What is Fuzzy Logic?
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[ed. : This article was posted to comp.ai.fuzzy on Fri, 15 Jan 1993 20:27:33
GMT]
I've seen a lot of confusion in the first few articles posted to this newsgroup
about what, exactly, fuzzy logic is. Since I've been working in the field for
five years, I thought I'd help get things started by posting some introductory
material. This article covers the question "What is Fuzzy Logic?" from a
mathematical point of view. Succeeding articles will cover the questions "What
is a Fuzzy Expert System?" and "What is Fuzzy Control?".
Warning: If you're not already familiar with fuzzy logic, you're going to see a
lot of new terms defined in this article. I'll try to put _underscores_ around
terms that are likely to be new. You may need to read it a few times, just to
pick up all the terms.
What is Fuzzy Logic?
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Fuzzy logic is a superset of conventional (Boolean) logic that has been
extended to handle the concept of partial truth - truth values between
"completely true" and "completely false". It was introduced by Dr. Lotfi
Zadeh of U.C. Berkeley in the 1960's.
Fuzzy Subsets
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There is a strong relationship between Boolean logic and the concept of a
subset. There is a similar strong relationship between fuzzy logic and fuzzy
subset theory (Note: there is no fuzzy set theory, as far as I am aware - only
a fuzzy subset theory).
A subset U of a set S can be defined as a set of ordered pairs, each with a
first element that is an element of the set S, and a second element that is
an element of the set { 0, 1 }, with exactly one ordered pair present for
each element of S. This defines a mapping between elements of S and elements
of the set { 0, 1 }. The value zero is used to represent non-membership, and
the value one is used to represent membership. The truth or falsity of the
statement
x is in U
is determined by finding the ordered pair whose first element is x. The
statement is true if the second element of the ordered pair is 1, and the
statement is false if it is 0.
Similarly, a fuzzy subset F of a set S can be defined as a set of ordered
pairs, each with a first element that is an element of the set S, and a second
element that is a value in the interval [ 0, 1 ], with exactly one ordered
pair present for each element of S. This defines a mapping between elements of
the set S and values in the interval [ 0, 1 ]. The value zero is used to
represent complete non-membership, the value one is used to represent complete
membership, and values in between are used to represent intermediate _degrees
of membership_. The set S is referred to as the _universe of discourse_ for
the fuzzy subset F. Frequently, the mapping is described as a function, the
_membership function_ of F. The degree to which the statement
x is in F
is true is determined by finding the ordered pair whose first element is x.
The _degree of truth_ of the statement is the second element of the ordered
pair.
That's a lot of mathematical baggage, so here's an example. Let's talk about
people and "tallness". In this case the set S (the universe of discourse) is
the set of people. Let's define a fuzzy subset TALL, which will answer the
question "to what degree is person x tall?" To each person in the universe of
discourse, we have to assign a degree of membership in the fuzzy subset TALL.
The easiest way to do this is with a membership function based on the person's
height.
[erik - I hope this notation is clear]
tall(x) = { 0, if height(x) < 5 ft.,
(height(x)-5ft.)/2ft., if 5 ft. <= height (x) <= 7 ft.,
1, if height(x) > 7 ft. }
A graph of this looks like:
Warning: put on your peril-sensitive sunglasses. Bad ASCII graphics
follow!
1.0 + +-!!!!!!!!!!!!!!!!!-
| /
| /
0.5 + /
| /
| /
0.0 +-!!!!!!!!!!!-+-!!!-+-!!!!!!!!!!!!!!!!!-
| |
5.0 7.0
height, ft. ->
Given this definition, here are some example values:
Person Height degree of tallness
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Billy 3' 2" 0.00 [I think]
Yoke 5' 5" 0.21
Drew 5' 9" 0.38
Erik 5' 10" 0.42
Mark 6' 1" 0.54
Kareem 7' 2" 1.00 [depends on who you ask]
So given this definition, we'd say that the degree of truth of the statement
"Drew is TALL" is 0.38.
Note: Membership functions almost never have as simple a shape as tall(x). At
minimum, they tend to be triangles pointing up, and they can be much more
complex than that. Also, I've discussed membership functions as if they always
are based on a single criterion, but this isn't always the case, although it
is the most common case. One could, for example, want to have the membership
function for TALL depend on both a person's height and their age (he's tall for
his age). This is perfectly legitimate, and occasionally used in practice.
It's referred to as a two-dimensional membership function. It's also possible
to have even more criteria, or to have the membership function depend on
elements from two completely different universes of discourse.
Logic Operations
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Ok, we now know what a statement like
X is LOW
means in fuzzy logic. The question now arises, how do we interpret a statement
like
X is LOW and Y is HIGH or (not Z is MEDIUM)
The standard definitions in fuzzy logic are:
truth (not x) = 1.0 - truth (x)
truth (x and y) = minimum (truth(x), truth(y))
truth (x or y) = maximum (truth(x), truth(y))
which are simple enough. Some researchers in fuzzy logic have explored the
use of other interpretations of the AND and OR operations, but the definition
for the NOT operation seems to be safe. Note that if you plug just the values
zero and one into these definitions, you get the same truth tables as you
would expect from conventional Boolean logic.
Some examples - assume the same definition of TALL as above, and in addition,
assume that we have a fuzzy subset OLD defined by the membership function:
old (x) = { 0, if age(x) < 18 yr.
(age(x)-18 yr.)/42 yr., if 18 yr. <= age(x) <= 60 yr.
1, if age(x) > 60 yr. }
And for compactness, let
a = X is TALL and X is OLD
b = X is TALL or X is OLD
c = not X is TALL
Then we can compute the following values.
height age X is TALL X is OLD a b c
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3' 2" 65? 0.00 1.00 0.00 1.00 1.00
5' 5" 30 0.21 0.29 0.21 0.29 0.79
5' 9" 27 0.38 0.21 0.21 0.38 0.62
5' 10" 32 0.42 0.33 0.33 0.42 0.58
6' 1" 31 0.54 0.31 0.31 0.54 0.46
7' 2" 45? 1.00 0.64 0.64 1.00 0.00
3' 4" 4 0.00 0.00 0.00 0.00 1.00
Where is Fuzzy Logic Used?
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Directly, very few places. The only pure fuzzy logic application I know of
is the Sony PalmTop, which apparently used a fuzzy logic decision tree
algorithm to perform handwritten (well, computer lightpen) Kanji character
recognition.
The only common use of fuzzy logic, to my knowledge, is as the underlying
logic system for fuzzy expert systems, which will be the subject of the next
article in this series, titled "What is a Fuzzy Expert System?".
Whew. A lot of words, wasn't it?
--
Copyright (c) Togai InfraLogic, 1993. All rights reserved.
Permission to freely distribute this document, provided that it remains
complete and intact, is hereby granted.