Fuzzy Logic
By Andrew Pro
References
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Alexander, Thor, “An Optimized Fuzzy Logic Architecture For Decision Making”,
AI Game Wisdom.
Bonissone, P. Piero and Cheetham, William, “Fuzzy Case Based Reasoning For
Residential Property Valuation” (http://www.rpi.edu/~bonisp/fuzzycourse/99/L10/fuzzycbr4realestate.pdf)
Hansen, Bjarne K., and Riordan, Denis, “Weather Prediction Using Case-Based
Reasoning and Fuzzy Set Theory”
(http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.16.3814)
Hellman, M., “Fuzzy Logic Introduction”
(http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.85.9757&rep=rep
1&type=pdf)
Shwab, Brian, AI Game Engine Programming.
(http://site.ebrary.com/lib/lehighlibrary/docDetail.action?docID=10074871&f
orce=1)
Zarozinski, Michael, “Imploding Combinatorial Explosion In A Fuzzy System”,
Game Programming Gems 2.
http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/sbaa/report.html
Outline
• Introduction to Fuzzy Logic
• Fuzzy Game AI
• Fuzzy Logic in CBR
Introducing Fuzzy Logic
• Takes into account the approximate nature of
human reasoning
– If the light is turned off, the room will be dark.
How dark?
• Everything in Fuzzy logic is a matter of degree
– Binary logic is fixed at a degree of 0 or 1, where
fuzzy truth values can be anywhere in the range of
[0,1]
Paradox of Self-Reference (just for fun)
• Liar paradox – “This sentence is false.”
– In binary logic, X cannot equal ¬X.
– We can use fuzzy logic and declare X to be (.5)
false, so ¬((.5)X) = (.5)X. “This sentence is (half)
false.” is (half) true.
• Epimenedes – A Cretan says: “All Cretans are
liars.”
– Perhaps he means with a truth value of .999
Fuzzy Membership Functions
• Fuzzy Logic extends multi-valued logic
• A function is defined for each logic value
• Instead of values being mutually exclusive like
in a normal logic system, the logic values
overlap .
Fuzzy Membership Functions
• Functions can be simple or complex (fuzzily
speaking of course)
• Which functions could describe the fuzzy
value, Tall?
Fuzzy Sets and Operations
• Fuzzy Set defined by function.
– COLD(x) =
– Negation
¬COLD(x) =
1 if temp(x)<= 50
(60 – temp(x))/10 if 50<temp(x)<60
0 if temp(x)>=60
0 if temp(x)<= 50
(temp(x)-60)/10 if 50<temp(x)<60
1 if temp(x)>=60
• The function and its inverse overlap!
Fuzzy Set Operation
• A AND B
at x=4.75
A AND B = min(A,B)
= .25
• A OR B
at x=4.75
A AND B = max(A,B)
= .75
Fuzzy Logic or Probability Theory
• Fuzzy logic deals with the measure of how
much a variable is in a particular set.
• Probability theory handles subjective
probability where a function may map how
likely a variable is to be in a particular set.
Fuzzy Logic in Game AI
• Fuzzy logic can be used in a variety of ways to
help game AI seem more human-like
• NPC Finite State Machines can be improved by
making fuzzy determinations for state
transition logic
Example NPC FSM
• Character stays in state until distance value is
crossed
FSM Rules
• FSM transition rules: State Chase
– IF player is close THEN attack
– ELSE IF player is far THEN returnHome
– ELSE chase
• Close and far are crisply defined
– Is necessary if distance is the only state transition
measure
Fuzzy Rules cont’d
• Fuzzy rules might be
– IF player is close THEN attack
– IF player is far THEN returnHome
– IF player is inbetween THEN chase
• Combined with other factors (weapon range,
time of day, weather condition) a distance
measured close may not be close enough
• The NPC now has more varied, interesting
behavior
Fuzzy AI Decision
• Rules are created using
every combination of 1
set from each variable.
• Rules may be difficult to
produce. Variables can
be given weights or rule
base can be generated
by expert.
Fuzzy AI Decision cont’d
• Each rule has an aggressiveness value (or truth
value) using the Fuzzy AND operation.
• All-out attack is the best choice because is has
the highest degree of truth, but there still is a
small part of Fight defensively that wants to be
considered. So…
Defuzzification
• Want to derive a numeric output value for
aggressiveness. One method is the center-ofmass method
• Take the maximum value from each output
variable set. (attack – 53, defend – 18, run – 0)
• Create a full output block by capping each
membership function at the final output
value, then OR the functions.
• Find the center of mass.
Combinatorial Explosion
• For real time AI, this becomes unpractical at a
certain point.
• Can compromise some of the functionality to
significantly reduce the number of rules.
The Combs Method
• Use rules based on each set’s relationship to
the output.
• Take output from each rule fired, OR the
matching sets to get the overall truth value for
each output set.
– (Run – 0, Defense – 83, Attack – 60)
• Different, but still reasonable
– Aggressiveness values fairly close after
defuzzification
Fuzzy AI
• Keep fuzziness in game AI on a small scale
• FuSM can be used where states don’t interfere
with eachother.
– Face modeling can be fuzzy to look more natural
Fuzzy CBR
• Think about CBR using fuzzy logic
– Case is similar or not similar
– Retrieval can be more accurate with fuzzy
attributes (…symbolic attributes)
– Case can be adapted using fuzzy set functions
based on differences between problem case and
retrieved case
Restaurant Example
Ex’ple
Bar
Fri
Hun
x1
x4
x5
x6
no
no
no
yes
no
yes
yes
no
yes
yes
no
yes
x7
x8
yes
no
no
no
x9
yes
x10
x11
yes
no
Pat
Alt
Type
wait
some
full
full
some
yes
yes
yes
no
French yes
Thai
yes
French no
Italian yes
no
yes
none
some
no
no
Burger no
Thai
yes
yes
no
full
no
Burger no
yes
No
yes
no
full
none
yes
no
Italian
Thai
no
no
Restaurant Example
• Bar / No Bar doesn’t really work, but
fuzzifying Hungry and Patrons could help
– (starving, pretty hungry, a little hungry)
• Patrons seems to be begging for fuzziness.
– Is the difference between None and Some really
just 1 patron?
– Does full actually mean there are no open tables
whatsoever?
Fuzzy CBR Systems
• WIND-1
– Weather prediction for airports. Expert identifies
fuzzy attributes (cloud ceiling, visibility, etc)
– Expert defines similarity thresholds to measure
sim between cases. Outputs are fuzzy. (Very near,
near, slightly near)
– Fuzzy predictor performs more accurately than
system finding k-NN using crisp attribute values.
Fuzzy CBR Systems
• PROFIT
– Fuzzy CBR system for real estate value estimation
that uses fuzzy logic in similarity computation,
case adaptation, and confidence value generation
– Sale comparison approach: “finding the most similar
houses, located close to the subject property, sold not too long ago;
and selecting a balanced subset of the most promising comparables to
derive the final estimate.”
Fuzzy CBR Systems cont’d (PROFIT)
• Similarity functions describing living area, lot
size, preferred #bedrooms, and preferred
#bathrooms
Fuzzy CBR Systems cont’d (PROFIT)
• Finds most similar cases, applies adjustment
rule set using differences to subject case.
Similarity rank could differ from net
adjustment rank. Discard cases with too much
price adjustment.
Final estimate = sum(adjPrice*sim)/sum(sim)
• Confidence assessment using aggregation of
fuzzy functions related to cases retrieved,
similarity, average price adjustment, etc.
Conclusion
• Fuzzy logic helps represent vague variables
and sets in a more natural
• Game AI can be altered using fuzzy state
transitions and/or fuzzy states to add variance
to NPCs
• In Case-Based Reasoning, the Retrieve and
Revise steps can benefit from having fuzzy
membership functions of case features.