificial intelligence has made great and important applications to provide fuzzy languages ​​and programs (Boehm, 1999; Zu-Guo Yu, 2000, Wei-Yi Liu, 2000) [7].

Here are some critical reflections about the applicability of fuzzy logic and epistemology to the social sciences. A first positive assessment on the theory of blurring is directly related to the "mismatch problem" (Kosko) [8]: Traditionally, science has posed problems addressing the dichotomy as "white or black" to refer to a world that is "gray." In this sense, the new paradigm would more accurately reflect reality, because as stated by Kosko:

"When you talk (scientifically), simplifying. And when you simplify, you lie."

In this sense, fuzzy logic provides an enriching nuances epistemological perspective, understanding and hermeneutics, as an alternative to positivist explanation that is still the dominant approach to social sciences. Fuzzy logic contributes to the holistic thinking and facilitates analysis and understanding of entropic phases of any social reality because it gives meaning and relevance to the subtle but important differences that emerge from the gray magma and quantitatively inaccurate perceptions, feelings and particularities.

2 .- Method of approximate reasoning.

As my readers may infer, the desert epistemological continues in this section. I understand your anguish suffered because the 'first hand', and now I come to cross this desert oasis I can assure you the practical applications of the theory of fuzzy logic as a tool for the analysis of Social Chaos Theory is close behind the dune, which although not the last, at least is more imminent. So I suggest you not 'jump' this chapter (the temptation is great, I promise) because here you will find the methodological elements that made the argument about the ideal instrument for the understanding of social chaos. Drink a glass of water, stretch your legs or update your annotations, after this pause, breathe deeply, because what is coming is even denser. Ready? Go on.

As we have mentioned in previous paragraphs, Zadeh introduced the theory of approximate reasoning and many other authors have made important contributions to this field. Although superficially it may seem that the theory of approximate reasoning and logic differ greatly classical, classical logic can be viewed as a special case, an 'exception' shown from a positivist explanatory reasoning. In both systems, you can see the premises as inducers of subsets of possible worlds that meet, although in the case of the theory of approximate reasoning, these sets are composed of overlapping fuzzy subsets, and shed new constructs overlap information , scalar and subjective, but no less important or less significant than the responses obtained from the classical logic.

The implication in both systems is based on a rule of inclusion: a hypothesis can be inferred from a collection of premises if the subset of possible worlds that satisfy the conjunction of the premises is contained in the subset of possible worlds that satisfy the hypothesis.

Variables and quantifiers of approximate reasoning

The main contribution of approximate reasoning is its use of variables and the representation of propositions in terms of linguistic truth-values ​​of fuzzy subsets, as values ​​of these variables. Classical logic uses only so variable implicit idea in the real sense of value associated with a proposal. However, their binary nature allows you to hide this fact, we can refer to a proposition that is true for its denotation, p, and one that is false simply by its negation, ¬ p, thus avoiding the introduction of a variable Vp whose value is the value of the proposition p. The use of the concept of variable in the theory of approximate reasoning leads to treat domains that are not within the scope of classical logic, such as the problems that deal with fuzzy expert systems or fuzzy controllers.

The theory of approximate reasoning to represent linguistic quantifiers also located between two fuzzy scales. Zadeh said a quantifier like "most" can be represented as a fuzzy subset of a universe of discourse. Approximate quantifiers are used to represent common sense knowledge.

An interesting extension of the theory of approximate reasoning is the ability to deal with it prototypical knowledge. Reiter [9] suggested an approach to the representation of common sense knowledge using default rules and Yager tested it in the context of the theory of approximate reasoning. According to Reiter, a default rule as "typically birds fly" can be interpreted as follows: If an object is a bird and our knowledge available is not incompatible with the object fly, then we assume that the bird flies.

The binary logic can be viewed as a special case of approximate reasoning theory in which the basis sets have two elements {T, F} and the degree of membership is restricted to 1 or 0. Possibilistic logic can be seen as an extension of it, while, although the basis sets are restricted to two values, T and F, we allow degrees of membership are numbers in the unit interval.

It is for this perception quantum fuzzy logic goes beyond the binary logic allowing its formalization in terms of the theory of approximate reasoning. So, what is true reach several representations and what is assumed to be false too, both indicate that the value of absolute truth of the proposition is unknown.

The main rule of inference in classical logic, the mode of reasoning introduced by the Megarians and Stoics in Aristotle's time, is the Modus Ponens (name assigned in the Middle Ages), which is that if you have the rule A → B and For example, the rule is "If it rains then I get wet", if that is the very fact that "it rains", then I can conclude that "get wet". In fuzzy logic can generalize this rule, leaving his scheme as follows: For example, the rule might be "If the city is large (x is A), the traffic is very dense (and B)," the fact could be "the city is not very large (x is A ')," What could be said of traffic (B' (x))?.

Suppose that the variables are related, and not necessarily a function, but for any relationship. Suppose that is a fuzzy binary relation R in the universe XxY. A 'and B' are fuzzy sets in X and Y respectively. If we know R and A 'could know B' by the so-called compositional rule of inference:

B '= A' (x) R (x, y)

B '(y) = Xmin SUPX ∈ [A' (x), R (x, y)]

Where R (x, y) = I (A (x), B (y)) (implication function).

This is possible because an inference is an assessment that the mind makes between concepts, to interact, show their properties in a discreet, necessitating use abstraction to gain an understanding of the units that compose the problem, or creating an axiomatic point or circumstantial, that allows you to draw a logical line of cause and effect, between the various points inferred in solving the problem. After solving the problem arises what we know as a postulate, or a transform of the original premise, which by being framed in a different referential context, you get an equivalent meaning.

An example that clarifies it all

Everything we set out to explain it here with an example which I am sure, will understand immediately, and then criticize me for not having used "before -" the theoretical approach.

A Venezuelan middle class family enters a crisis scenario. A crisis whose variables I am going to expose a row, and having a clear relationship of 'cause and effect': After 25 years of marriage, Mom and Dad are facing the possibility of a divorce. His three children (the twins 25 years and under 21) are single and live in the family home, a comfortable and spacious villa of three levels in the east of Caracas, the family inherited from the maternal grandparents. Although 'dad is a democrat and former PDVSA worker who' already-not-being '(actively participated in the "general strike" for some o' oil boycott 'to others) the twins are Chavistas unruly. 'Mom' has decided to return to the university classroom (you choose a second mastery) and the youngest son has decided to form a band he calls 'ethnic cleansing' with five other residents of the exclusive area where you live, and every night out with his friends to place homeless people in the center of the capital, and film their outrages to hang YouTube videos.

Any social or psychological explanation is possible to expose the likely epicenter and the possible causes of the crisis of the family. One could infer hundreds of logical lines of cause and effect and any of them would have high contingencies of success in the diagnosis of this family trance, because the analysis binary 'good - bad', 'correct - incorrect', 'true - false' would the scale to use. However, the reality is imprecise and vague, even in the more solid appearance. If you have already 'psyched' cause of the crisis in the family, do the following exercise: Write them down ... Yes, stop reading, find pen and paper and write down what in his view, are the causes of the crisis in the family. Do not write many, only five.

Did he? Do you have on hand? Now try to point out their successes with these linguistic quantifiers located between fuzzy scales 'Mom' returns to university classrooms, believing that "sometimes" is not as well tested and accepted in their social dintorno as her husband. 'Dad', after his expulsion from PDVSA, found that a leader is more democratic than a technocrat (¿'much' more will be significantly 'more'?). The 'twin' entered the PSUV induced by a friend who also became fond of the two and 'minor' sale 'almost' every night 'more or less' to locate the homeless, and when I shoot with a rifle paint ball tell the stories as part of the field work of social research on people's reaction to the unexpected assault, and that includes your thesis.

What were the causes of this family crisis? Is not that lacked data and information? What happened is that by applying the inference of classical logic in a stage built on concrete but partial information, you are obviously the impact of causalities that are generated in the vortex of the undefinable. The little or almost no inter relationship between the desires of 'Mom' with the discovery of new social skills 'daddy' is as elusive as associate homosexual exploration of the twins with the thesis of the youngest child. So one might infer that the crisis of marriage (if there is or there) has nothing to do with the activities or new meanings of the members of this family. Binary logic, linear, no longer relevant, then there Approximate Reasoning Theory as the most suitable for the analysis of chaos in which the family has subsided.

Now extrapolate this analysis to the global situational Are successful country analysis and insights you hear on the street and in the media about the political and social events? Do you envisage this analysis the imperceptible, but very important linguistic differences between the scales 'ends'?

Although the original intention was to create a Professor Zadeh formalism to more efficiently handle the imprecision and vagueness of human reasoning expressed linguistically, caused some surprise that the success of fuzzy logic came first to the field of automatic control of processes. This was mainly due to the "boom" of blur in Japan, launched in 1987 and reached its peak in the early nineties. Since then, many have been released to market products using fuzzy technology, many of them using the label "fuzzy" as a symbol of quality and advanced features. An example is the advertisement of the Bosch washing machine fuzzy eco-system.

Professor Mamdani in 1974, successfully tested a fuzzy controller in a steam engine, but the first real implementation of a controller of this type was conducted in 1980 by F. L. Smidth & Co. in a cement plant in Denmark. In 1983, Fuji applies fuzzy logic to control chemical injection for water treatment plants for the first time in Japan. In 1987 the company developed the first drivers OMRON fuzzy business with Prof. Yamakawa.

From that point, the fuzzy control has been successfully applied in many different technological branches, for example, in metallurgy, manufacturing robots in maneuvering of aircraft controls, elevators or trains (train-meter Sendai, Japan , 1987), in picture and sound sensors (system of image stabilization in video cameras and Sony, Sanyo, Cannon ...), in appliances such as Panasonic or Bosch washing machines, or air conditioning Mitsubishi rice-cooker system. Also in the car industry, whose most notable example is the ABS systems of Mazda or Nissan, or automatic gearboxes for Renault and a long list of commercial applications.

But where lies the success of the applications of control? Success is in the simplicity, both conceptual and development. The two classical paradigms of fuzzy control are the focus of Mamdani and Takagi-Sugeno I describe briefly below.

Linguistic labels Mamdani and Takagi Sugeno

Mamdani's approach is to specify expert knowledge in the form of linguistic rules, must define the linguistic labels that will describe the states of the variables. For each entry (input information) is to specify the corresponding linguistic label that defines the output information. Each of the n input variables and output has been divided into specific fuzzy sets with some meaning. This may be defined several fuzzy sets, all different, the output variable. The same can be done with the rest of the other variables involved in the process and its outputs. Each fuzzy set associated with it should take a linguistic label.

Here is an example of using the Mamdani fuzzy approach to apply in a restaurant. If we capture the essence of the problem and we put aside all the factors that may be arbitrary, we could stay with the following set of rules:

If the service is poor or the food is bad, then the tip will be small

If the service is good, then the tip will be half

If the service is excellent and the food is delicious, then tip is generous

In these three rules underlying the solution. We have just defined the rules of a fuzzy inference system. If we were mathematical meaning to the linguistic variables, we would have a complete fuzzy inference system. Not go into detail in this section, we are interested in is what we have just shown, fuzzy logic is adaptable, simple and easily applied.



This is the graph associated with the fuzzy inference system, and generated by the three rules above. With just these three rules we have been able to generate a solution in a way much closer to human thinking and therefore easier for us. Quite the opposite occurs in the classical approach, in which we have to adapt to the modus operandi of the machine.

Finally it is noteworthy that if we add or modify certain considerations of the problem, it would suffice to include or change rules. This form is much less expensive than that used in the classical approach.

In the Takagi-Sugeno approach remains the same specification of fuzzy partitions of the domains of the entries in the Mamdani model, but does not require a partition output fuzzy domain. Control rules should contain a function that is generally linear: The degree of applicability is obtained in the same way that the Mamdani model.

3 .- Stages of a fuzzy controller

And for what?