1. Ragionamento in condizioni di incertezza: Approccio fuzzy Paolo Radaelli Corso di Inelligenza Articifiale - Elementi

2. 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?

3. 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

4. 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

5. Fuzzy sets

6. Complementary set 0 1 0,5

7. Union and Intersection 0 1 0,5

8. Fuzzy Set properties These properties are true either in classical and fuzzy set theory:  Symmetric law  Associative law  De Morgan's laws  Distributive law

9. 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

10. Subsethood and Entropy Subsethood: measure “how much” a set A is a subset of B Entropy: measure the “fuzziness” of a fuzzy set

11. 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

12. Concentrators/ Dilators “very”, “extremely” (concentrators) “quite”, “a little” (dilators) Proposed way to handle concentrators:

13. 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)

14. Contrast intesifier formula

15. 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

16. 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”

17. T-Norms A family of mathematical functions Properties:  Symmetry  Associativity  Limit  Monotonicity

18. S-norms S-norms (or T-conorms) generalize union Properties:  Symmetry  Associativity  Limit  Monotonicity For each norm, there is an associated conorm

19. T-Norms :some example Minimum norm Probabilistic norm Lukasiewic's norm

20. 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”

21. 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