Preference relation in pliant system

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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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Elements of pliant system

• Conjunction, disjunction, negation
• Aggregation
• Preference relation
• Distending function
• Distending function as preference

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• Conjunction, disjunction, negation

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Conjunctive and disjunctive operator

• We shall be looking for the general form of
  c(x,y) and
• d(x,y)
• is continuous
• Strict monotonous increasing
• Compatible with the two valued logic
• Associative
• Archimedian

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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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Operators and DeMorgan law

• Lets generalize tha conjunctive and disjunctive
• operators and let
• where

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Negation

• Definition
• ?(x) is a negation iff satisfies the
  following
• conditions
• ?(x) is continuous
• Boundary conditions are and
• Monotonicity for
• Involutivness

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Negation

• Other properties
• ? fix point of the negation, where
• The decision value

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Negation

• On Figure there are some negation functions with
  different ? and ? values

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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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Operators and DeMorgan law

• Theorem (DeMorgan law)
• The generalized DeMorgan law is valid iff
• where

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Negation and DeMorgan law

• Parametrical form of the negation is

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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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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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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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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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Multiplicative pliant system

• If f fc , then depending on thevalue of ?
  the
• operator is

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Dombi operator system

• Let choose then we get

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• Aggregation

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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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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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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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Aggregative operators and representable uninorms

• Definition
• An aggregative operator is a strictly increasing
• function with the properties
• Continuous on
• Boundary conditions are and
• Associativity
• There exists a strong negation ? such that
• (self DeMorgan identity)

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Aggregative operators and representable uninorms

• Definition
• A uninorm U is a mapping having the
• following properties
• Commutativity
• Monotonicity if and
• Associativity
• Neutral element

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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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Aggregation

• Theorem
• It holds that
• Theorem
• It holds for the aggregative operator that
• 1.
• 2.
• 3.
• 4.

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Aggregation
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The neutral value

• Theorem (Additive form of negations)
• Let be a continuous function, then the
• following are equivalent
• ? is a negation with neutral value ?.
• There exists a continuous and strictly monotone
• function and such that for
• all

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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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Conjunctive, disjunctive and aggregative operators

• Theorem
• The following are equivalent
• is a logical operator.
• is an aggregative aoperator.

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Aggregation and Pan operators

• Pan operator
• Theorem
• Let c and d be a conjunctive and a disjunctive
  opera-
• tor with additive generator functions fc and
  fd .
• 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 fc(x)fd(x) 1 .

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Conjunctive, disjunctive and aggregative operators

• Definition
• Let f be the additive generator of a logical
  operator.
• The aggregative operator is called
• the corresponding aggregative operator of the
• conjunctive or disjunctive operator, and vice
  versa.

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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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Pliant operators

• Theorem
• Let c and d be a conjunctive and disjunctive
  operator
• with additive generator functions fc and fd .
  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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Unary operators

• The general form of the unary operator
• Special case of the function
• if ?1 and ? gt ?0 then
  concentration operator
• if ?1 and ? lt ?0 then dilutor
  operator
• if ?-1 then negation
  operator
• if f(?0) f(?) 1 then sharpness
  operator

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Pliant operators

• The Dombi operator case

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Pliant operators

• Modifier
• if ?1 and ? gt ?0 then
  concentration operator
• if ?1 and ? lt ?0 then dilutor
  operator
• if ?-1 then negation
  operator
• Negation

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• Preference relation

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Preference operator on the 0,1 interval

• We define the preference function in the
  following
• way

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Properties of preference operator

• Theorem
• Let the pliant operations
• and the preference operator

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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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Properties of preference operator

• 4. Boundary conditions if then
• 5. Neutrality
• 6. Preference property

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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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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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Properties of preference operator

• 5. Exchangeability
• 6. Preference of aggregation

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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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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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Preference and multicriteria decision making

• In pliant concept and 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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• Distending function

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Distending function instead of membership function

• Let choose an often used one the term old. The
  same
• example exist in Zadehs 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 (a63).
• 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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Distending function instead of membership function

• We can write this in an inequality form, using a
• characteristic function
• The expression altx is equivalent with the
  expression
• 0 lt x-a , so the above form could be written as

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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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General form of the distending function

• Let start with the aggregation concept. The
  weighted
• aggregation operator is
• where xi 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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Distending function

• Theorem
• Using the aggregation
• if and only if
• --------------------------------------------------
  --------------------
• Dombi operator case

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Sigmoid function and logistic regression

• The sigmoid function has the following
  properties.
• The sigmoid function is able to modelize
  inequality.

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• Distending function as preference

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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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Distending function as preferences on the real
line
17-Sept-2009

• The following properties hold for .
• I. Preference properties
• 1. Continuity
• 2. Monotonicity
• 3. Limes property

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Distending function as preferences on the real
line
17-Sept-2009

• 4. Boundary conditions
• 5. Neutrality
• 6. Preference property

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Distending function as preferences on the real
line
17-Sept-2009

• 7. Translation property
• II. Preference and negation operator
• III. Preference and aggregation
• 1. Transitivity with aggregation
• 2. Common basis principles

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Distending function as preferences on the real
line
17-Sept-2009

• 4. Neutrality
• IV. Threshold property
• 1. Threshold transitivity
• P(?) is threshold transitiv if
• 2. Strongly complete

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Distending function as preferences on the real
line
17-Sept-2009

• 3. Antisymmetric

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Animation
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Animation
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Animation
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Animation
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Animation
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Animation
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• Thank you for your attention!