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Preference relation in pliant system http://www.inf.u-szeged.hu/~dombi/dr University of Szeged Department of Informatics Pamplona 2009

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2 17-Sept-2009 Elements of pliant system 1.Conjunction, disjunction, negation 2.Aggregation 3.Preference relation 4.Distending function 5.Distending function as preference

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3 17-Sept-2009 Conjunction, disjunction, negation

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4 17-Sept-2009 Conjunctive and disjunctive operator We shall be looking for the general form of c(x,y) and d(x,y) : 1. is continuous 2.Strict monotonous increasing 3.Compatible with the two valued logic 4.Associative 5.Archimedian

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5 17-Sept-2009 Conjunctive and disjunctive operator Theorem: (Aczél) If with u and v, h(u,v) also always lies in a given (possibly infinite) interval and h(u,v) is reducible on both sides, then

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6 17-Sept-2009 Operators and DeMorgan law Let’s generalize tha conjunctive and disjunctive operators and let: where

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7 17-Sept-2009 Negation Definition: (x) is a negation iff satisfies the following conditions: 1. (x) is continuous 2.Boundary conditions are and 3.Monotonicity:for 4.Involutivness:

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8 17-Sept-2009 Negation Other properties: - * fix point of the negation, where - The decision value:

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9 17-Sept-2009 Negation -On Figure there are some negation functions with different * and values:

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10 17-Sept-2009 Operators and DeMorgan law Definition: The DeMorgan law for general conjunctive and disjunctive operator is: where (x) is the negation function.

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11 17-Sept-2009 Operators and DeMorgan law Theorem: (DeMorgan law) The generalized DeMorgan law is valid iff where

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12 17-Sept-2009 Negation and DeMorgan law Parametrical form of the negation is:

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13 17-Sept-2009 Representation theorem of negation For all given (x) there exist an f(x) such that where k(x) is a strictly decreasing function with the property and f is the generator function of a conjunctive, or disjunctive operator. ---------------------------------------------------------------------- Trillas’ result:

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14 17-Sept-2009 Operator with various negations Theorem: c(x,y) and d(x,y) build DeMorgan system for where if and only if

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15 17-Sept-2009 Multiplicative pliant system Definition: If k(x) = 1/x, i.e. and then we call the generated connectives multiplicative pliant system.

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16 17-Sept-2009 Multiplicative pliant system Theorem: The general form of the multiplicative pliant system is where f(x) is the generator function of either the conjunctive or the disjunctive operator.

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17 17-Sept-2009 Multiplicative pliant system If f = f c, then depending on thevalue of the operator is

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18 17-Sept-2009 Dombi operator system Let choose then we get

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19 17-Sept-2009 Aggregation

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20 17-Sept-2009 Aggregation Let us consider a set of objects.Let us characterize every object with a number m of its properties,where and i = 1,…,n. Thus, if the aggregative operator as denoted as, for a decision level we have

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21 17-Sept-2009 Aggregation Let us next substitute every property by its antithetic one (in the following its negative form and carry out division into classes at the level:

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22 17-Sept-2009 Aggregative operators and representable uninorms Definition: (of correct decision formation) The condition of correct formation is thus Theorem: It is necessary and sufficient condition of the aggre- gative operator satisfying correct decision formation that should hold.

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23 17-Sept-2009 Aggregative operators and representable uninorms Definition: An aggregative operator is a strictly increasing function with the properties: 1.Continuous on 2.Boundary conditions are and 3.Associativity: 4.There exists a strong negation such that (self DeMorgan identity)

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24 17-Sept-2009 Aggregative operators and representable uninorms Definition: A uninorm U is a mapping having the following properties : 1.Commutativity: 2.Monotonicity: if and 3.Associativity: 4.Neutral element:

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25 17-Sept-2009 Aggregative operators and representable uninorms Theorem: Let be a function. It is an aggregative operator if and only if there exists a continuous and strictly monotone function with such that for all

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26 17-Sept-2009 Aggregation Theorem: It holds that: Theorem: It holds for the aggregative operator that 1. 2. 3. 4.

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27 17-Sept-2009 Aggregation

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28 17-Sept-2009 The neutral value Theorem: (Additive form of negations) Let be a continuous function, then the following are equivalent: 1. is a negation with neutral value *. 2.There exists a continuous and strictly monotone function and such that for all

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29 17-Sept-2009 Conjunctive, disjunctive and aggregative operators Definition: We will use the term conjunctive operator for strict, continuous t-norms, and disjunctive operator for strict, continuous t-conorms. The expression logical operators will refer to both of them.

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30 17-Sept-2009 Conjunctive, disjunctive and aggregative operators Theorem: The following are equivalent: 1. is a logical operator. 2. is an aggregative aoperator.

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31 17-Sept-2009 Aggregation and Pan operators Pan operator: Theorem: Let c and d be a conjunctive and a disjunctive opera- tor with additive generator functions f c and f d. Suppose their corresponding negations are equivalent (i.e. ), denoted by ( ( * ) = * ). The three connectives c, d and form a De Morgan triplet if and only if f c (x)f d (x) = 1.

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32 17-Sept-2009 Conjunctive, disjunctive and aggregative operators Definition: Let f be the additive generator of a logical operator. The aggregative operatoris called the corresponding aggregative operator of the conjunctive or disjunctive operator, and vice versa.

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33 17-Sept-2009 Conjunctive, disjunctive and aggregative operators Multiplicative form of negations: The function is a negation with neutral value if and only if where f is a generator function of a logical operator.

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34 17-Sept-2009 Pliant operators Theorem: Let c and d be a conjunctive and disjunctive operator with additive generator functions f c and f d. Suppose their corresponding negations are equivalent ( i.e. ), denoted by. The three connectives c, d and n form a DeMorgan triplet if and only if

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35 17-Sept-2009 Unary operators The general form of the unary operator: Special case of thefunction: if =1 and > 0 then concentration operator if =1 and < 0 then dilutor operator if =-1 then negation operator if f( 0 )= f( ) = 1 then sharpness operator

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36 17-Sept-2009 Pliant operators The Dombi operator case:

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37 17-Sept-2009 Pliant operators Modifier: if =1 and > 0 then concentration operator if =1 and < 0 then dilutor operator if =-1 then negation operator Negation:

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38 17-Sept-2009 Preference relation

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39 17-Sept-2009 Preference operator on the [0,1] interval We define the preference function in the following way:

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40 17-Sept-2009 Properties of preference operator Theorem: Let the pliant operations: and the preference operator

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41 17-Sept-2009 Properties of preference operator The following properties hold for the preference rela- tions: I. Preference properties 1. Continuity: 2. Monotonicity: 3.Compatibility conditions:

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42 17-Sept-2009 Properties of preference operator 4. Boundary conditions: if then 5. Neutrality: 6. Preference property:

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43 17-Sept-2009 Properties of preference operator 7. Bisymmetric property: 8. Common basis property: for all z II. Preference and negation operator 1. 2. 3.

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44 17-Sept-2009 Properties of preference operator III. Preference and aggregation 1. Transitivity with aggregation: 2. Common basis principles 3. Inverse property: 4. Neutrality:

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45 17-Sept-2009 Properties of preference operator 5. Exchangeability: 6. Preference of aggregation:

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46 17-Sept-2009 Properties of preference operator IV. Threshold property 1. Threshold transitivity: p(x,y) is threshold transitiv if: 2. Strong completeness: 3. Antisymmetricity:

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47 17-Sept-2009 Preference and multicriteria decision making We can express the preference relation in additive form: where g(x)=ln(f(x)). In multicriteria decision the preference is

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48 17-Sept-2009 Preference and multicriteria decision making In pliant conceptand so (1) and (2) are the same. Most cases in the framework of multicriteria decision (3) are used. We can approximate (3) using Rolle theorem: i.e. Substituting it into (1) where i.e. the preference depends on y and x.

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49 17-Sept-2009 Distending function

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50 17-Sept-2009 Distending function instead of membership function Let choose an often used one the term “old”. The same example exist in Zadeh’s seminal paper. We suppose now that the term “old” depends only on age, and we do not care that most polar terms are always context dependant i.e. old professor is defined in an other domain than old student. In classical logic we have to fix a dividing line, in our case let it be 63 years (a=63). If somebody is older than 63 years then he/she belongs to the class (set) of old people, otherwise does not.

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51 17-Sept-2009 Distending function instead of membership function We can write this in an inequality form, using a characteristic function: The expression a

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52 17-Sept-2009 Distending function instead of membership function Generaly, on the left side of the inequality could be any g(x) function. In the pliant concept we introduce the distending function. We will use the notation We can generalize this in the following way:

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53 17-Sept-2009 General form of the distending function Let start with the aggregation concept. The weighted aggregation operator is: where x i are the distending values and f is the generator function of the logical operator. Intuitively aggregation is a weighted average of the values, The following theorem gives the exact description of

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54 17-Sept-2009 Distending function Theorem: Using the aggregation if and only if ---------------------------------------------------------------------- Dombi operator case:

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55 17-Sept-2009 Sigmoid function and logistic regression The sigmoid function has the following properties. The sigmoid function is able to modelize inequality.

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56 17-Sept-2009 Distending function as preference

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57 17-Sept-2009 Distending function as preferences on the real line The distending function has the following form: We can define a preference function:

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58 17-Sept-2009 Distending function as preferences on the real line The following properties hold for. I. Preference properties 1. Continuity: 2. Monotonicity: 3. Limes property:

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59 17-Sept-2009 Distending function as preferences on the real line 4. Boundary conditions: 5. Neutrality: 6. Preference property:

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60 17-Sept-2009 Distending function as preferences on the real line 7. Translation property: II. Preference and negation operator III. Preference and aggregation 1. Transitivity with aggregation: 2. Common basis principles

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61 17-Sept-2009 Distending function as preferences on the real line 4. Neutrality: IV. Threshold property 1. Threshold transitivity: P (λ) is threshold transitiv if: 2. Strongly complete:

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62 17-Sept-2009 Distending function as preferences on the real line 3. Antisymmetric:

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63 17-Sept-2009 Animation

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64 17-Sept-2009 Animation

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65 17-Sept-2009 Animation

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66 17-Sept-2009 Animation

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67 17-Sept-2009 Animation

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17-Sept-2009

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69 Thank you for your attention! 17-Sept-2009

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