Fuzzy LogicFuzzy Logic

Logical and Artificial Intelligence in Games

Lecture 13

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Set Logic ExtendedSet Logic Extended

Fuzziness can be seen as an extension to Boolean set theory

In classic logic, all set are crisp (true or false)

Numbers are crisp also – meaning each concept is perfectly known and expressed as such

E.g., health = 68, armor = 41

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When is a pile of chalk big?When is a pile of chalk big?

If we have three pieces of chalk in the room is that considered a big pile of chalk?

Some people might say, yes that is a big pile and some would not.

Someplace between those three pieces of chalk and a whole room full of chalk the pile of chalk turns from a small pile into a big pile.

This could be a different spot for different people.

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Fuzzy Set TheoryFuzzy Set Theory

In ordinary set theory every element “x” from a given universe is either in or out of a set S

x Sx S

In fuzzy set theory set membership is not so easily determined

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Membership FunctionMembership Function

A: X [0,1]

x A(x) x X

A is the membership function

X is universe of discourse

Diagram below shows different fuzzy membership functions

A – triangular

B – Curve

C – Crisp Singleton

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Collection of SingletonsCollection of Singletons

Collection of ordered pairs (x, A(x)) known as a fuzzy singleton

Consider a set A defined as the set of hostile game objects from X

X = {door, gun, fred, bot1}

a = [0 0 0.3 0.8]

A = {(door,0), (gun,0), (fred, 0.3), (bot1, 0.8)}

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  Fuzzy SetTheoretic

Operations

Probability TheoryIndependent

Events

A(x) a a

B(x) b b

not A(x)complememt

1-a 1 - a

A and Bintersection

min(a,b) a * b

A or Bunion

max(a,b) a + b – a*b

A = Bequality

x: A(x) = B(x)

A Bsubset

x: A(x) < B(x)

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Fuzzy Modifiers - 1Fuzzy Modifiers - 1

In “very dangerous” the word “very” could be thought of as a fuzzy linguistic modifier

dangerous = [0 0.1 0.4 0.8 1]

very dangerous = [0 0.01 0.16 0.64 1]

More formally we could think of a fuzzy modifier m as a function that converts one fuzzy set a into another set m(a)

m: X F(X)

a m(a) | a F(X)

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Fuzzy Modifiers - 2Fuzzy Modifiers - 2

Possible fuzzy definitions (restrictive)

very(a) = a2

extremely(a) = a3

moreorless(a) = a1/2

slightly(a) = a1/3

exactly(a) = a+

always(a) = a0

Adverbs might be used as offset modifiers

A(x) = m(A(x + k))

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Fuzzy Variables and ValuesFuzzy Variables and Values

Fuzzy variables can be considered as a term associated with a floating point value

A collection of fuzzy variables could be used to define an enemy profile

{(healthy 0.9) (dangerous 0.3) (moving 0.3)}

Linguistic variables is a collection of fuzzy variables used to de

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Liguistic VariablesLiguistic Variables

Linguistic variables is a collection of fuzzy variables used to define a high level concept like “allegiance”

term set: {enemy,neutral,friendly}

values: [0.0, 0.4, 0.9]

A linguistic variable is defined over a base variable

Base variables can be either crisp or fuzzy values

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FuzzificationFuzzification

Creating a fuzzy variable from a crisp one

Define a membership function with the same range as the crisp variable

Evaluating the membership based on the crisp variable value indicates degree of set membership

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Defuzzification - 1Defuzzification - 1

Converts a fuzzy value to a crisp value

Fairly easy for fuzzy variables

Can be tricky for linguistic values

Process often involves information loss, unless you are doing simple conversions

There is no unique equation, since humans use problem-dependent techniques to defuzzify linguistic concepts

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Defuzzification - 2Defuzzification - 2

Maximum Approach

Uses highest points in membership function graph

First maxima – picks crisp value from highest of all membership function

Average maxima – weighted or unweighted average of all crisp values

Centroid Approach

based on finding center of gravity of the membership function graph

Very computationally intensive

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Fuzzy Logical OperatorsFuzzy Logical Operators

Operator Notation Equation

NOT 1 – x

AND min(x, y)

OR max(x, y)

IMPLIES max(1 - x, y)

max(0, x + y – 1)

The concepts of fuzzy logic are similar to those used in fuzzy set theory.

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Linguistic ExpressionsLinguistic Expressions

Dead or alive

Extremely nasty and more or less dead

Shot and bleeding badly implies dying

d a

n3 d1/2

s b2 d

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Fuzzy LogicFuzzy Logic

Fuzziness is really deterministic uncertainty and depends on probability assumptions which may not be met

Fuzzy logic contains inconsistencies

(e.g. the set of animats that are both dead and alive is not empty)

Seems to model human knowledge and reasoning better than standard truth functional logic

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Fuzzy Control SystemsFuzzy Control SystemsDecision MakingDecision MakingWorking Memory

Similar to RBS, except fuzzy variables used in place of crisp symbols

Knowledge Base

Rules containing fuzzy variables are used, e.g.,

IF very confident or somewhat vulnerable

THEN attack

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Rule ExtensionsRule Extensions

Linguistic rules

Keyword is used to refer to fuzzy terms

if allegiance is friendly then happy

Fuzzy modifiers

Can be applied to both antecedent and consequent variables

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Creating Membership FunctionsCreating Membership Functions

Knowledge engineer can try to extract information and heuristics from human experts

Survey can be used to collect information regarding several people’s understanding and membership function reflects average

Incremental tuning will be required for any approach chosen

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Fuzzy InterpreterFuzzy InterpreterMatchingMatchingIn crisp systems it is easy to identify rules that match WM, facts are present or they are not

In fuzzy systems, fuzzy variables are rarely zero so every rule needs to be checked during every match cycle

To reduce impact of combinatorial explosion, thresholds will be used to reduce the number of low probability matches

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Fuzzy InterpreterFuzzy InterpreterInferenceInferencePurpose is to determine the degree of truth for the consequent part of each rule

Two techniques used

Min – uses the minimum of the rule antecedent values and treats this as the value for the output set – possibilistic reasoning

Product – uses the product of the rule antecedent values and treats this as the value for the output set – probabilistic reasoning

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Fuzzy InterpreterFuzzy InterpreterCompositionCompositionMany rules have the same fuzzy variable in the rule body (consequent)

To combine support shown by multiple rules

Max – uses the maximum of the rule output values and uses this as the fuzzy variable value in WM – possibilistic reasoning

Sum – uses the sum of the rule output values and uses this as the fuzzy variable value in WM – probabilistic reasoning

A B

Sum

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AlgorithmAlgorithm

// current WM with latest fuzzy variable values

// previous is copy of most recent WM

for each value in WM

{

previous[value] = current[value];

current[value] = 0;

}

for each rule

{

result = evaluate(rule.expression,previous);

current[rule/body] = max(current[rule.body},result);

}

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Discussion – 1Discussion – 1

Fuzzy systems are surprisingly robust

Can cope with noisy and incomplete data

Requires little feedback to perform well

Fuzzy expert systems mimic human reasoning surprisingly well

Fuzzy RBS are easy to extend incrementally

Rule syntax is easy enough for non-programmers to write rules

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Discussion – 2Discussion – 2

Fuzzy systems use rules of thumb (heuristics) and not mathematical models of the problem

Every rule needs to be evaluated during every match cycle which is computationally expensive (for a large number of rules)

Creating membership functions is not easy

Combinatorial explosion of rules is possible for large numbers of input combinations, unless knowledge is represented implicitly