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: VERY(Truth(A)) = Truth(A)2

NOT_VERY(Truth(A)) = Truth(A)0.5

(Truth(A) is the degree of membership of A in some fuzzy set)

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 OR B) = MAX(Truth(A), Truth(B))

Logical AND (conjunction) Truth(A 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: 0.2Do nothing to degree: 0.4Flee 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)

…

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Probabilities and Uncertainty Techniques Tactical and Strategic AI