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)

Upcoming lectures
 Probabilities
and Uncertainty Techniques
 Tactical and Strategic AI