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SPRI Purpose and Review:
Many look for answers to be handed to them.
They record something and see only static and simply say there is nothing.
But a person who truly seeks, will look at the differences of the static and list and study each, and also the way they interact between each other and even study the spaces between the static and the differences between each space while still listing and reviewing. Heat, sound, space, time and vacume all that will be studied.
The Purpose of this site is to begin the process to make humanly visible all motion, existence and visibility of the "Existing" cheifly God. If there are other spirits, angels etc., to reveal them visibly.
The Biblical command is to seek his face: Ps: 34:14 "Seek ye my face; thy face will I seek." ; 2 Chr.15:2 "if thou seek him he will be found";"Even to "his" haitation shall ye seek and come" Deut. 23:6; Ps 9:10 "Has not forsaken them that seek thee"; PS 69:32 Your heart shall live that seek God"
For why would God command this of living man to seek and find, if in death God will be found anyway, at judgement. Or is that which is ment, to find at death? or at the time lifted up?
Jesus says pray you will "come upon" God "before" he himself returns for at the time he returns it will be a terrible time.This means one these things, that you die and therfore come upon God, or be lifted up, or Physically come upon him - find him.
Jesus hates death (Revelations).
What is the soul and the spirit in man which we all have. Life is in the blood, but what of spirit and soul.
By either computer meter assist or to the eye by use of necessary filters in a form easily used i.e. by glasses constructed for such use, somewhat like infrared field glasses.
To find the unseen these parameters shall be used to see and recognize:
Paul: Galations 3:11: "The just shall live by "faith"". But since Jesus has not put away any commandments of God, are we not to seek? And if we seek, are we not to find? And are not those that seek "not forsaken"?
Infinity:
1.) finite (structured and apart)
2.) infinite (structured in some way but surrounding - and / or - fully involved in all things)
3.) Partitioned multi existent.
Size: (as in size of total or part of)
1.) Sound Wave (frequency high to low and of new type either infinite or finite)
2.) Magnetic wave
3.) Gravitational wave
4.) Sub Particle
5.) Micro Particle (Quark etc.)
6.) Minute Particle
7.) Small Particle
8.) Particle
9.) Visible
10.) Visible in Part
11.) Visible in Action
12.) Heat or cold
Color:
1. Spectrum meter (a single color or multi color)
2. clear, transparent
Action:
1.) Movement
2.) Stationary and Surrounding
3.) Stationary and Apart
4.) Active Manipulative
5.) Active Non-Manipulative
6.) Active Responsive
7.) Active Non-Responsive
8.) Inactive
9.) Inactive in Motion but Active in infinite operation
10.) Inactive but active, Responsive in response
11.) Inactive but active, Non-Responsive
Human Involvement:
1.) Within
2.) Without
3.) To Monitor, use:
Human Interaction:
1.) Within
2.) Without
3.) To Monitor, use:
Home
Contents
Basic Methods
Logic
Fuzzy
Fuzzy Logic
How do you handle the ambiguities in your world? What logic do you apply, if any? Traditional logic? Fuzzy logic? Can you tell which you are using? ...
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Index of Page Topics
Traditional Logic
Fuzzy Logic
The New Logic
Fuzzy Theory
Dealing with Ambiguity
Fuzzy Engineering
Image Recognition
Ordinary Logic
Applications
Computer Logic
References-1
Computer Games
Fuzzy Numbers
Fuzzy Sets
Fuzzy Software
Logic
Using a Roster
You could still do that, of course, in what is called a roster format, like listing the members of a club. In a set you can specify the members explicitly, by saying who or what they are. For example, the set, A, might consist of the numbers: 1, 3, 7, 8. We write the set symbolically as:
A = {1, 3, 7, 8}
What is a Set?
To paraphrase a famous poet, a set is a set is a set.
A set is a class, or group of things taken together and sharing a common property. This may not seem like much, but it's high-class stuff and eventually gets into things like infinity and transfinite numbers. You can count on it!
Transfinite numbers are way beyond what we can study here. Our aim is to develop the basics of set theory, and that's a pretty tall order in itself.
Set theory originated at the turn of the century, because a sharp German mathematician named George Cantor needed to compare infinite sets of numbers. Set theory has become the very foundation of math, and so also of modeling. It's also the foundation of object-oriented programming.
Cantor's Definition
This is okay for a small number of members. But it's different for a large number, especially an infinite number. For how then could you list them, or tell someone how to build the sets?
According to Cantor, a set is any collection of distinguishable objects of any kind considered as a whole. In other words, consider a set as a something by itself. To be in the swing of things, call the set S. Then the objects themselves are members of S.
Members are indicated by the membership symbol, Î, read as "a member of." For example, if a is a member of S, you say a Î S. But if a isn't a member of S, you say a Ï S, just like in No Parking signs.
Using 'membership,' you can say when different sets are equal. They would be the same if they had the same members. In the jargon of math, two sets are equal if and only if they have the same members.
The 'if and only if' part of the sentence specifies the sufficient and necessary conditions for sets being the same. The if condition is enough ammo to say the sets are the same, and the only if condition tells you what has to be the case for the sets to be the same.
Using a Formula
It would help to have a general way to define membership and use the definition to identify the members. Take an infinite set, A, for instance. How might you say what they are? My guess is you'd be hard-pressed to do so, at least at first.
But the trick is really simple: you define the set as the elements, x, that meet a certain condition, f(x), whatever that condition might be. This was Cantor's idea, in a nutshell. He defined the sets by using formulas. More precisely, he used formulas to define the membership in sets.
A formula gives you a rule for building the membership. In terms of a variable, x, for members, the principle is to define a formula in x, as the mathematician says. The formula is a rule that lets you select the members. That formula is a function of x, or f(x), which you can read as the function, f, of x, or simply f of x. Seen as a rule, the formula says:
If x meets "a certain condition," then it is a member of the set.
An example of a formula is:
x is 5 feet tall.
Using this formula, you would build the set, S, say, by including everybody five feet tall. So, the rule is simply:
If x is five feet tall, then x is a member of S.
Speaking more generally, the members of a set, A, are the objects that satisfy the formula, f(x). For example, if you select the Laker's big guy Shaq O'Neal as a candidate for membership in the set, you would test the statement Shaq O'Neal is five feet tall against the facts, ... and then toss him out on his ear -- if you could! -- 'cause you know Shaq ain't 5 feet tall!
Using the idea of a formula, we have a way, now, to deal with infinite sets. You would identify the set, A, using the notation:
A = {x | f(x)},
which is pretty neat. The vertical line is just short-hand for 'such that.' So the form can be read:
A is the set of elements, x, such that x satisfies the function, f(x).
The element, x, represents the members of A, whether they are finite, or infinite. If x satisfies f(x), then it's a member of A. Otherwise it isn't.
Back to Index You can even include the empty set, Æ, in your set of sets. (Compare Æ, in sets, with 0, in numbers. Just as you can add 0 to any number without changing the number, so you can include Æ in any set without changing the set.) So you could have:
Ssets+ = {Sapples, Soranges, Sempty}
Contents of Sets
Almost anything you can think of can go into a set, and that even includes other sets. So, for example, you could have the set made up of apples, or the set consisting of oranges. But you could also have the set consisting of the two sets, themselves -- the two members in that case would be the set of apples and the set of oranges. Going a step further, you can include the empty set -- that's the set with no members, and it usually carries the symbol, Æ.
Written in neat math form, the set of apples, the set of oranges, the set of sets of apples and oranges, and the empty set would be, respectively:
The Universal Set
Cantor also defined a universal set. This is the opposite of the empty set, in a sense -- i.e., it includes everything, instead of nothing. It generally takes the symbol U and has to do with a universe of discourse and its subsets. For instance, the universe could be the officers in your company, and the subsets could be all the possible groupings of the officers. That is, U consists of all the subsets of the sets under consideration, whatever the sets might be.
Back to Index
.Sapples = {apple1, apple2, ...}
Soranges = {orange1, orange2, ... }
Ssets = {Sapples, Soranges}
Sempty = Æ = { }
---------------------------------------------
Ways To Define Sets
It's okay to put sets together using apples and oranges, tables and chairs, different animals, and all that, but it can be awkward to list all the members, especially if you have many of them.
------------------------------------------------
Traditional Logic
Logic deals with matters of truth. Science does, too. So what's the difference?
Let's be more logical -- or should that be 'more scientific'? Logic deals with truth -- that's clear from the fact that 'truth' appears in most logic books. And truth involves the use of statements, or what we understand to be propositions, things we normally presume to express information and so to be true or false. On that basis you'd think that it's logic's job to establish the truth (or falsity) of propositions. But that's not quite the case.
Rather, the aim of logic is to establish the truth of statements relative to other statements. In other words, it is concerned with the validity of arguments, the truths that can be deduced from other truths -- not the truth of statements taken independently.
Science, too, deals with truth, but its aim is to establish the truth of statements relative to the methods of measurement, as facts. It's concerned with the validity of observation. And interpretation of the observations depends on the theoretical structure that defines it. One theory -- like Newton's -- leads to one set of observations and information, and another -- like Einstein's -- leads to another system of measurements and information, and still another -- say Quantum Mechanics -- leads to yet another system. Similar relationships occur in most other fields. I refer to the various structures as repositories of information. The dependence on theory is crucial.
In this respect, both logic and science use injunction to derive specific results. By following the rules of injunction, you can obtain the same results: The results are repeatable. Do this and that, take this sequence of steps, follow these rules and directives, and you will see what happens!
By virtue of their structure, propositions provide us with information about objects or events and associated data. They are the means to express this information about observed phenomena -- like the height or color of a table. And it is the propositions themselves that we claim to be true or false. (The table is, or is not, three feet high and colored red.) If you identify information truth-values differently -- say as you'd identify them in Fuzzy logic -- you change the information and alter the rules of injunction.
Models spell out the framework for information. They take the form of equations, formulas, tables, graphs, rules, or the like. Via the modeling process we manufacture the information. This is what makes theories of science so valuable -- each theory yields new quantities of information. For more, see here.
Deductions
Consider what happens when we deduce consequences from axioms. We first construct axioms (statements we presume true). Then we use rules of deduction (inference) to prove theorems from the axioms. When a theorem is deduced from the axioms in accordance with the deduction rules, we say the information contained in the theorem is true. The idea is that true statements imply certain other true statements. You see the connection when you "read" the underlying pattern. It requires understanding.
In the deduction, information contained "implicitly" in the axioms is "drawn out," or made explicit by the theorem. Information present but unexpressed is given form in the theorem. (In the same way, information contained in experience, as fact, is unexpressed until a statement is formed. In the same way, too, you can know something implicitly without verbalization.) The information is drawn out by the rules by rearranging the symbols or substituting equivalent symbols. (This is called calculation, so we speak of a calculus.) The axioms are shorthand for all the information they contain (repositories).
The Syllogism
To see what deduction means, consider a familiar syllogism. Take as a basic premise the statement:
All men are mortal.
Now let's add the statement:
John is a man.
What can we deduce? Well, because we've asserted that John is a member of the class of men, and that everyone in the class is mortal, it follows that:
John is mortal.
So this is a "new" piece of information that comes out of the premises. Grasping the idea is the understanding.
Information Formatting
Information can be expressed in different ways, using different formats. This is common with math functions. For example, the simple linear function:
y = 2x + 3
holds the special class of information that relates the value of y to the value of x in a special (linear) way. Substituting a specific value of x (numerical data) into the function yields a specific value of y. For instance, if x is 2, then y is 7. We plug in the value for x, and compute the value of y. In English we get the sentence, or proposition:
The sum of three plus twice two equals seven.
The proposition informs you about a specific numerical connection between 3, 2, and 7.
That same linear function can also be expressed in a tabular format:
X
...
-3
-2
-1
0
1
2
3
...
Y
...
-3
-1
1
3
5
7
9
...
To draw out the information from this data table, you need to understand how tables are used and apply the underlying rule for interpreting its format -- call it the rule of tabular grammar, if you like. It says that each column in the table relates a particular value of x to a particular value of y. Given the value of x, you read down to get the value of y. For example, you can get information if x equals 1. In that case, y equals 5. (For more details of this view, see my book.)
The Law of the Excluded Middle
This kind of structural formatting, or modeling, has certain fundamental characteristics that determine the way information is defined. Just as with the method of deduction, it presumes the law of the excluded middle, which says that a (meaningful) statement is either true or false, but not both. This is characteristic of traditional logic. It renders the logic two-valued.
Consider the following simple logical function, for example:
If P then Q.
This function is false -- i.e., has the truth-value F -- whenever P is true and Q is false. Otherwise it is true -- i.e., has the truth-value T. The only possible values are T and F. Expressing this in tabular form, we get the truth table:
P
Q
If P then Q
T
T
T
T
F
F
F
T
T
F
F
T
Back to Index
Membership and Subsets
Sets are defined in terms of their members. But some of their more interesting properties have less to do with members than with collections of the members, or what are known as subsets of the sets. It is the subsets and the operations performed on them that make up most of the ideas of set theory. You can think of subsets as a way to divide up the various elements of the sets into different categories.
As an example of a subset, take the red apples in the set of apples, say. Another subset would be all the green apples. Still another would be the sweet apples. And yet another set would isolate all apples that are "bigger than a major league baseball," as if you really cared! Indeed, you can dream up as many such formulas of subsets as you like.
The interesting thing about subsets of sets is that they can also be subsets of other subsets, which is to say all the members of one of the subsets can also be members of the other one. The term used to talk about this aspect of sets is called set inclusion. And the symbol that represents set inclusion is the right-facing horseshoe, Ì. So, for example, if the set, A, is entirely contained in set, B, or if all the members of A are in B, we write:
A Ì B
The meaning is, if x is a member of A, then it is also a member of B.
There can be subsets that are so different they have no members in common at all, or have only some members in common but are otherwise completely different. These relationships can be defined using just a few simple operators.
Two sets of special interest are the Null (or empty) set, Æ, and the Universal (or all-inclusive) set, U. The Null set has the strange property that it is contained in all sets -- nothing is in everything! And the universal set has the property that it contains all sets -- all the sets are in it. The universality refers to all of the sets that you are concerned with at the moment, or what is within your universe of discourse.
Example
It's important to see the difference between the idea of membership in a set and containment in a set, so let's look at an example. Suppose we let A be the set consisting of the members a, b, and c, or:
A = {a, b, c}
By agreement, then, A has three members. But how many subsets does it have?
Well, for one, we can isolate a and form the subset, {a}, by considering it as a distinct category. We can do the same thing with b to get {b}. And c to get {c}. That gives us three subsets, so far.
Now we can isolate a and b together as a collective to get {a, b}. We can do the same with a and c, and with b and c. That gives us additional subsets {a, b}, {a, c}, and {b, c}, so we'd now have a total of six subsets.
Also, we can group all three of the members together to form the subset {a, b, c}, giving us a sub-total of seven subsets.
But wait! Isn't {a, b, c} the original set? ... Well, yes, it is. In fact, it's the universal set, U. It defines our universe of discourse -- the elements of interest, a, b, and c. But it's also a subset, because every one of its members is a member of the original set. The difference is that it's not a proper subset, like the other six. But it's still a subset. (To be a proper subset, the set has to be smaller.) So we have seven subsets.
But we're not done yet. There's still the Null set, Æ, to add. It consists of none of the members of the set, and it's contained in all sets. For that reason, it, too, is a subset of A, and of course it's a proper subset. And that, finally, gives us a total of eight subsets.
The set of all possible subsets of a set has a special name, i.e., the power set.
----------------------------------
Operations for Sets
The operations for sets are just ways to combine sets, or ways to generate new sets out of old ones. Operations on sets correspond to the operations you find in arithmetic, which you know as addition, subtraction, multiplication, and division. Operations in logic do similar kinds of things, but they work on statements.
Just as logical operations have to do with combining statements, set operations deal with sets, whereas arithmetic operations are performed on numbers.
In set theory, then, you find the operators, È (read as union, join, or sum), Ç (read as intersection, meet, or product), and Comp (read as complement).
The union of sets A and B is the set of all members either in A or in B and is written as:
A È B
So, if x is a member of A, then x is a member of A È B, and if y is a member of B, then y is a member of A È B.
Similarly, the intersection of A and B is the set of all members of A that are also members of B and is written as:
A Ç B
So, if x is a member of A and if x is a member of B, then x is a member of A Ç B.
You can also define the difference between sets A and B:
A - B
as the elements in B not in A So if A = {Ann. Arnie, Amos) and B = {Ann, Arnie, Ben), then A-B = {Ben}. Similarly, B-A = {Amos}.
There is also the operation performed only on one set -- not two, as in union and intersection. The operation is complementation, Comp(A). Given a set, A, you take its complement relative to the universe of discourse, U. So, if A consists of the set {x}, with x drawn from U, then the complement of A is:
Comp(A) = {x | x Ï A}
which consists of all the members of U that are not in A.
Back to Index
---------------------------------------------
The New Logic
The new logic turns the tables on the old logic and considers the fuzziness in things foremost. As Bart Kosko says, in Fuzzy Thinking: The New Science of Fuzzy Logic:
Some things are not fuzzy no matter how closely you look at them. These things tend to come from the world of math. Here by design man or God has kept fuzziness out of the picture. We agree that "Two plus two equals four" is 100% true. But when we move out of the artificial world of math, fuzziness reigns. It blurs borders and deadlines as if our words cut the universe into pieces with a blunt knife.
By contrast with traditional logic, fuzzy logic does not assume the law of the excluded middle. This is to say that propositions can now be both true and false. A new understanding -- a new form -- is now required.
The reason that propositions in this view can be both true and false is that truth-value is considered to be a matter of degree. That is, propositions are true to a certain degree, and false to a certain degree. In the extreme case, if a proposition happens to be completely true -- i.e., to a maximum degree -- then it can not be false in any amount -- i.e., it is false to a minimum degree. One value is the opposite of the other.
It may be helpful to compare the notion of degree with terms in other theories. In statistics, for example, we use the term 'probability.' The corresponding term in neural networks is 'weight.' In fractal theory, the word 'fractal' is used. And multi-valued logic identifies a corresponding 'range' of truth-values.
For convenience, the maximum degree is specified as 1, and the minimum degree is set at 0. This means that propositions can have truth-values anywhere between 1 and 0, so we are now dealing with so-called multi-valued logic.
As a modeling device, this logic changes the way environments of skills are perceived.
Back to Index
-----------------------------------------------
Dealing with Ambiguity
In the real world, problems often arise because of the ambiguity of language -- because of the way we model or form our experience. Many statements that may seem perfectly clear, lose their clarity when examined closely. Propositions may be ambiguous, or have a multiplicity of meaning. These propositions are natural candidates for fuzzy treatment.
See examples discussed by Ruggiero, Rao & Rao, and Terano, et. al.
Back to Index
-----------------------------------------------
Image Recognition
A prominent feature of all behavioral environments is that they involve objects and perception. The objects are seen, heard, touched, etc., and their differentiable characteristics recognized. So our experiential basket includes myriad properties, among which are: height, weight, size, color, speed, and quickness. You can grind out your own long list of properties. As descriptors for such characteristics, we use words like tall, very short, heavy, mammoth, light green, crimson, fast, awesome, and so on and on.
Each of these terms has a certain amount of ambiguity -- the terms mean different things for different people. So the information containing such data elements has no clear and unambiguous truth-value, and likewise perception would be less than perfectly clear.
Now enter fuzzy theory!
With this new approach we can say that descriptive terms have a certain degree of truth, and the recognition would acquire degrees of certainty. Take 'mammoth,' for instance. Say the reference is a dog. Now consider who might be seeing the dog, and compare the degree of certainty in the proposition that the dog is mammoth. For a child, the term would likely carry a high degree of truth. But for the 330 pound ex L. A. Lakers' center, Shaquille O'Neal, for example, the dog might not only not be mammoth, but may even be seen as tiny, so the proposition would be true only to a slight degree.
Back to Index
----------------------------------------------
Applications
Fuzzy theory has found application both in analyzing the stock market and in the design of trading strategies. Among the more obvious visual pattern recognition problems is that of recognizing price history patterns in the time series charts of stocks -- patterns such as triangles, double tops, head and shoulders tops, runaway gaps, and so on, not to mention the many patterns described by the Elliott Wave theory. These patterns are generally subject to interpretation and thus don't always provide perfectly clear trading signals. By developing the fuzzy definitions of these patterns, the trading strategies could be improved.
Commercial Applications
Fuzzy logic has already been used commercially, particularly in Japan. It has been applied particularly to controllers and robotics, as in mechanisms to control a subway car, and also for different controls in automobiles -- like braking systems, transmission controls, and fuel injectors. Perhaps more familiarly, fuzzy logic is used in cameras and camcorders for different control applications.
Neural Networks
As defined by Rao & Rao, in their book on neural networks, a neural network is a group of processing elements where typically a subgroup makes independent computations and passes the results to another subgroup. After a number of such transmissions, a subgroup of one or more processing elements determines the output from the networks. Click here for more stuff on neural networks.
The authors discuss several ways in which fuzzy theory can be applied to the networks. One way is to use a fuzzifier function to pre-process or post-process data for a neural network. Such processing may be necessary to convert small-difference, hard-to-differentiate inputs to larger-difference, easier-to-differentiate inputs. The fuzzifier function converts data into fuzzy data for application to the networks. Another way is to use fuzzy sets to define the weights on which basis the independent elements make their computations.
Back to Index
-------------------------------------------------
Home | Top of Page | Contents
Basic Methods | Logic | Fuzzy
Traditional Logic
Logic deals with matters of truth. Science does, too. So what's the difference?
Let's be more logical -- or should that be 'more scientific'? Logic deals with truth -- that's clear from the fact that 'truth' appears in most logic books. And truth involves the use of statements, or what we understand to be propositions, things we normally presume to express information and so to be true or false. On that basis you'd think that it's logic's job to establish the truth (or falsity) of propositions. But that's not quite the case.
Rather, the aim of logic is to establish the truth of statements relative to other statements. In other words, it is concerned with the validity of arguments, the truths that can be deduced from other truths -- not the truth of statements taken independently.
Science, too, deals with truth, but its aim is to establish the truth of statements relative to the methods of measurement, as facts. It's concerned with the validity of observation. And interpretation of the observations depends on the theoretical structure that defines it. One theory -- like Newton's -- leads to one set of observations and information, and another -- like Einstein's -- leads to another system of measurements and information, and still another -- say Quantum Mechanics -- leads to yet another system. Similar relationships occur in most other fields. I refer to the various structures as repositories of information. The dependence on theory is crucial.
In this respect, both logic and science use injunction to derive specific results. By following the rules of injunction, you can obtain the same results: The results are repeatable. Do this and that, take this sequence of steps, follow these rules and directives, and you will see what happens!
By virtue of their structure, propositions provide us with information about objects or events and associated data. They are the means to express this information about observed phenomena -- like the height or color of a table. And it is the propositions themselves that we claim to be true or false. (The table is, or is not, three feet high and colored red.) If you identify information truth-values differently -- say as you'd identify them in Fuzzy logic -- you change the information and alter the rules of injunction.
Models spell out the framework for information. They take the form of equations, formulas, tables, graphs, rules, or the like. Via the modeling process we manufacture the information. This is what makes theories of science so valuable -- each theory yields new quantities of information. For more, see here.
Deductions
Consider what happens when we deduce consequences from axioms. We first construct axioms (statements we presume true). Then we use rules of deduction (inference) to prove theorems from the axioms. When a theorem is deduced from the axioms in accordance with the deduction rules, we say the information contained in the theorem is true. The idea is that true statements imply certain other true statements. You see the connection when you "read" the underlying pattern. It requires understanding.
In the deduction, information contained "implicitly" in the axioms is "drawn out," or made explicit by the theorem. Information present but unexpressed is given form in the theorem. (In the same way, information contained in experience, as fact, is unexpressed until a statement is formed. In the same way, too, you can know something implicitly without verbalization.) The information is drawn out by the rules by rearranging the symbols or substituting equivalent symbols. (This is called calculation, so we speak of a calculus.) The axioms are shorthand for all the information they contain (repositories).
The Syllogism
To see what deduction means, consider a familiar syllogism. Take as a basic premise the statement:
All men are mortal.
Now let's add the statement:
John is a man.
What can we deduce? Well, because we've asserted that John is a member of the class of men, and that everyone in the class is mortal, it follows that:
John is mortal.
So this is a "new" piece of information that comes out of the premises. Grasping the idea is the understanding.
Information Formatting
Information can be expressed in different ways, using different formats. This is common with math functions. For example, the simple linear function:
y = 2x + 3
holds the special class of information that relates the value of y to the value of x in a special (linear) way. Substituting a specific value of x (numerical data) into the function yields a specific value of y. For instance, if x is 2, then y is 7. We plug in the value for x, and compute the value of y. In English we get the sentence, or proposition:
The sum of three plus twice two equals seven.
The proposition informs you about a specific numerical connection between 3, 2, and 7.
That same linear function can also be expressed in a tabular format:
X
...
-3
-2
-1
0
1
2
3
...
Y
...
-3
-1
1
3
5
7
9
...
To draw out the information from this data table, you need to understand how tables are used and apply the underlying rule for interpreting its format -- call it the rule of tabular grammar, if you like. It says that each column in the table relates a particular value of x to a particular value of y. Given the value of x, you read down to get the value of y. For example, you can get information if x equals 1. In that case, y equals 5. (For more details of this view, see my book.)
The Law of the Excluded Middle
This kind of structural formatting, or modeling, has certain fundamental characteristics that determine the way information is defined. Just as with the method of deduction, it presumes the law of the excluded middle, which says that a (meaningful) statement is either true or false, but not both. This is characteristic of traditional logic. It renders the logic two-valued.
Consider the following simple logical function, for example:
If P then Q.
This function is false -- i.e., has the truth-value F -- whenever P is true and Q is false. Otherwise it is true -- i.e., has the truth-value T. The only possible values are T and F. Expressing this in tabular form, we get the truth table:
P
Q
If P then Q
T
T
T
T
F
F
F
T
T
F
F
T
Back to Index
---------------------------------------------
The New Logic
The new logic turns the tables on the old logic and considers the fuzziness in things foremost. As Bart Kosko says, in Fuzzy Thinking: The New Science of Fuzzy Logic:
Some things are not fuzzy no matter how closely you look at them. These things tend to come from the world of math. Here by design man or God has kept fuzziness out of the picture. We agree that "Two plus two equals four" is 100% true. But when we move out of the artificial world of math, fuzziness reigns. It blurs borders and deadlines as if our words cut the universe into pieces with a blunt knife.
By contrast with traditional logic, fuzzy logic does not assume the law of the excluded middle. This is to say that propositions can now be both true and false. A new understanding -- a new form -- is now required.
The reason that propositions in this view can be both true and false is that truth-value is considered to be a matter of degree. That is, propositions are true to a certain degree, and false to a certain degree. In the extreme case, if a proposition happens to be completely true -- i.e., to a maximum degree -- then it can not be false in any amount -- i.e., it is false to a minimum degree. One value is the opposite of the other.
It may be helpful to compare the notion of degree with terms in other theories. In statistics, for example, we use the term 'probability.' The corresponding term in neural networks is 'weight.' In fractal theory, the word 'fractal' is used. And multi-valued logic identifies a corresponding 'range' of truth-values.
For convenience, the maximum degree is specified as 1, and the minimum degree is set at 0. This means that propositions can have truth-values anywhere between 1 and 0, so we are now dealing with so-called multi-valued logic.
As a modeling device, this logic changes the way environments of skills are perceived.
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Dealing with Ambiguity
In the real world, problems often arise because of the ambiguity of language -- because of the way we model or form our experience. Many statements that may seem perfectly clear, lose their clarity when examined closely. Propositions may be ambiguous, or have a multiplicity of meaning. These propositions are natural candidates for fuzzy treatment.
See examples discussed by Ruggiero, Rao & Rao, and Terano, et. al.
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Image Recognition
A prominent feature of all behavioral environments is that they involve objects and perception. The objects are seen, heard, touched, etc., and their differentiable characteristics recognized. So our experiential basket includes myriad properties, among which are: height, weight, size, color, speed, and quickness. You can grind out your own long list of properties. As descriptors for such characteristics, we use words like tall, very short, heavy, mammoth, light green, crimson, fast, awesome, and so on and on.
Each of these terms has a certain amount of ambiguity -- the terms mean different things for different people. So the information containing such data elements has no clear and unambiguous truth-value, and likewise perception would be less than perfectly clear.
Now enter fuzzy theory!
With this new approach we can say that descriptive terms have a certain degree of truth, and the recognition would acquire degrees of certainty. Take 'mammoth,' for instance. Say the reference is a dog. Now consider who might be seeing the dog, and compare the degree of certainty in the proposition that the dog is mammoth. For a child, the term would likely carry a high degree of truth. But for the 330 pound ex L. A. Lakers' center, Shaquille O'Neal, for example, the dog might not only not be mammoth, but may even be seen as tiny, so the proposition would be true only to a slight degree.
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Applications
Fuzzy theory has found application both in analyzing the stock market and in the design of trading strategies. Among the more obvious visual pattern recognition problems is that of recognizing price history patterns in the time series charts of stocks -- patterns such as triangles, double tops, head and shoulders tops, runaway gaps, and so on, not to mention the many patterns described by the Elliott Wave theory. These patterns are generally subject to interpretation and thus don't always provide perfectly clear trading signals. By developing the fuzzy definitions of these patterns, the trading strategies could be improved.
Commercial Applications
Fuzzy logic has already been used commercially, particularly in Japan. It has been applied particularly to controllers and robotics, as in mechanisms to control a subway car, and also for different controls in automobiles -- like braking systems, transmission controls, and fuel injectors. Perhaps more familiarly, fuzzy logic is used in cameras and camcorders for different control applications.
Neural Networks
As defined by Rao & Rao, in their book on neural networks, a neural network is a group of processing elements where typically a subgroup makes independent computations and passes the results to another subgroup. After a number of such transmissions, a subgroup of one or more processing elements determines the output from the networks. Click here for more stuff on neural networks.
The authors discuss several ways in which fuzzy theory can be applied to the networks. One way is to use a fuzzifier function to pre-process or post-process data for a neural network. Such processing may be necessary to convert small-difference, hard-to-differentiate inputs to larger-difference, easier-to-differentiate inputs. The fuzzifier function converts data into fuzzy data for application to the networks. Another way is to use fuzzy sets to define the weights on which basis the independent elements make their computations.
Fuzzy Cold Dark Matter: The Wave Properties of Ultralight Particles
Wayne Hu, Rennan Barkana, and Andrei Gruzinov
Institute for Advanced Study, Princeton, New Jersey 08540
Received 27 March 2000
Cold dark matter (CDM) models predict small-scale structure in excess of observations of the cores and abundance of dwarf galaxies. These problems might be solved, and the virtues of CDM models retained, even without postulating ad hoc dark matter particle or field interactions, if the dark matter is composed of ultralight scalar particles (m~10-22 eV), initially in a (cold) Bose-Einstein condensate, similar to axion dark matter models. The wave properties of the dark matter stabilize gravitational collapse, providing halo cores and sharply suppressing small-scale linear power.
©2000 The American Physical Society
URL: http://link.aps.org/abstract/PRL/v85/p1158
DOI: 10.1103/PhysRevLett.85.1158
PACS: 95.35.+d, 98.80.Cq
It was never supposed that cogitation is inherent in matter, or that every particle
is a thinking being. Yet if any part of matter be devoid of thought, what part can we
suppose to think? Matter can differ from matter only in form, bulk, density, motion and
direction of motion: to which of these, however varied or combined, can consciousness
be annexed? To be round or square, to be solid of fluid, to be great or little, to be
moved slowly or swiftly one way or another, are modes of material existence, all
equally alien from the nature of cogitation. If matter be once without thought, it can
only be made to think by some new modification, but all the modifications which it can
admit are equally unconnected with cogitative powers.
-Samual Johnson
What is this project about?
[You may download my research funding proposal here]
Cognition, emotions, and that which is spiritual, is what many argue machines will never have.
But rocks, trees, water, and people are all made of the same matter, governed by
the same laws of physics, and exist in the same world of limited resources.
So why should our human matter be considered 'conscious' while rocks and water not?
Fuzzy, short for fuzzy logic, is my attempt to emulate the above said cognition in a machine.
The goal would be to prove that machines, governed by the same rule of physics and built
of the same matter, could equally have a 'spiritual' side.
Videos
Mobility Test
first mobility test
(2.8mb) Circle Test
angle calculation test
(.6mb) Object Avoidance Test
laser vision and object
avoidance test (3.7mb)
Production Images
First Layer Second Layer Sonar Added Top View
IR Added Front Closeup View
Current State of Completion
* ran out of ROM on the PIC16F877 so the project is now on hold
* everything that can be finished with the memory limitation is complete
Fuzzy System Design
Mechanical
process of design, omni 4 wheel system, servos, traction Electrical
3 scanning IR sensors, 3 sonar, comparative photoresistors/IR, digital compass, and cerebellum Computational
mention crazy angle software, trig lookup table, vision systems, and AI emotion emulation Parts List
(3) Devantech SRF04 Sonar Ranger $34.50 each
(1) Devantech Digital Compass $50 each
(3) Sharp GP2D12 Detector Package $11.50 each
(4) Kornylak 2051-1/4-X Cat-Trak Transwheel Omni-wheel $3.42 each
(4) HS-225MG Hitec Servo $32 each
(1) HS-311 Standard Hitec Servo $9
(1) Cerebellum $95 (I got a $55 special)
(2) NiMH 6V 2100mAh Battery $21 each (I got a $10 special)
High Density Polyurethane $10
misc fasteners, wires, etc. $5
total cost: ~$450 (including shipping)
List of Emulated Emotions:
tired: slow speed
bored: randomly activate other emotions
afraid: hide, avoid
claustrophobic: stay out in lighted, open areas
hungry: low battery alert, be more efficient
wanderlust: explore a lot
frustration: try something new to acheive a goal
created, designed, and built by John Palmisano
project done at Carnegie Mellon University
with the support of the CMU Robotics Club
and funding from SURG
Characteristic Function of a Set
The ideas of set theory form the basis for modern math, but there's a connecting link between the algebra of sets and logic. This link is established by a characteristic function, which uses the fact that logicians assign truth-values to predicate statements.
For instance, in the predicate calculus -- which is the modern version of the traditional logic of syllogisms -- the sentence 'The table is red' would be considered to be 'true' when the object denoted by 'table' satisfies the condition that it is red, i.e., that it's in the set of red tables.
Whether or not something is in a particular set becomes the test for truth. The test is made through a special kind of set specification called the characteristic function of the set. This is really a third way to specify a set.
The procedure is to first define a universe of discourse, U. This universe is the collection of items from which you will construct your sets. This is similar to defining sets by formula. Now, though, for any given set, A, the characteristic function defines whether or not any specific element in the universe is in A.
A function, CA(x), is a characteristic function if it takes the value, 1, when x is in A, and the value, 0, when x is not in A. The function is defined for all the elements of the universe, and it maps all of U to the set of two elements {0, 1}. It is written formally as:
CA(x) = 1 if x is in the set A.
CA(x) = 0 if x is not in A.
The connection with two-valued logic is now direct -- you need only associate the truth value, false, to 0, and the truth value, true, to 1. That is, you identify the set {0, 1} with the set {false, true}.
For other forms of logic, however, this characteristic function doesn't apply. In one four-valued logic, for instance, a function has to be defined to map the whole of the universe, U, to the set of four elements 0, 1, imaginary, and uncertain. In other multi-valued logics, other functions have to be devised. Yet again, for fuzzy logic, the function has to take the fuzziness of sets into account. ------------------------------------------------
Ordinary Functions
What in the world is a fuzzy function? For that matter, what is an ordinary function? Ordinary functions are a bit easier to deal with, so it's worth looking to see if they help understand the fuzzy kind.
Speaking in modern lingo, an ordinary function is a mapping from the elements, x, of a number set, A, to the elements, y, of a number set, B. This is really a function of a single variable (x) -- which means that only the elements of the set, A, are associated with elements of the set, B. To each element in A we hook an element from B. (Note that nothing says A and B can't actually be the same set.) The function defines the rule for making the connection.
We normally think of this type of function as a single-valued function, meaning to say that only one member of B can be associated with any given member of A. That's the rule. Stated another way, the rule says that a given number in A can only go to a single number in B. If x1 is in A, it maps uniquely to y1. (This doesn't mean, however, that different numbers in A can't go to the same number in B.) Such a function is normally expressed as:
f(x)
This relationship between members of specific sets being the case, you've probably already figured out that fuzzy functions have something to do with fuzzy sets, which indeed they do. These fuzzy sets involve fuzzy variables. Strangely enough, the fuzzy variables could be fuzzy numbers or even linguistic variables, which are inherently fuzzy.
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Fuzzy Variables
Yikes! Out of the frying pan into the fire! What in the world are fuzzy variables? To see, let's take a look at fuzzy numbers and at linguistic variables.
Fuzzy Numbers
Spikes! That's how Bart Kosko views traditional, ordinary, what-you-see-is-what-you-get numbers. He graphs them as spikes.
In fuzzy set terms, the graph is intended to show that an ordinary number is a (limiting case) fuzzy number that is 100% in the fuzzy set of that number. Like 0 in the fuzzy set ZERO. Or 1 in the fuzzy set ONE. And so on. What's more, no other number is in the limiting set. This is like a boundary for fuzzy number sets -- a limiting condition.
For most fuzzy numbers, though, close clusters are involved. For instance, numbers close to 0 can be included in the fuzzy set ZERO, but they would have different degrees of membership. We commonly say, for instance, that "there is essentially no money left in the bank -- a few bucks, maybe, but that's about it." In this case, as Kosko says:
The number zero belongs 100% to the set ZERO, but close numbers may belong only 80% or 50% or 10%.
The more general graph of degree of membership for fuzzy numbers might look like the following, using straight lines to define the membership degrees. (Different problem situations might call for different ways to represent closeness.)
The graph for ZERO could take any shape, but in any case zero is 100% in ZERO, whereas numbers like 1 and -1 are only partially in the set -- their degree of membership is less than one. And in this case numbers like 2 and -2 aren't in the set at all -- their degree of membership is zero. When the graph shrinks so that only 0 is in the set, we get back to a correspondence with the ordinary number 0. The same thing applies to all the numbers. So ordinary arithmetic becomes a special case of fuzzy arithmetic.
Linguistic Variables
The situation really gets bizarre when we start talking about variables in terms of ordinary words of the language. (Hey! In Visual Basic you can even have variable objects!) What sense can be made of linguistic variables?
In fact, the idea of a linguistic variable makes good sense and is really very clever, because its underlying utility is to solve long-standing insoluble problems. Using the precise techniques in math and logic, we've made fairly good progress, relatively speaking, in the physical sciences, but the methods have been woefully unsuited to deal with ordinary human interactions -- I might say underwhelming. Even if fuzzy theory can only inch us forward in the social arena, they're still worth considering. The use of linguistic variables is one approach.
Linguistic variables are related to, but different from fuzzy sets -- they have the relationship of goal to instrument. The relationship is neatly spelled out by Schmucker, who says:
[Having] precisely manipulatable natural language expressions is the goal, and fuzzy set theory is a tool to achieve that goal.
---
A linguistic variable is a variable whose values are natural language expressions referring to some quantity of interest. These natural language expressions are then, in turn, names for fuzzy sets composed of the possible numerical values that the quantity of interest can assume. (My emphasis)
We use linguistic variables without realizing it, even though we may not formalize the process. Think about 'pressure', for instance. Language allows us to refer to different pressure states. That is, we have many descriptive terms for pressure. We can say the pressure is very low, or low, or medium low, or moderate, or average, or moderately high, or high, or very high, or explosively high, or anything else you can think of. And we relate the terms to different numerical value ranges of the pressure. Bottom line is, these terms make up values of the 'pressure' variable.
As another example, Schmucker defines a linguistic variable he calls number. He uses the integers between 1 and 10 for the illustration and identifies the set of values {few, several, many} as the natural language expressions that number can take. The expressions are names for the number sets as follows:
few = {.4/1, .8/2, 1/3, .4/4}
several = {.5/3, .8/4, 1/5, 1/6, .8/7, .5/8}
many = {.4/6, .6/7, .8/8, .9/9, 1/10}
In other words, 'few', for example, can mean 1, or 2, or 3, or 4, but to different degrees.
See Fuzzy Sets for details of degree of membership in sets.
In graphical form, we have the schematic:
From this diagram you can see that 1 is a member of the fuzzy set 'few' with degree .4. 2 is a member of 'few' with degree .8. 3 is a member of 'few' with degree 1 and a member of the set 'several' with degree .5. And so on.
In practical applications it would be natural to design the language expressions to maintain a rich but finite set. And because it is likely to be large, the set would best be specified by rule rather than by listing them, as a roster. You would need to have a way to specify the different (non-numerical) values of the variable.
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Building the Fuzzy Functions
Traditionally, functions are rules that define mappings from ordinary numbers to ordinary numbers -- from independent to dependent variable. Fuzzy functions, by contrast, are rules mapping fuzzy variables to fuzzy variables. To see what a fuzzy function might look like, let's develop one that uses the fuzzy variable defined for pressure. (You can find other examples in Kosko, Schmucker, and Pappo.)
To this end, assume we wish to simulate an individual's behavior and would like to model the way that person responds to social pressure. Specifically, let's say we need to model the effect of pressure on the person's motivation in specific situations. To keep it simple we will use only the values low, medium, and high as our linguistic representations for pressure, and we'll use goes down, remains the same, and goes up as our linguistic representations for motivation. We will, further, accept the following rules as the applicable conditionals for the individual:
Rule 1: If the social pressure is low, the person's motivation goes down.
Rule 2: If the pressure is medium, the person's motivation remains the same.
Rule 3: If the pressure is high, the person's motivation goes up.
The rules thus define our function. They provide the connection between the social condition and the person's response and form the basis for a fuzzy inference. Between the input (the social pressure) and the output (the response). Between the independent variable (pressure) and the dependent variable (response).
By defining numerical fuzzy subsets for each of the linguistic representations of the two fuzzy variables, the rules can be embedded in a traditional function to model a specific connection between numerical values for pressure and numerical values for motivation:
Pressure ® Fuzzy Function ® Motivation
For more details of these relationships, you might consult my book. ------------------------------------------------
Beyond Ordinary Operations
Part 1 of the Theory of Fuzzy Sets deals with fuzzy extensions of ordinary set operations and includes fuzzy union, intersection, and complement of sets. When working with fuzzy sets, however, additional operations can be defined. These operations are uniquely fuzzy operations and have no counterpart in ordinary set theory.
The operations, as discussed by Kurt Schmucker, are:
o Concentration
o Dilation
o Normalization
o Intensification
o Fuzzification
Concentration is an operation that reduces even further the degree of membership of already peripheral members of a set and thus takes them well out of the picture, in effect concentrating on the central characters.
Dilation has the opposite effect of concentration in that it heightens the degree of membership of the peripherals and brings them more into the picture.
Normalization changes the degree of set membership of elements across the board in order to raise the membership status of at least one of them to the maximum of 1, rendering the element(s) totally in the set.
Intensification increases the degree of membership of all the elements of a set that are already half members or better and decreases the degree of those members less than half in the set. That is, it makes a fuzzy set less fuzzy.
Fuzzification operates in reverse of intensification and makes the set more fuzzy.
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Concentration
The purpose of the concentration operator is to compress the members of a fuzzy set that are already practically full members and to separate further away the members that are already pretty much out of it -- strengthening the strong members and weakening the weak members. . This means increasing the degree of membership for the top bananas of the set and decreasing the degree of membership for the untouchables. You can do this by squaring the degree of membership of each member in the set.
Note that the degree values range from 0 on the low end to 1 on the high end. So when you square a degree it stays in the same range. But more importantly, if the degree is either 1 or 0, it remains the same, and if it lies in between 1 and 0 it gets smaller. That is:
x2 < x for 0 < x < 1.
For example, (1/2)2 = 1/4 < 1/2.
Now, values at a higher degree decrease less than values at a lower degree. For instance, .7 goes to .49, when squared, so it drops less than 30%. But .2 goes to .04, which is 80% less than .2. Squaring pushes the lower members practically out of the set.
The concentrated form of fuzzy set, A, is thus given as:
CON(A) = {a2(x)/x | x is an element of U}
An example of a fuzzy set and its concentrated set is as follows:
In this diagram the black line is the original fuzzy set, A, and the blue line is (an approximation to) the concentrated fuzzy set of A. It's an approximation because I only plotted concentrated points derived from two values, namely .89 and .50. Plotting all the points will yield a smoothly continuous curve. In any case, you can see that squaring brings the top bananas closer together.
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Dilation
The procedure for dilation is the same as for concentration, except it inverts the order. Instead of squaring the degree of membership for each member, you take the positive square root. This goes in the opposite direction from concentration. If you start with 1/2 and square it, you get 1/4. Start with 1/4, take the positive square root, you get 1/2.
Now you can see that for all values except 1 and 0, which remain the same in the operation, the result of taking the square root is to increase the degree of membership for all members. The positive square root of .01 is .1. The positive square root of 1/9 is 1/3. The positive square root of .25 is .5. The positive square root of .49 is .7. The positive square root of .81 is .9. And so on. Each positive square root is bigger than the original number.
The dilation set of a set, A, is:
DIL(A) = {SQRT(a(x))/x | x is an element of U}
And a sample graph approximating -- because I only plotted two dilatable points -- the dilated set (in blue) of A is:
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Normalization
Suppose you have two sets of numbers that you'd like to compare but they are expressed in different scales so a direct comparison is difficult. One thing you can do is normalize the sets. This process reduces the sets to the same standard. It standardizes them. Now you can compare them more directly. But what is the process?
Say you have a number, a, that's less than 1. If you divide that number by itself you get a/a for your effort, and this is simply the value, 1. OK? Similarly, if you divide each member of a set by the largest number in the set, you get the value 1 as the new largest number in the set -- the maximum. So, for example, if you have the set:
{3, .6, and 1.2}
you can divide through by 3 and get the set:
{1, .2, .4).
In just this way, if you divide the degree of membership of the elements of a fuzzy set by the maximum -- the largest -- of all the degrees of membership in the set, you normalize the set. If the maximum degree of membership already happens to be 1, you of course change nothing by the division. Doing this to both of your sets thus puts them in the same scale for comparison. The formula for normalization is:
NORM(A) = {a(x)/m(x))/x | x is an element of U}
where m(x) = max{a(x)} for all x in U.
The graph of the normalized version (in blue) of a non-normalized set (in black) might be as follows:
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Intensification
Combining aspects of concentration and dilation, intensification raises the degree of membership of some elements of a fuzzy set and lowers the degree of membership of other elements of the set. In particular it raises the degree of membership of elements already having a degree of membership greater than .5, and it lowers the degree of membership of elements already having a degree of membership less than .5. In other words it heightens the contrast between the "haves" and the "have nots."
The intensification formula is:
INT(A) = {m(x)/x | x is an element of U}
where
m(x) = 2a2(x) for a(x) for x greater than or equal to 0 and less or equal to .5
m(x) = 1 - 2(1 - a(x))2 for x greater than .5 and less than or equal to 1.0
The graph for an (approximately) intensified set (blue) might look like the following:
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Fuzzification
Fuzzification of a fuzzy set, A, produces the fuzzy union of certain subsets of A. The result is a new set made up of possibly different elements of U, each of which may have a new degree of membership in the fuzzified A.
Each one of the subsets of A has selected elements from U, and each element in each subset has its degree of membership in the subset. The membership and the corresponding degree of membership in each subset are determined by the product of the degree of membership of the element in A and a function, K(x), which thereby acquires special importance. The formula representing this operation is:
Fuzzification of A = È {a(x)*K(x)}
Note that È is the fuzzy union, a(x) is a real number, and K(x) is a set. The rule for the multiplication, *, is for a(x) to form the product of the degree of membership of each x in A with the degree of membership of each occurrence of x in the K set. The idea is to fuzzify even further each fuzzy element of each particular subset.
Using an example from Zadeh, if U is the set:
U = {1, 2, 3, 4}
and the original fuzzy set is:
A = {.8/1, .6/2}
we can get another specific subset by applying a particular function, K(x). For example, K(1) maps the element 1 into A as follows:
K(1) = {1/1, .4/2}
This says that K maps 1 into 1 with degree of membership 1, and it also maps 1 into 2 with degree of membership .4.
Using the real number/set multiplication, *, we can multiply the degree of membership of the elements in A by the degree of membership of the elements of the subset K(1). That is,
a(1)*K(1) = .8{1/1, .4/2} = {.8/1, .32/2}
We can do the same thing with:
K(2) = {.4/1, 1/2, .4/3}
which maps the element, 2, into 1, 2, and 3. The result is:
a(2)*K(2) = .6{.4/1, 1/2, .4/3} = {.24/1, .6/2, .24/3}
Combining the two sets using fuzzy union, we get:
Fuzzified A = {.8/1, .6/2, .24/3}
Note that the degree of membership of 2 is determined by taking the larger of the two degrees of membership of 2 -- i.e., the maximum.
What is a Fuzzy Set?
Let me quote from a source -- Bart Kosko -- to explain what a fuzzy set is. In his book, Fuzzy Thinking: The New Science of Fuzzy Logic, he writes:
... Fuzzy sets arise when a set partially contains an element, as when an audience [a group of people in a room, say] contains a somewhat happily employed person or when a barrel contains a somewhat rotten apple or when a chromosome contains a somewhat mutated gene. In some sense the set is not fuzzy but its elements are fuzzy. The elements all have some property to some degree. I call this elementhood. The old fuzzy or multivalued logic is all about elementhood.
What happens when one set contains another set? Put a little box in a big box. Then the big box contains the little box. It contains the little box 100%. Can that containment take on degrees? Sure it can. Put the little box only half way in the big box. Then the big box contains the little box only 50%. The little box is both in and outside the big box.
I called this fuzzy containment subsethood, the degree to which one set is a subset of another set. Traditional fuzzy theory assumed subsethood was bivalent, all or none, 100% or 0%. That seemed as extreme as any other black-white claim. Very tall men made up a 100% subset of tall men. That I could buy. Every very tall man is tall. But the old view said that tall men made up only a 0% subset of very tall men. That I could not buy. It was a matter of degree. Every tall man is very tall to some degree, often to a very small degree.
In hindsight subsethood was the next step from the elementhood of fuzzy sets. Elementhood puts balls in boxes. It puts them in to some degree. Subsethood puts boxes in boxes. In the special case the input box shrinks to a ball and you get back the old idea of a vague or fuzzy set whose elements belong to it to some degree. So subsethood subsumes elementhood as a special case. Yet subsethood differs from elementhood. Subsethood holds between sets. Elementhood holds within sets. (My coloring)
According to the dictionary, to contain is to hold or accommodate. That's the key. All three terms -- contain, hold, accommodate -- mean to have within or to have the capacity for having within. But there is nothing explicit in the meaning that says that holding within has to be totally within. The holding could be only partial and still satisfy the definition. So you can readily accept the idea of one set being only partially in another set. The range from none to all would then be from 0% to 100%. Therefore the degree of containment would range from 0 for 0% to 1 for 100%).
Another way of looking at this is that you don't have to round off. This is turning the issue around. You don't need to have a sharp dividing line between in and out. If a number is 23.7, you don't have to round off and say the number is 24. That is, you don't have to force the number into the class 24. If 87% of the elements of a set are in another set, you don't have to resort to a bivalent truth-value system that rounds up to say it's true that the first set is entirely in the second set. You can say that it's partially true that 23.7 is 24 or that it's partially true the first set is in the second set. This makes the idea of a fuzzy set much more palatable.
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How to Define Fuzzy Sets
In ordinary set theory a set is defined in terms of its members. We say that a set, A, is the set of elements, x -- given the proviso that x satisfies some function, f(x). That is:
A = {x | f(x)}
Or A is the set of elements, x, such that f(x).
Finite Sets
To deal with fuzzy sets, the idea of partial containment has to be considered. An example is given by Kurt J. Schmucker (in Fuzzy Sets, Natural Language, Computations, and Risk Analysis). In his example the universe is the set {a, b, c, d, e, f}, and one possible fuzzy subset, A, could be defined in the following way:
a is present with degree of membership 1.0
b is present with degree of membership .9
c is present with degree of membership .2
d is present with degree of membership .8
e is present with degree of membership 1.0
f is present with degree of membership 0
This longwinded expression can be written equivalently as:
A = {1/a, .9/b, .2/c, .8/d, 1/e}
where each element is juxtaposed next to its degree of containment and elements with 0 degree containment are omitted. For example:
Infinite Sets
The same general idea applies to infinite sets as applies to finite sets. But now, of course, the elements and their degree of membership can't be listed, so they have to be defined implicitly, using functions. This can be written as the set:
Y = {m(x)/x | f(x)}
In this definition, m(x) is the function -- the rule -- identifying the degree of membership of each x in Y. And f(x) is the function that defines the universe of elements making up the membership of x.
One example, borrowed from Kurt Schmucker, is as follows:
Y = {m(x)/x | x > 0}
where
m(x) = 1.0 for x greater than 0 and less than or equal to 25
m(x) = 1/(1 + ((x - 25)/5)) for x greater than 25
and f(x) specifies the positive real numbers:
x > 0
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Operations for Fuzzy Sets
As with ordinary sets, the operations involving fuzzy sets include the union, intersection, and complement, but now it is less clear how the operations are to be defined.
Set Inclusion
First let's consider the general problem of set inclusion. What does it mean for a fuzzy set to be included in another fuzzy set?
Sets are defined in terms of their members, as we see here, and subsets are collections of the members. This is the case whether we're dealing with ordinary sets or fuzzy sets, but with fuzzy sets we have to take the degree of membership into account. Say the set, A, is contained in set, B. If an element, a, is a member of A, to degree m(a), is it a member of B to the same degree? Not necessarily.
Let's take as an example the set of rose-colored apples, A, and the set of red apples, B. In ordinary sets, rose colored apples would form a subset of red apples and we would say that the former is included in the latter. All rose apples are red apples. A Ì B. This is the traditional form of set inclusion. A is entirely contained in B. All the members of A are in B. If x is a member of A, x is also a member of B.
In fuzzy sets, however, one set, A, may be only partially included in another set, B. So a member with a certain degree of membership in A would be likely to have a different degree of membership in B. A light rose apple might be in A to a medium degree, but in B only to a very slight degree. Light rose apples are OK in A, but they barely qualify for membership in B.
Union of Fuzzy Sets
The operations for sets are ways to combine sets, or ways to generate new sets out of old ones. Operations on sets correspond to the operations you find in arithmetic, which you know as addition, subtraction, multiplication, and division. There are also operations in logic that do similar kinds of things. The difference is that set operations deal with sets, and arithmetic operations work on numbers, and logic operations have to do with combining true or false statements. In set theory, then, you find the operators, È (read as union, join, or sum), Ç (read as intersection, meet, or product), and Comp (or complement).
In ordinary sets the union of sets A and B is the set of all members that are either in A or in B and is written:
A È B
So, if x is a member of A, then x is a member of A È B, and if y is a member of B, then y is a member of A È B.
With fuzzy sets, though, x and y would each have their own degree of membership in their respective sets, so their degree of membership in the union isn't predetermined. If x is in A to degree .7, it needn't be in A È B to the same degree. Indeed, A itself could be in A È B only to some degree. Similarly with B in A È B.
Following Schmucker in a formulation proposed by L. A. Zadeh, the union is expressed as:
A È B = {max(a(x), b(x))/x | x is an element of U}
In this formulation a(x) is the degree of membership of x in A, and b(x) is the degree of membership of x in B. To get the membership of x in A È B, you have to compare the membership of x in each of the two sets, A and B, and take the larger of the two values as the degree of membership of x in A È B.
Intersection of Fuzzy Sets
Similarly, the intersection of A and B in ordinary sets is the set of all members of A that are also members of B and is written as:
A Ç B
So, if x is a member of A and if x is a member of B, then x is a member of A Ç B.
In fuzzy sets, however, x may not have the same degree of membership in the two sets taken separately, so what might be the degree of membership in their intersection? The answer isn't obvious. But here's how Zadeh proposed to define it:
A Ç B = {min(a(x), b(x))/ | x is an element of U}
Since the degree of membership of x in A could be different than the degree of membership of x in B, you have to compare them and use the smaller of the two values to define the degree of membership of x in the intersection.
Complement of a Fuzzy Set
The complement of a set is an operation that's performed only on one set -- not two, as in the union and intersection. That unary operator is known as complementation, Comp(A). Given a set, A, you take its complement relative to the universe of discourse, U. So, if A consists of the set {x}, with x drawn from U, then the complement of A is:
Comp(A) = {x | x Ï A}
which consists of all the members of U that are not in A.
That's okay for ordinary sets, but what about fuzzy sets? If x is a member of A to some degree a(x), then it has to be in the complement of A for the remaining degree. That is, x would be in Comp (A) to degree (1 - a(x)). So the complement of A would be the set:
Comp(A) = {(1 - a(x))/x | x is in A}
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Characteristic Function
The characteristic function of fuzzy sets generalizes on the characteristic function of ordinary sets. Like that for ordinary sets, it is a mapping from the whole of the universe, U. But it differs from the function for ordinary sets in that, now, the mapping goes to a portion of the real line from 0 to 1 instead of just to the points 0 and 1. For a fuzzy subset A, the line portion is expressed using solid brackets:
CA(x): U ® [0, 1]
So the characteristic function is the degree of membership function, m(x).
Recall that 1 and 0 were associated with the truth values true and false. Since any segment of the real line has an infinite number of points, the mapping of the fuzzy characteristic function goes to an infinite number of points, each of which defines a degree of membership in the set.
A similar association with truth values therefore gives us an infinite number of them, so we now have an infinite number of possible truth values, which can range anywhere from 1 to 0.
A modified fuzzy clustering algorithm for operator independent brain tissue classification of dual echo MR images.
Suckling J, Sigmundsson T, Greenwood K, Bullmore ET.
Department of Health Care of the Elderly, King's College School Medicine and Dentistry, London, UK. j.suckling@iop.kcl.ac.uk
Methods for brain tissue classification or segmentation of structural magnetic resonance imaging (MRI) data should ideally be independent of human operators for reasons of reliability and tractability. An algorithm is described for fully automated segmentation of dual echo, fast spin-echo MRI data. The method is used to assign fuzzy-membership values for each of four tissue classes (gray matter, white matter, cerebrospinal fluid and dura) to each voxel based on partition of a two dimensional feature space. Fuzzy clustering is modified for this application in two ways. First, a two component normal mixture model is initially fitted to the thresholded feature space to identify exemplary gray and white matter voxels. These exemplary data protect subsequently estimated cluster means against the tendency of unmodified fuzzy clustering to equalize the number of voxels in each class. Second, fuzzy clustering is implemented in a moving window scheme that accommodates reduced image contrast at the axial extremes of the transmitting/receiving coil. MRI data acquired from 5 normal volunteers were used to identify stable values for three arbitrary parameters of the algorithm: feature space threshold, relative weight of exemplary gray and white matter voxels, and moving window size. The modified algorithm incorporating these parameter values was then used to classify data from simulated images of the brain, validating the use of fuzzy-membership values as estimates of partial volume. Gray:white matter ratios were estimated from 20 twenty normal volunteers (mean age 32.8 years). Processing time for each three-dimensional image was approximately 30 min on a 170 MHz workstation. Mean cerebral gray and white matter volumes estimated from these automatically segmented images were very similar to comparable results previously obtained by operator dependent methods, but without their inherent unreliability.
PMID: 10463658 [PubMed - indexed for MEDLINE] What do ya mean fuzzy ??!!
Before illustrating the mechanisms which make fuzzy logic machines work, it is important to realize what fuzzy logic actually is. 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". As its name suggests, it is the logic underlying modes of reasoning which are approximate rather than exact. The importance of fuzzy logic derives from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature.
The essential characteristics of fuzzy logic as founded by Zader Lotfi are as follows.
* In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning.
* In fuzzy logic everything is a matter of degree.
* Any logical system can be fuzzified
* In fuzzy logic, knowledge is interpreted as a collection of elastic or, equivalently , fuzzy constraint on a collection of variables
* Inference is viewed as a process of propagation of elastic constraints.
The third statement hence, define Boolean logic as a subset of Fuzzy logic.
Fuzzy Sets
Fuzzy Set Theory was formalised by Professor Lofti Zadeh at the University of California in 1965. What Zadeh proposed is very much a paradigm shift that first gained acceptance in the Far East and its successful application has ensured its adoption around the world.
A paradigm is a set of rules and regulations which defines boundaries and tells us what to do to be successful in solving problems within these boundaries. For example the use of transistors instead of vacuum tubes is a paradigm shift - likewise the development of Fuzzy Set Theory from conventional bivalent set theory is a paradigm shift.
Bivalent Set Theory can be somewhat limiting if we wish to describe a 'humanistic' problem mathematically. For example, Fig 1 below illustrates bivalent sets to characterise the temperature of a room.
The most obvious limiting feature of bivalent sets that can be seen clearly from the diagram is that they are mutually exclusive - it is not possible to have membership of more than one set ( opinion would widely vary as to whether 50 degrees Fahrenheit is 'cold' or 'cool' hence the expert knowledge we need to define our system is mathematically at odds with the humanistic world). Clearly, it is not accurate to define a transiton from a quantity such as 'warm' to 'hot' by the application of one degree Fahrenheit of heat. In the real world a smooth (unnoticeable) drift from warm to hot would occur.
This natural phenomenon can be described more accurately by Fuzzy Set Theory. Fig.2 below shows how fuzzy sets quantifying the same information can describe this natural drift.
The whole concept can be illustrated with this example. Let's talk about people and "youthness". In this case the set S (the universe of discourse) is the set of people. A fuzzy subset YOUNG is also defined, which answers the question "to what degree is person x young?" To each person in the universe of discourse, we have to assign a degree of membership in the fuzzy subset YOUNG. The easiest way to do this is with a membership function based on the person's age.
young(x) = { 1, if age(x) <= 20,
(30-age(x))/10, if 20 < age(x) <= 30,
0, if age(x) > 30 }
A graph of this looks like:
Given this definition, here are some example values:
Person Age degree of youth
--------------------------------------
Johan 10 1.00
Edwin 21 0.90
Parthiban 25 0.50
Arosha 26 0.40
Chin Wei 28 0.20
Rajkumar 83 0.00
So given this definition, we'd say that the degree of truth of the statement "Parthiban is YOUNG" is 0.50.
Note: Membership functions almost never have as simple a shape as age(x). They will at least tend to be triangles pointing up, and they can be much more complex than that. Furthermore, membership functions so far is discussed 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 YOUNG depend on both a person's age and their height (Arosha's short 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.
Fuzzy Set Operations.
Union
The membership function of the Union of two fuzzy sets A and B with membership functions and respectively is defined as the maximum of the two individual membership functions. This is called the maximum criterion.
The Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebra.
Intersection
The membership function of the Intersection of two fuzzy sets A and B with membership functions and respectively is defined as the minimum of the two individual membership functions. This is called the minimum criterion.
The Intersection operation in Fuzzy set theory is the equivalent of the AND operation in Boolean algebra.
Complement
The membership function of the Complement of a Fuzzy set A with membership function is defined as the negation of the specified membership function. This is caleed the negation criterion.
The Complement operation in Fuzzy set theory is the equivalent of the NOT operation in Boolean algebra.
The following rules which are common in classical set theory also apply to Fuzzy set theory.
De Morgans law
,
Associativity
Commutativity
Distributivity
Glossary
Universe of Discourse
The Universe of Discourse is the range of all possible values for an input to a fuzzy system.
Fuzzy Set
A Fuzzy Set is any set that allows its members to have different grades of membership (membership function) in the interval [0,1].
Support
The Support of a fuzzy set F is the crisp set of all points in the Universe of Discourse U such that the membership function of F is non-zero.
Crossover point
The Crossover point of a fuzzy set is the element in U at which its membership function is 0.5.
Fuzzy Singleton
A Fuzzy singleton is a fuzzy set whose support is a single point in U with a membership function of one.
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