fuzzy reasoningmark shelton | merrick cloete saman majrouh | sahithi jadav

this presentation

fuzzy set theory

graduation

granulation

fuzzy control

strengths

limitations

applications(to be continued)

fuzzy logics

“as the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until […] precision and significance become almost mutually exclusive characteristics.” - Zadeh, 1965

fuzzy set theory

fuzzy sets

‘classical’ sets are called crisp sets - membership values of 0 or 1 only

a set where each element has a degree of membership

a membership function converts values into grades of membership

fuzzy logic is a more ‘human’ approach to computation.it involves two main concepts:

graduationgranulation

granulation

inputs are then grouped together, e.g. cold, lukewarm, warm, hot

inputs are drawn together by similarity, proximity or functionality

we don’t know exactly where each object starts and ends

graduation

the designer decides what constitutes as ‘cold’, as well as all degrees of it.

everything is a matter of degree, e.g. not cold, a bit cold, a lot cold

we assign a value between 0 and 1, e.g 0.7 is hot, 0.3 is cold

fuzzy controlIn a single cycle, the system read all inputs

each option is weighted and used to output the result

rather than select a single option to evaluate, the system evaluates all options

fuzzy operators

conjunction(P AND Q)

union(P OR Q)

zadeh operator

probabilisticoperator

boundedoperator

min(P, Q) P x Qmax

(0, P + Q - 1)

max(P, Q) P + Q – P x Q

min(1, P + Q)

1 if tv(P) ≤ tv(Q) else 0

min(1, 1 – tv(P) + tv(Q))

max(1- tv(P), min(tv(P),

tv(Q)))implication

(IF P THEN Q)

example

system reads the temperature as 0.9 cold, 0.1 warm, 0.0 hot

if cold, set heater to onif warm, set heater to off

system sets heater to on 90% of the time and off 10% of the time within a cycle

fuzzy logicsMost fuzzy logic systems are variations on t-norm fuzzy logics. A t-norm is a continuous function that satisfies the following properties between 0 and 1:

commutativityT(a, b) = T(b, a)

monotonicityT(a, b) ≤ T(c, d) if a≤ c and b ≤ d

associativityT(a, T(b, c)) = T(T(a,b), c)

identity T(a, 1) = a

fuzzy logicsSome of the types of fuzzy logics are:

monoidal left continuous t-norms

basic continuous t-norms

product for strong conjunction: Tprod(a, b) = a b

pavelka’s stems from Lukasiewicz, each formula has an evaluation

But today we are going to focus on two key types – lukasiewicz and godel

fuzzy logicslukasiewicz logic is similar to a basic t-norm

Tluk(a, b) = max(0, a+b-1)

https://en.wikipedia.org/wiki/T-norm

fuzzy logicsgodel is the minimum t-norm and is the standard for weak conjunction.

Tmin(a, b) = min(a,b)

https://en.wikipedia.org/wiki/T-norm

strengths

convenient user interface with easy end-user interpretation

can model problems with imprecise and incomplete data, and nonlinear functions of arbitrary complexity

corresponds well withhuman perceptions

limitationsrequires ad-hoc tuning of

membership functions

may not scale well to large or complex problems

deals with imprecision and vagueness, but not uncertainty

applications

coal powerplant

refuse disposalplant

water treatmentsystem

ac inductionmotor

frauddetection

conclusion

fuzzy reasoning

binarycomputation

humanexperience

natural languageartificial intelligencebiotechnology

fuzzy reasoningmark shelton | merrick cloete saman majrouh | sahithi jadav