# Fuzzy logic

Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. Compared to traditional binary sets (where variables may take on true or false values), fuzzy logic variables may have a truth value that ranges in degree between 0 and 1. Fuzzy logic has been extended to handle the concept of partial truth, where the truth value may range between completely true and completely false.1 Furthermore, when linguistic variables are used, these degrees may be managed by specific functions. Irrationality can be described in terms of what is known as the fuzzjective.citation needed

The term "fuzzy logic" was introduced with the 1965 proposal of fuzzy set theory by Lotfi A. Zadeh.23 Fuzzy logic has been applied to many fields, from control theory to artificial intelligence. Fuzzy logics however had been studied since the 1920s as infinite-valued logics notably by Łukasiewicz and Tarski.4

## Overview

Classical logic only permits propositions having a value of truth or falsity. The notion of whether 1+1=2 is absolute, immutable, mathematical truth. However, there exist certain propositions with variable answers, such as asking various people to identify a color. The notion of truth doesn't fall by the wayside, but rather a means of representing and reasoning over partial knowledge is afforded, by aggregating all possible outcomes into a dimensional spectrum.

Both degrees of truth and probabilities range between 0 and 1 and hence may seem similar at first. For example, let a 100 ml glass contain 30 ml of water. Then we may consider two concepts: Empty and Full. The meaning of each of them can be represented by a certain fuzzy set. Then one might define the glass as being 0.7 empty and 0.3 full. Note that the concept of emptiness would be subjective and thus would depend on the observer or designer. Another designer might equally well design a set membership function where the glass would be considered full for all values down to 50 ml. It is essential to realize that fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon while probability is a mathematical model of ignorance.

### Applying truth values

A basic application might characterize subranges of a continuous variable. For instance, a temperature measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled.

Fuzzy logic temperature

In this image, the meanings of the expressions cold, warm, and hot are represented by functions mapping a temperature scale. A point on that scale has three "truth values"—one for each of the three functions. The vertical line in the image represents a particular temperature that the three arrows (truth values) gauge. Since the red arrow points to zero, this temperature may be interpreted as "not hot". The orange arrow (pointing at 0.2) may describe it as "slightly warm" and the blue arrow (pointing at 0.8) "fairly cold".

### Linguistic variables

While variables in mathematics usually take numerical values, in fuzzy logic applications, the non-numeric are often used to facilitate the expression of rules and facts.5

A linguistic variable such as age may have a value such as young or its antonym old. However, the great utility of linguistic variables is that they can be modified via linguistic hedges applied to primary terms. The linguistic hedges can be associated with certain functions.

## Early applications

The Japanese were the first to utilize fuzzy logic for practical applications. The first notable application was on the high-speed train in Sendai, in which fuzzy logic was able to improve the economy, comfort, and precision of the ride.6 It has also been used in recognition of hand written symbols in Sony pocket computerscitation needed, Canon auto-focus technologycitation needed, Omron auto-aiming camerascitation needed, earthquake prediction and modeling at the Institute of Seismology Bureau of Metrology in Japancitation needed, etc.

## Example

### Hard science with IF-THEN rules

Fuzzy set theory defines fuzzy operators on fuzzy sets. The problem in applying this is that the appropriate fuzzy operator may not be known. For this reason, fuzzy logic usually uses IF-THEN rules, or constructs that are equivalent, such as fuzzy associative matrices.

Rules are usually expressed in the form:
IF variable IS property THEN action

For example, a simple temperature regulator that uses a fan might look like this:

IF temperature IS very cold THEN stop fan
IF temperature IS cold THEN turn down fan
IF temperature IS normal THEN maintain level
IF temperature IS hot THEN speed up fan


There is no "ELSE" – all of the rules are evaluated, because the temperature might be "cold" and "normal" at the same time to different degrees.

The AND, OR, and NOT operators of boolean logic exist in fuzzy logic, usually defined as the minimum, maximum, and complement; when they are defined this way, they are called the Zadeh operators. So for the fuzzy variables x and y:

NOT x = (1 - truth(x))
x AND y = minimum(truth(x), truth(y))
x OR y = maximum(truth(x), truth(y))


There are also other operators, more linguistic in nature, called hedges that can be applied. These are generally adverbs such as "very", or "somewhat", which modify the meaning of a set using a mathematical formula.

## Logical analysis

In mathematical logic, there are several formal systems of "fuzzy logic"; most of them belong among so-called t-norm fuzzy logics.

### Propositional fuzzy logics

The most important propositional fuzzy logics are:

• Monoidal t-norm-based propositional fuzzy logic MTL is an axiomatization of logic where conjunction is defined by a left continuous t-norm, and implication is defined as the residuum of the t-norm. Its models correspond to MTL-algebras that are prelinear commutative bounded integral residuated lattices.
• Basic propositional fuzzy logic BL is an extension of MTL logic where conjunction is defined by a continuous t-norm, and implication is also defined as the residuum of the t-norm. Its models correspond to BL-algebras.
• Łukasiewicz fuzzy logic is the extension of basic fuzzy logic BL where standard conjunction is the Łukasiewicz t-norm. It has the axioms of basic fuzzy logic plus an axiom of double negation, and its models correspond to MV-algebras.
• Gödel fuzzy logic is the extension of basic fuzzy logic BL where conjunction is Gödel t-norm. It has the axioms of BL plus an axiom of idempotence of conjunction, and its models are called G-algebras.
• Product fuzzy logic is the extension of basic fuzzy logic BL where conjunction is product t-norm. It has the axioms of BL plus another axiom for cancellativity of conjunction, and its models are called product algebras.
• Fuzzy logic with evaluated syntax (sometimes also called Pavelka's logic), denoted by EVŁ, is a further generalization of mathematical fuzzy logic. While the above kinds of fuzzy logic have traditional syntax and many-valued semantics, in EVŁ is evaluated also syntax. This means that each formula has an evaluation. Axiomatization of EVŁ stems from Łukasziewicz fuzzy logic. A generalization of classical Gödel completeness theorem is provable in EVŁ.

### Predicate fuzzy logics

These extend the above-mentioned fuzzy logics by adding universal and existential quantifiers in a manner similar to the way that predicate logic is created from propositional logic. The semantics of the universal (resp. existential) quantifier in t-norm fuzzy logics is the infimum (resp. supremum) of the truth degrees of the instances of the quantified subformula.

### Decidability issues for fuzzy logic

The notions of a "decidable subset" and "recursively enumerable subset" are basic ones for classical mathematics and classical logic. Thus the question of a suitable extension of these concepts to fuzzy set theory arises. A first proposal in such a direction was made by E.S. Santos by the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program (see Santos 1970). Successively, L. Biacino and G. Gerla argued that the proposed definitions are rather questionable and therefore they proposed the following ones. Denote by Ü the set of rational numbers in [0,1]. Then a fuzzy subset s : S $\rightarrow$[0,1] of a set S is recursively enumerable if a recursive map h : S×N $\rightarrow$Ü exists such that, for every x in S, the function h(x,n) is increasing with respect to n and s(x) = lim h(x,n). We say that s is decidable if both s and its complement –s are recursively enumerable. An extension of such a theory to the general case of the L-subsets is possible (see Gerla 2006). The proposed definitions are well related with fuzzy logic. Indeed, the following theorem holds true (provided that the deduction apparatus of the considered fuzzy logic satisfies some obvious effectiveness property).

Theorem. Any axiomatizable fuzzy theory is recursively enumerable. In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general. Moreover, any axiomatizable and complete theory is decidable.

It is an open question to give supports for a Church thesis for fuzzy mathematics the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. To this aim, an extension of the notions of fuzzy grammar and fuzzy Turing machine should be necessary (see for example Wiedermann's paper). Another open question is to start from this notion to find an extension of Gödel's theorems to fuzzy logic.

### Synthesis of fuzzy logic functions given in tabular form

It is known that any boolean logic function could be represented using a truth table mapping each set of variable values into set of values {0,1}. The task of synthesis of boolean logic function given in tabular form is one of basic tasks in traditional logic that is solved via disjunctive (conjunctive) perfect normal form.

Each fuzzy (continuous) logic function could be represented by a choice table containing all possible variants of comparing arguments and their negations. A choice table maps each variant into value of an argument or a negation of an argument. For instance, for two arguments a row of choice table contains a variant of comparing values x1, ¬x1, x2, ¬x2 and the corresponding function value

f( x 2 ≤ ¬x1 ≤ x1 ≤ ¬x2 ) = ¬x1

The task of synthesis of fuzzy logic function given in tabular form was solved in.7 New concepts of constituents of minimum and maximum were introduced. The sufficient and necessary conditions that a choice table defines a fuzzy logic function were derived.

## Fuzzy databases

Once fuzzy relations are defined, it is possible to develop fuzzy relational databases. The first fuzzy relational database, FRDB, appeared in Maria Zemankova's dissertation. Later, some other models arose like the Buckles-Petry model, the Prade-Testemale Model, the Umano-Fukami model or the GEFRED model by J.M. Medina, M.A. Vila et al. In the context of fuzzy databases, some fuzzy querying languages have been defined, highlighting the SQLf by P. Bosc et al. and the FSQL by J. Galindo et al. These languages define some structures in order to include fuzzy aspects in the SQL statements, like fuzzy conditions, fuzzy comparators, fuzzy constants, fuzzy constraints, fuzzy thresholds, linguistic labels and so on.

Much progress has been made to take fuzzy logic database applications to the web and let the world easily use them, for example: http://sullivansoftwaresystems.com/cgi-bin/fuzzy-logic-match-algorithm.cgi?SearchString=garia This enables fuzzy logic matching to be incorporated into a database system or application.

## Comparison to probability

Fuzzy logic and probability are different ways of expressing uncertainty. While both fuzzy logic and probability theory can be used to represent subjective belief, fuzzy set theory uses the concept of fuzzy set membership (i.e., how much a variable is in a set), and probability theory uses the concept of subjective probability (i.e., how probable do I think that a variable is in a set). While this distinction is mostly philosophical, the fuzzy-logic-derived possibility measure is inherently different from the probability measure, hence they are not directly equivalent. However, many statisticians are persuaded by the work of Bruno de Finetti that only one kind of mathematical uncertainty is needed and thus fuzzy logic is unnecessary. On the other hand, Bart Kosko arguescitation needed that probability is a subtheory of fuzzy logic, as probability only handles one kind of uncertainty. He also claimscitation needed to have proven a derivation of Bayes' theorem from the concept of fuzzy subsethood. Lotfi A. Zadeh argues that fuzzy logic is different in character from probability, and is not a replacement for it. He fuzzified probability to fuzzy probability and also generalized it to what is called possibility theory. (cf.8) More generally, fuzzy logic is one of many different proposed extensions to classical logic, known as probabilistic logics, intended to deal with issues of uncertainty in classical logic, the inapplicability of probability theory in many domains, and the paradoxes of Dempster-Shafer theory.

## Relation to ecorithms

Harvard's Dr. Leslie Valiant, co-author of the Valiant-Vazirani theorem, uses the term "ecorithms" to describe how many less exact systems and techniques like fuzzy logic (and "less robust" logic) can be applied to learning algorithms. Valiant essentially redefines machine learning as evolutionary. Ecorithms and fuzzy logic also have the common property of dealing with possibilities more than probabilities, although feedback and feedforward, basically stochastic "weights," are a feature of both when dealing with, for example, dynamical systems.

In general use, ecorithms are algorithms that learn from their more complex environments (hence eco) to generalize, approximate and simplify solution logic. Like fuzzy logic, they are methods used to overcome continuous variables or systems too complex to completely enumerate or understand discretely or exactly. See in particular p. 58 of the reference comparing induction/invariance, robust, mathematical and other logical limits in computing, where techniques including fuzzy logic and natural data selection (ala "computational Darwinism") can be used to shortcut computational complexity and limits in a "practical" way (such as the brake temperature example in this article).9

## References

1. ^ Novák, V., Perfilieva, I. and Močkoř, J. (1999) Mathematical principles of fuzzy logic Dodrecht: Kluwer Academic. ISBN 0-7923-8595-0
2. ^ "Fuzzy Logic". Stanford Encyclopedia of Philosophy. Stanford University. 2006-07-23. Retrieved 2008-09-30.
3. ^ Zadeh, L.A. (1965). "Fuzzy sets", Information and Control 8 (3): 338–353.
4. ^ Francis Jeffry Pelletier, Review of Metamathematics of fuzzy logics in The Bulletin of Symbolic Logic, Vol. 6, No.3, (Sep. 2000), 342-346, JSTOR 421060
5. ^ Zadeh, L. A. et al. 1996 Fuzzy Sets, Fuzzy Logic, Fuzzy Systems, World Scientific Press, ISBN 981-02-2421-4
6. ^ Kosko, B (June 1, 1994). "Fuzzy Thinking: The New Science of Fuzzy Logic". Hyperion.
7. ^
8. ^ Novák, V. "Are fuzzy sets a reasonable tool for modeling vague phenomena?", Fuzzy Sets and Systems 156 (2005) 341—348.
9. ^ Valiant, Leslie, (2013) Probably Approximately Correct: Nature's Algorithms for Learning and Prospering in a Complex World New York: Basic Books. ISBN 978-0465032716

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