Transcript Fuzzy Logic
FUZZY LOGIC
Fuzzy Logic
Lotfi Zadeh (professor at UC Berkeley)
wrote his original paper on fuzzy set
theory. In various occasions, this is what
he said…
“Fuzzy
logic is a means of presenting
problems to computers in a way akin to the
way humans solve them”
“The essence of fuzzy logic is that
everything is a matter of degree”
What do these statements really mean?
Fuzzy Logic
Very often, we humans analyze situations and solve
problems in a rather imprecise manner
Do
not have all the facts
Facts might be uncertain
Maybe we only generalize facts without having the
precise data or measurements…
Real-life example: Playing a game of basketball
Everything is a matter of degree?
Is your basketball opponent tall, or average or
short? (use of linguistic terms to measure degree)
Is 7 feet tall? Is 6 feet 10 inches tall? Are they both
considered tall? (overlapping degrees)
Problem with traditional Boolean logic
You
are forced to define a point above which we will
consider the guy to be tall or just average, e.g. > 7 ft
Fuzzy Logic allows gray areas or degrees of being
considered “tall”
The degree of truth
So…you can think of fuzzy logic as classifying
something as being TRUE, but to varying degrees
Real-life control applications (air-conditioning,
household appliances):
Traditional
Boolean logic will result in abrupt switching
of response functions
Fuzzy logic alleviates this problem Responses will
vary smoothly given the degree of truth or strength of
the input conditions
Fuzzy logic for games
A previous game AI example…
An
AI character makes his decision to chase (using FSM
or DT) based on traditional Boolean logic, e.g. distance
of player < 20 units, and player health < 50%
In fuzzy logic, we can represent these input conditions
using a few “membership” degrees of measure
Distance:
(“Far”, “Average”, “Near”)
Health: (“Good”, “Normal”, “Poor”)
The
output actions can also be represented with
different membership degrees (“Chase Fast”, “Chase
Slow”)
How to use Fuzzy Logic in Games?
3 possible ways how fuzzy logic can be used in
games
Control
Modulating
steering forces, travelling/moving towards
target
Threat
Assessment
Assessing
player’s strengths/weaknesses for deploying units
and making moves
Classification
Identifying
the combat prowess of characters in the game
based on a variety of factors in order to choose opponent
There are many other possibilities…
Fuzzy Logic Basics
Fuzzy control or fuzzy inference process – 3 basic
steps
Step 1: Fuzzification
Fuzzification: Process of mapping/converting crisp
data (real numbers) to fuzzy data
Find
degree of membership of the crisp input in
predefined fuzzy sets
E.g. given a character’s health, determine the degree to
which it is “Good”, “Fair” or “Poor”.
Mapping is achieved using membership functions
Membership Functions
Membership Functions
Map
input variables to a degree of membership, in a
fuzzy set, between 0 and 1. Degree 1 absolutely
true, degree 0 absolutely false, any degree in
between true or false to a certain extent
“Boolean logic membership function”
Membership Functions
Fuzzy Membership Functions
Enables
us to transition gradually from false to true
Grade membership function
Membership Functions
Triangular m/f
Reverse grade m/f
Equations
are just the
inverse of the grade
m/f
Membership Functions
Trapezoid m/f
Other nonlinear m/f
Gaussian
or Sigmoid
‘S’-shaped curves
Membership Functions
Typically, we are interested in the degree of which
an input variable falls within a number of
qualitative sets
Membership Functions
Setting
up collections of fuzzy sets for an input variable
is a matter of judgment and trial-and-error not
uncommon to “tune” the sets
While tuning, one can try different membership
functions, increase or decrease number of sets
Some fuzzy practitioners recommend 7 fuzzy sets to
fully define a practical working range (?!?!?)
Membership Functions
One rule of thumb for ensuring smooth transitions (in
later steps) is to enforce overlapping between
neighboring sets
Hedge Functions
Hedge functions are sometimes used to modify the
degree of membership
Provide additional linguistic constructs that you can
use in conjunction with other logical operations.
Two common hedges:
= Truth(A)2
NOT_VERY(Truth(A)) = Truth(A)0.5
(Truth(A) is the degree of membership of A in some fuzzy
set)
VERY(Truth(A))
Step 2: Fuzzy Rules
Next, construct a set of rules, combining the input in
some logical manner, to yield some output
If-then style rules (if A then B) – A being the
antecedent/premise and B being the
consequent/conclusion
Fuzzy input variables are combined logically to
form premise
Conclusion will be the degree of membership of
some output fuzzy set
Fuzzy Axioms
Since we are writing “logical” rules with fuzzy input,
we need a way to apply logical operators to fuzzy
input (just like with Boolean input)
Logical OR (disjunction)
Truth(A
Logical AND (conjunction)
Truth(A
OR B) = MAX(Truth(A), Truth(B))
AND B) = MIN(Truth(A), Truth(B))
Logical NOT (negation)
Truth(NOT
A) = 1 – Truth(A)
Fuzzy Axioms
Example, given a person is overweight to the
degree of 0.7 and tall to the degree of 0.3:
Overweight
AND tall = MIN(0.7, 0.3) = 0.3
Overweight OR tall = MAX(0.7, 0.3) = 0.7
NOT overweight = 1 – 0.7 = 0.3
NOT tall = 1 – 0.3 = 0.7
NOT(overweight AND tall) = 1 – MIN(0.7, 0.3) = 0.7
There are other definitions for these logical
operators…
Rule Evaluation
Unlike traditional Boolean logic,
Rules
in fuzzy logic can evaluate into any number
between 0 and 1 (not just 0 or 1)
All rules are evaluated in parallel (not in series that the
first one that is true gets fired). Each rule always fires,
to various degrees
The strength of each rule represents the degree of
membership in the output fuzzy set
Rule Evaluation
Example: Evaluating
whether an AI should
attack player
Rules can be written like:
If
(in melee range AND
uninjured) AND NOT hard
then attack
Set up as many rules to
handle all possibilities in
the game
Rule Evaluation
Given specific degrees for the input variables, you
might get outputs (conclusions of the rules) that look
something like this:
Attack to degree: 0.2
Do nothing to degree: 0.4
Flee to degree: 0.7
The most straightforward way to interpret these
outputs is to take the action associated with the
highest degree (in this case, the action will be flee)
Step 3: Defuzzification
In some cases, you might want to use the fuzzy
output degree to determine a crisp value (real
number), which can be useful for further calculations
Defuzzification: Process of converting the results
from the fuzzy rules to get a crisp number as an
output
Opposite of fuzzification (you can say that,
although the purpose and methods are different!)
Step 3: Defuzzification
Previous example: Instead of determining some
finite action (do nothing, flee, attack), we also want
to use the output to determine the speed to take the
action
To get a crisp number, aggregate the output
strengths on the predefined output membership
functions
Step 3: Defuzzification
With the numerical output from the earlier example
(0.2 degree attack, 0.4 degree do nothing, 0.7
degree flee), we have the composite membership
function below
Defuzzifying composite m/f
Truncate each output set to the output degree of
membership for that set. Then combine all output sets
by disjunction
A crisp number can be arrived from such an output
fuzzy set in many ways
Geometric
centroid of the area under the output fuzzy set,
taking its horizontal axis coordinate as the crisp output
Using “predefuzzified” output
A less computationally expensive method is the use
of singleton output membership function or a
“predefuzzified” output function
Instead of doing lots of calculation, assign speeds to
each output action (-10 for flee, 1 for do nothing,
10 for attack).
E.g. The resulting speed for flee is simply the preset
value of -10 times the degree to which the output
action flee is true (-10 x 0.7 = -7)
Using “predefuzzified” output
Aggregate of all outputs with a simple weighted
average
In our example, we might have:
Output = [(0.7)(-10) + (0.4)(1) + (0.3)(10)] /
(0.7+0.4+0.3)
= -2.5
This output would result in the creature fleeing, but
not earnestly in full extent
Naturally, we can obtain various output (crisp)
values depending on the different input conditions
Further Examples
There are 2 good examples in the textbook,
showing the full process of using fuzzy logic to
model game AI characters
Using Fuzzy Logic in FSMs?
If we want to add some fuzzy logic into FSMs, how
can that be accomplish? Is it possible?
Remember:
Each state defines a behavior or action, and
each state is reached by transition from another state
on the basis of fulfilling some input conditions…
Conditions for transition are normally in Boolean logic,
how do we accommodate fuzzy logic?
Fuzzy State Machines
Different AI developers regard Fuzzy State
Machines differently
State
machine with fuzzy states
State transitions that use fuzzy logic to trigger
Both
Find out more about how these different variations
can be worked out and implemented (refer to
Millington book)
…
Next up
Homework
2 (due in Week 10, submission via mail)
Milestone #2 (due in Week 11, 11.00am 23/8, Thurs)
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