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

2. Fuzzy set applications Reasoning system using vague rules  Used to approximate a result in a process where formal rules aren't available Information evaluation and filtering, query answering  Allows the users to select a set of “soft constraints” about the information they need, and evaluate the adherence of the data on the users requests

3. Linguistic variable There are variables which can store a linguistic term as their value, instead of a numerical one Linguistic terms are translated into fuzzy-sets Main uses:  Formalize the meaning of linguistic expressions  Allow to formulate imprecise reasoning rules

4. Linguistic variable A 5-tuple X: The variable name T: terms set: all the values that the linguistic variable can assume U: domain of the discourse (the values that must be evaluated) G: syntactic rule (how to obtain T's elements M: semantic rule (hot to associate a fuzzy set to each element of T)

5. Example For a linguistic variable to store the price of an house  X: “house price”  T: cheap, quite cheap, a fair price, very expensive, etc...  U: The set of prices (or the set of houses)  G: There are 3 kind of linguistic components: Base terms (usually adjectives) Modifiers (usually adverbs) Connectives (and, or, not)

6. Syntactic rule  “cheap”, “expensive”  “very cheap”, “quite expensive”, “very very cheap”  “very cheap or quite expensive”, “not very cheap”

7. Semantic rule Base terms: each base term is associated to a fuzzy-set Example: cheap Modifiers: each modifier term is associated to a linguistic hedge. The resulting set is the application of the hedge to the original fuzzy-set. Example: very cheap Connectives: Each connective can be an union or a conjuntion. The resulting set is the application of the appropriate function to the two original sets Example: very cheap or quite expensive:

8. Fuzzy control systems They enhance a traditional control system Used when it is difficult to formalize a precise set of rule to model a problem The “rules” are represented as a sequence of relations between linguistic variables

9. Fuzzy control systems

10. Fuzzyfication Transform a numerical value into a fuzzy set It is based on a set of rules: IF val is F1, then vl is F2  val is the name of a (numeric) variable  F1 is a fuzzy set. Let's call a the degree of membership of val in F1  vl is the output linguistic variable  F2 is a fuzzy set related to the linguistic variable vl The output value for the variable vl will depend upon the values of F2 and a.

11. Min and product methods a

12. Inference In the fuzzyfication phase, all the rules are executed  It is possible to have more than one output value for each linguistic variables Different output values are joined together to obtain the final result  Max strategy: the result set is the union of the input fuzzy sets  Sum strategy: the output set is the set: (must be normalized)

13. Inference

14. De-fuzzyfication The last stage of the computation is used to transform a fuzzy-set into a numerical value  The value represents the most significative domain element, with respect to the fuzzy-set MAX method: the most significative value is the value with the highest membership values  some problems with sets that have more than one maximum element Center of mass method:

15. Fuzzy control system: IF Race.Track.Roughness is Rough AND Race.Day.SkyCondition is Sunny AND Race.Day,Temperature is High THEN Race.TSI is High IF Race.Track.Roughness is Smooth AND Race.Day.SkyCondition is Sunny AND Race.Track,Temperature is Low THEN Race.TSI is Low

16. Other examples Truck parking  http://www.iit.nrc.ca/IR_public/fuzzy/FuzzyTruck.html http://www.iit.nrc.ca/IR_public/fuzzy/FuzzyTruck.html Shower temperature control  http://www.iit.nrc.ca/IR_public/fuzzy/fuzzyShower.html Inverse pendulum  http://www.iit.nrc.ca/IR_public/fuzzy/FuzzyPendulum.html

17. Geographical entity City Site River GPS coordinates Belongs to Expansion Area Conservation state Distance From X near good farther than Y within Y Retrieve all sites in a good conservation state, located near a river and within 10 Km from an expansion area Query words are associated to the relevant ontology concepts Question answering and filtering

18. Fuzzy qualifications: are “elastic” constraint to pose upon the relevant concept properties Ontology concept: represent one of the concepts known by the system Geographical entity City Site River GPS coordinates Belongs to Expansion Area Conservation state Distance From X near good Basic Property: one of the attributes that characterize each concept's instance. Stored in the DB Basic Property: one of the attributes that characterize each concept's instance. Stored in the DB farther than Y within Y Fuzzy Ontology Complex Property: Attributes that are obtained on the basis of other properties. E.g. “distance from” will depend upon the coordinates of two entities Complex Property: Attributes that are obtained on the basis of other properties. E.g. “distance from” will depend upon the coordinates of two entities

19. Fuzzy control system:

20. Example – Data access and evaluation SELECT * FROM Races r INNER JOIN Tracks ON r.Track = t.ID INNER JOIN Weather w on r.wheater_cond = w.id WHERE w.SkyConditions = 'Rainy' AND t.Roughness >= 0.3 Those entities doesn't conform to the query Retrieve Races with Races.Wheater.Skycondition = Sunny AND Races.Track.Roughness = NotVeryRough AND 0%100% NotVeryRough SunnyRainy

21. Example – Data access and evaluation AND 0%100% NotVeryRough SunnyRainy MIN(1,0.12) 0.12 MIN(1,0.9) 0.9