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Ragionamento in condizioni di incertezza: Approccio fuzzy

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”

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Ragionamento in condizioni di incertezza: Approccio fuzzy

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  1. Ragionamento in condizioni di incertezza:Approccio fuzzy Paolo RadaelliCorso 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: • If I remove grain of sand from a mound, I obtain a mound again • But, if I remove all the sand's grains from the mound, I doesn't have a mound no more • 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 1 0,5 0

  7. Union and Intersection 1 0,5 0

  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

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