previously Copyright 2007 David Dodds incl equations

Fuzzy and Crisp Sets, Usuality

In most everyday kinds of thinking, and linguistic reflections of
that, people use crisp sets to categorize things. Being a member of a
crisp set is an all or nothing affair. A woman is pregnant or she
isn't, she's never "partly pregnant" or " a little pregnant".

Thinking with crisp sets makes everything simpler, because something
either is or is not a member of a crisp set. They can be used to
represent black and white conceptual thinking. Oftentimes too, when
something is a member of a given crisp set it is then (at the same
time) not a member of any other crisp set. Again this simplifies the
logic used with this kind of thinking process. Linguistic
constructions which reflect this kind of thinking can be quite useful.
Especially when crisp categories are used.

Fuzziness, or fuzzy sets are used frequently also by most people in
everyday thinking and their linguistic constructs reflect them as well.

In a society (like that of North America) where science and technology
are raised nearly to the level of a religion, it is an implicit
heinous crime to use terms like "fuzzy" because it seems to run
counter to the cherished principle of exactitude with which
engineering has made such great strides in the last two or three
centuries.

In this page the term fuzzy is not used pejoratively, rather, it is
used as the "father of fuzzy sets" (Lotfi Zadeh) himself has described
in many technical papers and presentations, as have a multitude of
authors since his initial development of the domain.

You wont have to be a rocket scientist to follow the math involved
(there has to be math and algorithms and such, we are dealing with
computers after all). While entire books filled with esoteric looking
mathematics have been written about the topic of fuzzy sets we will
simply refer you to them, rather than subject you to the lashings they
present to the more casual reader.



Fuzzy sets are very relevant to ordinary English speakers

Perhaps a collection of examples of fuzzy sets as they appear in
everyday language would be helpful.

Just as we dont have to be able to write down the grammar of English
to speak it we also do not have to be mathematicians to use fuzzy
terms in our everyday English. In fact almost no one, who speaks
English fairly fluently, is able to write down the entire grammar of
the English language, or even one fully describing his own subset of it.

Birds manage to fly without 4 years of university aeronautics training.

Before we launch into how to convey fuzzy set usage in natural
linguistics to the computer, let's look at a few linguistic examples
of fuzzy sets. The casual reader will be able to relate to these more
easily than the math.

* "How close is it to dinnertime, I'm hungry?"
* "We told them that we'd meet them around 9 o'clock."
* "Just before you start that let's see what almost happened."
* "When I was small he seemed nearly as tall as a giant."
* "A big dog is smaller than a small house."
* "Sometimes the weekend seems very far away."


These were some examples of relatively everyday linguistic expressions
which couched non-crisp set membership concepts into easy to
understand forms.

Let us now examine the fuzziness contained in each sentence in turn.

The examination will be conducted in both a linguistic and a
mathematical manner.

Keep in mind that categorization is a very important functionality of
human thinking and language usage, and that an awareness of Rosch'
Theory of Prototypes stands one in good stead for the appreciation of
what cognitive underpinnings we humans draw upon in that thinking and
natural language use.



In the case of the first statement

# In the case of the first statement "How close is it to dinnertime,
I'm hungry?"

we see that the word "close" is the fuzzy component. Close is a member
of the category "spatiality".

Now spatiality is nothing more than a collection of linguistic
representations which refer to the various concepts to do with
existential (or virtual) three dimensionality.

Three dimensionality refers to volumes, rather than planes (surfaces)
or lines. As a conscious adult part of our perception of ourselves is
that "I" am constituted by a continuous surface which is the outside
of an enclosed volume. I call this volume "me". Everything on one side
of this two dimensional continuous surface is "me" ( usually something
"fleshy, boney, or sinuey"), everything on the other side of this "me"
surface is "not me". There is a subjective sense of "space" largely
dependent on how we perceive our"height" vis-a-vis our surroundings,
and also the length of our limbs, our arms and our legs.

Objects, and people, which (who) reach up to "my waist" are "short" or
"small", while objects which tower over my head are "tall" or "big".
Objects which are within immediate reach of my outstretched arm are
"close" (to me), and objects which are progressively multiple
arm-lengths distant from "me" are "far away". An object which is but a
few steps with my feet away from where "I" am standing is "close"
while an object which is many steps is "far".

This subjectivity is called situated, and is an important concept for
you to remember. Natural language usage is from/between situated,
conscious sources, not unconnected ethereal capabilities such as
standard computer programs.

Notice how wonderfully well this measurement system is universal for
all people. It doesnt matter if the absolute length of my arm is x
centimetres and that of yours is y centimetres because these
linguistic spatial terms are all relative or subjective
(self-referential in a way).

Thus "near" to a baby means the same thing to him as "near" does to
you, even though your arm is likely much longer than his.

Once a language-using system (whether a person or a computer) has been
able to create the "thinking" facility called projection, the person
or system is able to make use of a very powerful tool. Projection,
sometimes called "extension", is the "mental" capability whereby the
thinker creates, in his mind's eye, a small "theatre in the round"
where he mentally maps his perceptions of the world into a time
varying subjective three dimensional space.

[In some (admittedly crude) ways computers are already able to compute
the translation of sensor data, such as a tv camera image, into an
eidetic image within the confines of a three dimensional coordinate
matrix. There are industrial robots,equipped with a tv camera, that
are able to locate simple objects on a simple surface solely by
"vision" through the camera. They have been programmed such that they
can examine the picture with a computer program and extract some
information from it, things like the edges of the objects in the
picture. The program is able to "map" these edges into a three
dimensional co-ordinate space in a mathematical "volume" inside the
computer (in it's memory). It is able to locate other objects there,
according to their visual edges and the various angles and rotations
of the camera. All this information is put together to allow the
system to "place" "seen" objects into the "visualization space". By
examining the visualization space, rather than laboriously rescanning
half the real world with its camera, the robot system is able to use
the visualization space for planning actions, in particular, seeing
where various objects "are" viv-a-vis their mapping.]

In humans, as we progress from infancy to toddler, we develop the
mental facility to use our "situated" or personally subjective "me"
space to relate to a bigger world. In the "theatre of the round" of
our mind's eye we place a "me", but we knowingly map some real world
thing into the place where "me" is. Now we can use all our skills of
relating to (spatialized) "me" on the rest of the universe! Metaphor.
Analogy.

Further down in this section we'll look at the (simple) mathematics of
these linguistic terms.



In the case of the second statement


# In the case of the second statement "We told them that we'd meet
them around 9 o'clock."

In this sentence "around" is the fuzzy component.Around is also a
member of the category "spatiality". It is in fact related to "close"
that we saw in the first sentence. It shares the feature of relating a
particular subset of "all spatiality" (such as the English word
"everywhere"), with "close" in the first sentence. What is different
about "around", from "close", is that "around" is used to signify a
different particular subset than "close" does. The particular subset
that "around" implies is that of proximity but in (possibly) two or
three dimensional axes. "Around" carries with its meaning the feature
of centrality coupled with proximity. "Close" is intended to denote
proximity but there is no intention to convey any sense of
dimensionality, whether it be 2d or 3d. "Around" is used with the
intention of suggesting a "set" which might be represented visually by
a volume (or a plane) within which at some location is a point, which
is used as the metaphorical "center" or "zero axis" for representing
the centrality.

As a simple first approximation, one might use a spheroid (sphere) to
represent the edge or border of the set (for "around"). This notion
actually corresponds well with how most people describe their use of
"around". Well, this simple circle/sphere represents "inside" and
"outside" the set. Or, "contained in" or not "contained in", the
set.This is like the use of the Venn diagram.

Well, of course, few people have such a black and white concept of
"around" as this. First order predicate logic loves it, and that's why
fopl does NOT adequately describe how people USE natural language.

We natural language users have a more sophisticated usage of "around".
Let's add some sophistication to our previous model. Let us say that
the notion of around can be visually represented by some sphere,
centered on our previous point of centrality AND which has a gradient,
running from the point to the outer shell. Perhaps if you think of a
series of concentric spheres radiating out from the center point. Each
sphere is infinitely thin and is coloured some colour of grey. Let's
say that the center point itself is the brightest possible grey (which
some folks call "white"). Each successive infinitely thin sphere away
from the center is slightly darker than the one it contains. At the
place where the shell exists, the outer part of our original sphere,
is darkest in (grey) colour.

Now we see that by refering to a particular level of grey (or
"brightness") we can signify a distance from the center point. If it
helps you visualize more easily, think of a ruler placed against the
center point, with the zero location of the ruler at the center
point.Now we can say that a distance from the center point is at so
many inches or millimetres on the ruler. We thus have the possibility
of saying that some spot is,say, 12.57 inches away from the center
point. If the ruler is a yardstick then we see that 12 inches is 1/3
of the distance from the center point along the ruler towards the end
of the ruler. For most of us 1/3 is "small", 1/2 is "even" and 2/3 is
"large". Since we are talking about "around" we translate "small", and
"large" into distance terms, and we get "near" and "far", respectively.



In the case of the third statement


# In the case of the third statement "Just before you start that let's
see what almost happened."

In this statement we see "just, before" and "almost" as the fuzzy
components. "Before" and "almost" are explained to the computer
(mathematically) in a way which is very similar to "close" in
statement one above. That's because these two terms, which are fuzzy,
are used by natural language speakers to convey yet further spatial
metaphors (just as "close" does). English usage is ripe with spatial
metaphors because they relate our common or shared conceptualizations
of space (the three dimensions x,y,z) and time. The conceptualizations
of space and time are common to all societies who are language using.
The details of their linguistic constructs of these shared concepts
varies.

The individuals in these societies are all (normally) equipped with
arms, legs, heads and bodies pretty much like ours. Since their
thinking and language usage is also situated they too relate to the
universe in terms of the length of their arms, their height, the
length of their stride and so on. It is not surprising that the
language they invent and use reflects the same subjective realities,
much as English does.

"Just" may be explained to a computer in the following way (via
spatial metaphor, using mathematical representation to convey the
spatiality.) In the context of the third statement, "just" may be
equitably represented by "delta", a mathematical construct, which
means an unstated very small magnitude value. In English, some people
might use the expression "a little bit" to similar advantage as "delta".

The statement then translates as: (using the real number line
visualization), that range of number values falling within the "width"
of the delta value, which is itself located leftward from the foot of
the function graphed by "before". "Before" is a graph (function) which
maps the degree of membership of values along the abscissa (x-line).
By definition, "before" 's "rightmost extent" occurs at the point (x
value) on the real number line where centrality is located. (If the
centrality point is at, say, x=500, then "before" is a range of x
values less than 500, extending "leftward" (aka toward zero) to a
value of x such that the membership function for "before" reaches
zero. Then "just" is a small range of x values less than this location
(where (the "before") membership is zero).



In the case of the fourth statement


# In the case of the fourth statement "When I was small he seemed
nearly as tall as a giant."

This might be the recollection of someone seeing the relative, or
subjective, height of his father, when he (the recollector) was a very
young person. "Small" and "nearly as tall as" are both fuzzy
components. That the recollector perceived himself/herself as "small"
was relative to the size of his/her father and mother. But not the
cat. Even babies are not "small" relative to (house) cats.

The perception of smallness was based upon the "mentalism" (operation
of the mind) which recognized that the general volume filled by
himself/herself was clearly much less in magnitude than the volume
filled by "dad" or "mom". The same argument goes for the kid's height
versus the height of "dad" and "mom". We adults take for granted that
there is some "device" (ie mind) which can apprehend that "people"
"have" "volumes" (or "height"), and that some magnitudes are "less"
than others. Perhaps at that age he/she perceived himself/herself to
be "the same size as Ralph" (the family's red and white handkerchief
wearing labrador dog).

Likewise, a "giant" would be several to many times his/her height (at
that age), and since "dad" was several times his/her height it wasnt
too much of a stretch to perceive him as being only a few kid heights
shorter than a giant, or "nearly(tall as(x))". This could be
translated as "height not quite = x". Alternatively, "as tall as"
means"height =", x.
nearly(height = x) could be restated as height = x - delta. another
restatement, or equivalency, is height = just (less than (x)). but
notice that this last equivalency says something slightly different.
This is the power of language, and sometimes its drawback.



In the case of the fifth statement


# In the case of the fifth statement "A big dog is smaller than a
small house."

You must remember that computers do not have "experiences" like we
have had. They have not experienced dogs or houses. A mindless parsing
of the sentence (which is what most computers do after all) provides
us with: indefinite count of big(x) = less than indefinite count of
small(x).

Isn't that meaningful to you? No? Well that is how computers "see"
natural language. And natural language is just as unclear to them as
that statement conveys. They have no "feel" for language because they
are not "situated", they do not "experience". They are like ghost
minds with no instrumentality to sense from. They are less than that
because (in my opinion) they don't (yet) have minds. They are less
organized than that. (in the sophistication/complexity of their
processing).

Now we'll return to something that is meaningful in regular English.
The key to comprehending the fifth statement is that the receiver of
the statement "knows" about dogs, and houses. That knowledge typically
is like the following. For dogs, we know their various shapes and
sizes, their colours, the fur texture and length (approximately of
course), their vocalizations, and their favourite fire hydrants. For
houses we know their various shapes and sizes, their colours, their
material components (brick, wood, plastic, etc) and many other
"typical descriptive details".

So! Since "everybody knows .." then a non-brain damaged speaker of
english knows that any sized dog is always smaller than any sized
house.(We're talking human-habitation houses; not dog, doll, or model
train houses.) This is what some people call common-sense. It is only
commonsense if you have a normal brain, didnt grow up in a cave, or a
society that doesnt have houses. (or dogs). [This would allow for
Igloos and Teepees.]

Can you see that the typical computer is such a disadvantaged
"person"? The mathematical section next up will show you how the
computer can be provided with an appreciation similar to what you have
just read that we humans have, and take for granted, never think about.



In the case of the sixth statement


# In the case of the sixth statement "Sometimes the weekend seems very
far away."

In this final sentence of this example set we see that "far" is the
fuzzy component. "Very" is what linguists call a hedge. Both of these
words are discussed next.

For most speakers of English "far" is just the simple opposite of
"close" (or "near"). In fact in the mathematical representation we
show in the next section we can see that one is treated as being the
complement of the other. It should be pointed out that some English
speakers use these words in such a way that they are NOT simply mirror
reflections of each other. In that case one must endeavour to discover
in what way their usage differs from the regular usage. (That is
covered elsewhere.)

"Very", as we said earlier, is a hedge. A hedge is a term or word that
is used to modify the meaning of a fuzzy term. "very far" modifies the
function "far" such that the grade of membership function amplifies
membership inside of "far" considerably more than plain "far" would
for the same range of x.

Perhaps you thought that I was going to ignore "sometimes". The most
likely usage of "sometimes" is to appeal to "some" and "times" as
semantic items. Another way of saying this is the construction
"sometimes" states "some(x)" where x is "times" in this case.
Depending on the context it is used in, "times" means instances of
datevalues (July 1, etc), or instances of timevalues (3pm, etc), or
relative times (before, then, now, soon, etc.)

"Some" means, make a new set or collection, populate this new set with
members from the set x ("times", in this case). The total count of
members used from the source set is fewer than the total population of
the entire source set. (The "definition" of "some".) In a way it means
randomly select a subset from the source set and use them. If, for
example, the source contains the digits 1 through 9, then "some" could
mean 1,3,7. It could also mean 3,5,9. It could also mean ... You get
the idea.

The mathematics of these linguistic terms.

Now we'll take a look at the (simple) mathematics of these linguistic
terms.

First we'll get three equations out of the way, and explain them in
"plain English".
The first two of these equations may be called "near" and "far",
("close" and "distant") while the third could reasonably be called
"around" (or "about").

The function (ie equation) for "far" is : (1.1)

g(x) = 1/2 + 1/PI*tan-1 [(x-k1)/k2]

The function (ie equation) for "near" is : (1.2)

g'(x) = 1 - g(x)


The function (ie equation) for "around" is : (1.3)

f(x) = exp - [(x-k1)/k2]2


In (1.1), the (arctan) function maps out a sigmoidal, or "S" shaped
curve (when plotted), having an intermediate K1 gradient and increases
in a fashion contextually dependent on object under reference.

(1.2), is simply the complementary function of (1.1). This is often
used to obtain a mirror reflection of a curve in fuzzy set systems.

In (1.3), the function is symetrical about K1, where it peaks, and has
a 'slope' of K2.

(1.1) maps a value of x into a grade of membership value g(x). For
values on the x-line (the abscissa) which are typically average or
middle values, dependent on the context, k1 has the value of x at that
point. Thus, x values less than then "middle" value of x (ie less than
k1) are lower in grade of membership (because they fall on the lower
tail of the function's curve). The "Q" or "width" of the curve is
controlled by k2. It is influenced strongly by the nature of the
context of the text using the term "far". The function "records"
values of x found in that contextual environment, thus a "middle"
value is known, and upper and lower occuring values as well (which
help define the value of K2).

If the context is referenced to "me" then the range of x might
typically be based on either "my" height, or the length of "my"
outstretched arm. The same process occurs if "me" is an infant, or an
elephant, having a different height and arm length than the author .

These (and other) functions can easily be included in the C++ or PHP
based XML-processing (and ontology-processing) software a system uses.
(We'll see in other postings that ontologies may include some
mathematical concepts (typically as so-called "restrictions" on the
classes (i.e. "concepts"). "Ranges" for "values" are common, for example.

[Since XML (document) processing seems to be about doing stuff with
character data (strings) one might consider whether SNOBOL IV or ICON
might be good to use for "XML processing" tasks.]




Usuality?! What the heck is Usuality?

"He's dead, Jim."*

Is this usuality? well, uh, not exactly. Let's look at what Lotfi
Zadeh meant by his term, usuality.

First of all, it is NOT the same as probability.

Read the extensive usuality feature. view the usuality file







(* International viewers of this web page may not have been watchers
of the 1960's TV program, Star Trek, from which this phrase was taken.
Trekkie fans around the world will recognize this oft said, and
punned, statement. If you don't see the relevancy or humour in this
reference please just bear with us anyway. (For the record, nobody
thinks dead people are funny.)