Ragionamento in condizioni di incertezza:

Approccio fuzzy

Paolo Radaelli

Corso di Inelligenza Articifiale - Elementi

Vagueness

Many expressions of natural language allows partial degree of truthfulness

“Lisa is quite tall” This expression is true if Lisa has an height of

150 cm? Or 170 cm? or 190 cm? “This hotel is very nice, but a quite

expensive” How to measure the niceness? How to average the two truthfulness values?

Classical logic's limits

Dichotomic logic: a predicate can only be totally false or totally true. Truthfulness of a formula is known ill-suited to handle vagueness or uncertainness

Uncertain: reasoning about facts that aren't known with certainty Probabilistic reasoning, bayesian networks,...

Vague: reasoning about facts that are partially true Fuzzy logic approaches

Classical logic's limits

mound's paradox:

1.If I remove grain of sand from a mound, I obtain a mound again

2.But, if I remove all the sand's grains from the mound, I doesn't have a mound no more

3.How many grains I have to remove to obtain a not-mound?

By induction axiom, either (1) or (2) are wrong Even an empty mound is a mound There is a threshold between mounds and not-

mounds

Fuzzy sets

Complementary set

0

1

0,5

Union and Intersection

0

1

0,5

Fuzzy Set properties

These properties are true either in classical and fuzzy set theory: Symmetric law

Associative law

De Morgan's laws

Distributive law

A∪B ∪C=A∪ B∪C

A∩B ∩C=A∩ B∩C

A∪B=B∪A

A∩B=B∩A

¬ A∪B =¬A∩¬B

¬ A∩B =¬A∪¬B

A∩ B∪C =A∪B ∩ A∪C

A∪ B∩C =A∩B ∪ A∩C

Fuzzy Set properties

Excluded middle and non-contradiction laws aren't valid in Fuzzy set theory

For example, consider the case where f(x)=0,5

∀ x,x∈ A∨x∉A

¬∃ x,x∈ A∧x∉A

Subsethood and Entropy

Subsethood: measure “how much” a set A is a subset of B

Entropy: measure the “fuzziness” of a fuzzy set

A⊆ B:∀ x,μ A x ≤μ B x

S A,B =∫ μA∩B x dx

∫ μA x dx

E A =S A∪¬A,A∩¬A =∫ μA∩¬A x dx

∫ μA∪¬A x dx

Linguistic Modifiers

Linguistic Modifiers (aka hedges) are unary operators which alter a fuzzy set membership function

Different modifiers are grouped in families on the basis of the kind of alteration they represent

Concentrator and Dilators

Contrast intensifiers/dilators

Approximation

Restriction

Each family is defined on the terms of axioms that the modified set must satisfy

Concentrators/ Dilators

“very”, “extremely” (concentrators)

“quite”, “a little” (dilators)

∀ x . μ x ≥ μ' x

∀ x . μ x ≤ μ' x

μ' x =μ2 x

μ' x=μ3 xμ' x =μ x−K

Proposed way to handle concentrators:

Contrast intensifier and dilators

Used to transform a fuzzy set into a “crispier” (intensifiers) or a less crisp one (dilators)

Contrast Intensifiers: The entropy of the modified set must be lower than the original

set's entropy

values higher than 0.5 are reduced, while values lower than 0.5 are augmented

Linguistic terms: Surely, absolutely (for contrast intensifiers)

Usually, generally (for contrast dilators)

Contrast intesifier formula

μ' x ={ 2μ 2 x se μ x ≤ 12

1−2⋅1− μ x 2 se μ x 12}

Approximation modifiers

They transform a single element into a symmetric set centred on the element (e.g. “about 170 cm tall”), or enlarge the support of a fuzzy set

They lack a formal semantic about the effects of this modifier

Their opposite modifier (“exactly”) doesn't exists in standard fuzzy logic theory

Restriction modifiers

“More than”, “higher than”, “less than”

Restriction modifiers lack a formal definition about their effects

Generally, those modifiers aren't implemented in applications nor used in theoretical researches

Needs a deeper study about the perceived semantics of phrases like “more than good”

T-Norms

A family of mathematical functions

Properties: Symmetry

Associativity

Limit

Monotonicity

¿ : [ 0,1 ]× [ 0,1 ] [ 0,1 ]

a∗1=a;a∗0=0

a∗b=b∗a

a∗b ∗c=a∗ b∗c

a≥b a∗c≥b∗c

a∗b≤min a,b

S-norms

S-norms (or T-conorms) generalize union

Properties: Symmetry

Associativity

Limit

Monotonicity

For each norm, there is an associated conorm

a° 1=1 ;a° 0=a

a°b=b°a

a°b °c=a° b°c

a≥b a°c≥b°c

a°b=¬¬b∗¬b

T-Norms :some example

Minimum norm

Probabilistic norm

Lukasiewic's norm

a∗b=min a,b

a°b=max a,b

a∗b=a⋅b

a°b=a+b−a⋅b

a∗b=max 0,a+b−1

a°b=min a+b,1

T-Norms: advantages and disvantages

Advantages: Well-known formalism

Properties of various t-norms have been extensively studied and are known to verify various theorems

Easily computable Their properties seems to model well the properties

of linguistic conjunctions

Disvantages Obtained values are somewhat “too low”

O.W.A

“Ordered Weighted Aggregators” n-ary operations that can replace norms or conorms

defined as a sequence of n values

Given the values ,

Disvantages: O.W.A.s break logic properties

w1, w 2, ,w n ∑i=0

n

w i=1

s1, s2, ,sn ,sa≥ sa+1

OWA s1, sn =∑i=0

n

w i⋅s i