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 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)
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”