Fuzzy logic is a form of
multi-valued logic derived from
fuzzy set theory to deal with
reasoning that is approximate rather than precise.
In contrast with "crisp logic", where
binary sets have
binary logic, the fuzzy logic variables may
have a
membership
value of not only 0 or 1 – that is, the
degree of
truth of a
statement can range between 0 and 1 and is
not constrained to the two truth values of classic
propositional logic. Furthermore, when
linguistic variables are used, these
degrees may be managed by specific functions.
Fuzzy logic emerged as a consequence of the 1965 proposal of
fuzzy set theory by
Lotfi Zadeh. Though fuzzy logic has been applied
to many fields, from
control theory
to
artificial intelligence,
it still remains controversial among most
statisticians, who prefer
Bayesian logic, and some
control engineers, who prefer traditional
two-valued logic.
Degrees of truth
Both degrees of truth and
probabilities
range between 0 and 1 and hence may seem similar at first. However,
they are distinct conceptually; truth represents
membership in vaguely
defined sets, not
likelihood of some event or condition as
in
probability theory. 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
randomness.A probabilistic setting would first define a
scalar variable for the fullness of the
glass, and second, conditional distributions describing the
probability that someone would call the glass full given a specific
fullness level. This model, however, has no sense without accepting
occurrence of some event, e.g. that after a few minutes, the glass
will be half empty. Note that the conditioning can be achieved by
having a specific observer that randomly selects the label for the
glass, a distribution over deterministic observers, or both.
Consequently, probability has nothing in common with fuzziness,
these are simply different concepts which superficially seem
similar because of using the same interval of real numbers [0,1].
Still, since theorems such as
De
Morgan's have dual applicability and properties of random
variables are analogous to properties of binary logic states, one
can see where the confusion might arise.
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 meaning of the expressions
cold,
warm, and
hot is 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
linguistic
variables are often used to facilitate the expression of rules
and facts.
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. For example, L. A. Zadeh
proposed to take the square of the membership function. This model,
however, does not work properly.For more details, see the
references.
Example
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.
In application, the
programming
language Prolog is well geared to
implementing fuzzy logic with its facilities to set up a database
of "rules" which are queried to deduct logic. This sort of
programming is known as
logic
programming.
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.
Mathematical fuzzy logic
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 model 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 model 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.
Higher-order fuzzy logics
These logics, called
fuzzy type
theories, extend predicate fuzzy logics to be able to quantify
also predicates and higher order objects. A fuzzy type theory is a
generalization of classical simple type theory introduced by B.
Russell
and mathematically elaborated by A. Church
and L. Henkin.
Decidability issues for fuzzy logic
The notions of a "decidable subset" and "
recursively enumerable subset" are
basic ones for
classical
mathematics and
classical logic.
Then, the question of a suitable extension of such 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 showed that such a definition
is not adequate and therefore proposed the following one.
Ü denotes the set of rational numbers in [0,1].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 proposed in Gerla
2006.The proposed definitions are well related with fuzzy logic.
Indeed, the following theorem holds true (provided that the
deduction apparatus of the 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 logic claiming that the proposed notion of
recursive enumerability for fuzzy subsets is the adequate one. To
this aim, further investigations on 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.
Application areas
Fuzzy logic is used in the operation or programming of:
Objections to fuzzy logic
Identical to "imprecise logic"
- Fuzzy logic is not any less precise than any other form of
logic: it is an organized and mathematical method of handling
inherently imprecise concepts. The concept of "coldness"
cannot be expressed in an equation, because although temperature is
a quantity, "coldness" is not. However, people have an idea of what
"cold" is, and agree that there is no sharp cutoff between "cold"
and "not cold", where something is "cold" at N degrees but "not
cold" at N+1 degrees — a concept classical logic cannot easily
handle due to the principle of
bivalence. The result has no set answer so it is believed to be
a 'fuzzy' answer. Fuzzy logic simply provides a mathematical model
of the vagueness which is manifested in the above example.
A new way of expressing 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), 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 argues that probability is a
subtheory of fuzzy logic, as probability only handles one kind of
uncertainty. He also claims to have proven a derivation of Bayes' theorem from the concept of fuzzy subsethood. Lotfi 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. Other approaches to uncertainty include Dempster-Shafer theory and rough sets.
- Note, however, that fuzzy logic is not controversial to
probability but rather complementary (cf. )
Difficult to scale to larger problems
- This criticism is mainly because there exist problems with
conditional possibility, the fuzzy set theory equivalent of
conditional probability (see Halpern (2003), Section 3.8). This
makes it difficult to perform inference. However there have not
been many studies comparing fuzzy-based systems with probabilistic
ones.
See also
Notes
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