Compensatory Fuzzy Logic Discovery of strategically useful knowledge

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Compensatory Fuzzy Logic Discovery of
strategically useful knowledge

• Prof. Dr. Rafael Alejandro Espin Andrade
• Management Technology Studies Center
• Industrial Engineering Faculty
• Technical University of Havana
• CUJAE
• espin_at_ind.cujae.edu.cu, rafaelespin_at_yahoo.com

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Why a new multivalued fuzzy logic

• Learning, Judgment, Reasoning and Decision
  Making are parts of a same process of thinking,
  and have to be studied and modeled as a hole.
• No compensation among truth value of basic
  predicates are an obstacle to model human
  judgment and decision making.
• Associativity is an obstacle to get compensatory
  operators with sensitivity to changes in truth
  values of basic predicates, and possibilities of
  interpretation of composed predicates truth
  values according a scale

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Fuzzy Logic based Decision Making Modeling

• It is not yet enough a formal field
• Bad behavior of multi-valued logic systems
• Pragmatic Combination of operators without
  axiomatic formalization
• Confluence of Objectives using only one operator

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Compensatory Logic

• It allows compensation among truth value of basic
  predicates inside the composed predicate.
• It is a not associative system.
• It is a sensitive and interpretable system
• It generalizes Classic Logic in a new and
  complete way.
• It is possible to model decision making problems
  under risk, in a compatible way with utility
  theory.
• It explains the experimental results of
  descriptive prospect theory as a rational way to
  think
• It allows a new mixed inference way using
  statistical and logical inference
• Its properties allows a better way to deal with
  modeling from natural and professional languages

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Existing efforts to create fuzzy semantics
standards

• Using min-max logic
• Using a pragmatic combination of operators

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Models

• Competitive Enterprises Evaluation from
  Secondary Sources. ()
• Analysis SWOT-OA (SWOTBSC) ()
• Competences Analysis
• Composed Inference from Compensatory Logic
  (Useful for Data Mining, Knowledge Discovering,
  Simulation) ()

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Models

• Integral Project Evaluation
• Negotiation
• New Theoretical Treatment of Cooperative
  n-person Games Theory Quantitative Indexes for
  Decision Making in Business Negotiation (Good
  Deal Index, Convenience Counterpart Index)
• SDIReadiness

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Compensatory Conjunction
Geometric Mean
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Negation
n(x) 1-x.
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Compensatory Disjunction
Dual of Geometric Mean
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Zadeh Implication i(x,y)d(n(x),c(x,y))
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Rule Definition of And
Operators of CFL using SWRL
Rule Definition of Negation
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Operators of CFL using SWRL
Rule Definition of Or
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Operators of CFL using SWRL
Rule Definition of Implication
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Operators of CFL using SWRL
Rule Definition of Equivalence
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Creating Ontologies from fuzzy trees

• Create the tree from formulation in natural
  language
• Create classes using OWL or SWRL (using built
  ins for membership functions)
• Use the created built ins to create the new
  classes inside SWRL

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No Associativity Level Properties
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As higher level a basic predicate be, more
influence it will has in the truth value of the
composed predicate.
c(c(x,y),z)
c(x,y,z)
c(x,y)
z
x
z
y
x
y
Both trees are the same for Associative Logic
Systems.
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Natural Implication i(x,y)d(n(x),y)
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Natural Implication
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Zadeh Implication i(x,y)d(n(x),c(x,y))
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Universal and Existential Quantifiers
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Universal and Existential Quantifiers over
bounded universes of Rn
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Compatibility with Propositional Classical
Calculus
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Compatibility with Propositional Classical
Calculus (Kleene Axioms)
Natural Zadeh Ax 1
0.5859 0.5685 Ax 2 0.5122 0.5073 Ax
3 0.5556 0.5669 Ax 4 0.5859
0.5661 Ax 5 0.8533 0.5859 Ax 6
0.5026 0.5038 Ax 7 0.5315 0.5137 Ax
8 0.5981 0.5981
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Theorem of Compatibility Exclusive Property of
CFL useful to get fuzzy ontologies and connected
it with non fuzzy ones
p is an only is a correct formula (tautology) of
Propositional Calculus according to bivalued
logic if it has truth value greater than 0.5 in
CFL
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Inference
Logic Inference
Statistical Inference
Composed Inference
Composed Inference It allows to make and to
model hypothesis using Background Knowledge, to
estimate truth value of hypothesis using a sample
and search in parameters space of the model
increasing truth
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Hypothesis

• 1. If past time t from t0 is short, PIB at t0 is
  high, and exchange rate peso-dollar is good, and
  inflation too, then inflation at t0t will be
  good. (sufficient condition for goodness of
  future inflation)
• 2. If past time t from t0 is short, PIB at t0 is
  high, and exchange rate peso-dollar is good, and
  inflation too, then exchange rate at t0t will be
  good. (sufficient condition for goodness of
  future exchange rate)
• 3. If past time t from t0 is short, PIB at t0 is
  high, and exchange rate peso-dollar is good, and
  inflation too, then PIB at t0t will be high.
  (sufficient condition for goodness of future PIB)

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Hypothesis 1 Hypothesis 2 Hypothesis 3 Hypothesis 1' Hypothesis 2' Hypothesis 3'
0,576160086 0,620615758 0,319922171 0,237583737 0,539710004 0,548489528
0,104861817 0,109086145 0,278621583 0,999987824 0,988994364 0,194511245
0,954676527 0,955654118 0,966207572 0,997833826 0,036824939 0,256004496
0,619516745 0,682054585 0,704510956 0,843278654 0,236556481 0,349107639
0,360116603 0,596395877 0,681417533 0,905856942 0,449708517 0,583310592
0,503173195 0,601806401 0,806558599 0,875167679 0,375314384 0,675469125
0,240645922 0,243684437 0,388070789 0,999988786 0,988986017 0,194481187
0,166064725 0,658210493 0,366102952 0,990618051 0,604258741 0,27023985
0,957234283 0,957808848 0,969335758 0,949513696 0,023763872 0,295017189
0,623564594 0,688110949 0,820055235 0,929246656 0,24713002 0,566004987
0,406582528 0,562805296 0,763656353 0,862017409 0,438632042 0,6827389
0,315003135 0,609940962 0,481272314 0,991513016 0,448097639 0,267486414
0,348881932 0,632987705 0,445100645 0,785234473 0,530719918 0,406855462
0,960632803 0,96120307 0,981985893 0,977613627 0,064500183 0,546627412
0,658386683 0,698912399 0,87122133 0,898146889 0,29047844 0,664809821
0,459222487 0,617040225 0,55840803 0,808412114 0,384684691 0,380968904
0,36295827 0,596407224 0,682984162 0,906088702 0,447004634 0,583111022
0,964920965 0,965265218 0,987462709 0,968219979 0,159229346 0,646455267
0,489849965 0,62196448 0,751265713 0,916814967 0,339829639 0,57450373
0,448296532 0,577103867 0,782469741 0,867623112 0,410028953 0,679530673
0,55878818 0,632516495 0,830278389 0,883071359 0,342921337 0,671518873
0,439202131 0,574390855 0,604586702 0,846168613 0,281018624 0,395033281
0,519845804 0,657303193 0,754040445 0,890357379 0,330713346 0,578926238
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Hypothesis 1 Hypothesis 2 Hypothesis 3 Hypothesis 1' Hypothesis 2' Hypothesis 3'
0,129937234 0,786977784 0,163182976 0,897313418 0,78995843 0,246740413
0,040471107 0,045576384 0,226920195 1 0,98899831 0,194313157
0,96601297 0,966586912 0,974686783 0,999999764 0,031185555 0,255212475
0,316375835 0,614728727 0,512356471 0,999226656 0,439429951 0,291353006
0,099650901 0,662430526 0,589342664 0,999838473 0,642938808 0,545153518
0,218548001 0,549156052 0,721617082 0,999637465 0,508526254 0,645300844
0,031375461 0,036578117 0,219591953 1 0,988998869 0,194313157
0,040525407 0,776150562 0,285353418 0,999998745 0,770101561 0,25525313
0,966018421 0,966594251 0,975835847 0,99982785 0,019121177 0,28913524
0,314847536 0,596608509 0,68759951 0,999859116 0,439114653 0,545002502
0,100646896 0,567339971 0,679511054 0,999611013 0,5903147 0,645417242
0,031430795 0,793473197 0,278579081 0,999998739 0,789884048 0,255253529
0,04487921 0,802329191 0,317793959 0,999085299 0,794414393 0,292286066
0,966014676 0,966568235 0,984519648 0,999968639 0,061925026 0,544546456
0,315423665 0,558446985 0,756193558 0,999660725 0,450390363 0,64520529
0,035867632 0,823077109 0,311327136 0,999080973 0,817720938 0,2923171
0,041885488 0,720353211 0,562956364 0,999833364 0,722064447 0,545194081
0,966016088 0,966509609 0,987918683 0,999924476 0,158754822 0,644565563
0,032816816 0,732504923 0,558813514 0,999832576 0,736638274 0,54520045
0,043013751 0,591969844 0,658918424 0,99959871 0,63696319 0,645474182
0,033966772 0,597042779 0,655685287 0,999596813 0,64484329 0,645483123
0,15293434 0,493060755 0,476469006 0,992585058 0,361083333 0,353165336
0,064133476 0,727586495 0,58911645 0,999644446 0,552232295 0,534562806
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Membership Functions
as true as false
almost false
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Membership function
As true as false10 Almost false5
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As true as false40 Almost false15
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Gamma Beta Gamma Beta
Inflation 11 5 10.3022482 5.30220976
GIP 2 0 2.70067599 0.12747186
Money Value 7 12 6.8657769 12.0650321
Future Inflation 11 5 6.39146712 6.35080391
Future GIP 2 0 2 0
Future Money Value 7 12 7 12
Time 2 4 1.19916547 4.14284971
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Relation between CFL and Utility Theory

• Two possible outlooks of Decision Making problem
  under risk using Compensatory Fuzzy Logic are
  possible
• First one
• Security All scenarios are convenient in
  correspondence with its probabilities of
  occurrence (Its is equivalent to be risk
  adverse)

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Hedges

• Operators which models words like very, more or
  least, enough, etc. They modifies the truth value
  intensifying or un-intensifying judgments.
• More used functions to define hedges are
  functions f(x)xa , a is an exponent greater or
  equal to cero. It is used to use 2 and 3 as
  exponents to define the words very and hyper
  respectively, and ½ for more or less.

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Function u(x)ln(v(x) have second diferential
positive (risk averse) when v es sigmoidal.
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Relation between CFL and Utility Theory

• Second outlook
• Opportunity There are convenient scenarios
  according with their probabilities (It is
  equivalent to be risk prone)

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These preferences are represented by
u(x)-ln(1-v(x) (It is proved by increasing
transformations). This function have negative
second differential (risk prone) when v is
sigmoidal.
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Teoremas

• Teorema 1
• Si f es un predicado difuso que representa la
  conveniencia de los premios. El punto de vista
  de la seguridad usando LDC representa las
  preferencias de un decisor averso al riesgo con
  función de utilidad u(x)ln(f(x)).
• El punto de vista de la oportunidad usando LDC
  representa las preferencias de un decisor
  propenso al riesgo con función utilidad
  u(x)-ln(1-f(x))

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Teoremas

• Teorema 2
• Dada un decisor con función utilidad u
  acotada en el intervalo (m,M).
• Si el decisor es averso al riesgo, el
  predicado de la Lógica Difusa Compensatoria que
  representa la conveniencia de los precios es
  v(x)exp(u(x)-M). Si es propenso al riesgo, el
  predicado de la Lógica Difusa Compensatoria que
  representa la conveniencia de los premios es
  v(x)1-exp(1-u(x)-m).

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Prospect Theory

• It is a descriptive decision making theory of
  decision making under risks, based on
  experiments. It deserved the nobel prize of
  Economy for Kahnemann and Tervsky in 2003.
• Individual decision makers are used to be risk
  averse attitude about benefits and risk prone
  attitude about loses
• More general
• There is a reference value a, satisfying
  for xlta that utility function is convex and for
  xgta is concave.

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Prospect Theory

• Differential of the function for loses is great
  than differential for benefits.
• Individual decision makers are used to attribute
  not linear weights to utilities using
  probabilities of the correspondent scenarios.
• That function are used to be concave in certain
  interval 0,b and convex in b,1 b is a real
  number greater than 0 and less than 1.

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Rational explanation of Experimental Results of
Kahnemann and Tervsky

• 56 lotteries and its experimental equivalents
  were used from experiments of Kahnemann and
  Tervsky.
• We estimated the truth value of the statement
  Every lottery is equivalent (in preference) to
  its experimental equivalent, according CFL for
  each preference model Universal (Risk Averse),
  Existential (Risk Prone), Conjunction Rule and
  Disjunction Rule. Best parameters of membership
  functions maximizing the statement truth value
  for all the models.

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Rational explanation of Experimental Results of
Kahnemann and Tervsky

• Result Experimental results of Kahnemann and
  Tervsky can be explained as result of a new
  based-CFL rationality working with no linear
  membership functions of probabilities and
  considering that security and opportunity are
  both desirables for individual decision makers.

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Prize1 Prize2 Prob1 Prob2 Equiv
Universal Existential Conjunction Disjunction
50 150 0.05 0.95 128 0.81046267 0.81043085 0.84579078 0.86240609
-50 -150 0.95 0.05 -60 0.98729588 0.98699553 0.98830472 0.988450983
-50 -150 0.75 0.25 -71 0.98987688 0.9898564 0.99040911 0.990496525
-50 -150 0.5 0.5 -92 0.99282463 0.99193467 0.99355369 0.993616354
-50 -150 0.25 0.75 -113 0.99547171 0.9930822 0.99578486 0.995822571
-50 -150 0.05 0.95 -132 0.99680682 0.98931435 0.99720726 0.997233981
100 200 0.95 0.05 118 0.80956738 0.76106093 0.85650544 0.874712043
100 200 0.75 0.25 130 0.79924863 0.77146557 0.8415966 0.86176461
100 200 0.5 0.5 141 0.77419623 0.76244496 0.82436946 0.845652063
100 200 0.25 0.75 162 0.75472897 0.75240339 0.79542526 0.822098935
100 200 0.05 0.95 178 0.72808136 0.72808118 0.77001401 0.801244793
-100 -200 0.95 0.05 -112 0.99547158 0.99535134 0.99569579 0.995718909
-100 -200 0.75 0.25 -121 0.99634217 0.99633821 0.99644367 0.996456576
-100 -200 0.5 0.5 -142 0.99762827 0.99734518 0.99777768 0.997786734
-100 -200 0.25 0.75 -158 0.99843469 0.99764225 0.99848705 0.998493062
-100 -200 0.05 0.95 -179 0.99905141 0.99589186 0.9991227 0.999126427
0.91683692 0.89594007 0.93079936 0.942636789
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Users and Groups
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Organizations and Matrices
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Organizations and Matrices
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Organization Matrices
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Job Parameters
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Parameters Configuration
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Parameters Configuration
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Compensatory Logic

• It allows compensation among truth value of basic
  predicates inside the composed predicate.
• It is a not associative system.
• It is a sensitive and interpretable system
• It generalizes Classic Logic in a new and
  complete way.
• It is possible to model decision making problems
  under risk, in a compatible way with utility
  theory.
• It explains the experimental results of
  descriptive prospect theory as a rational way to
  think
• It allows a new mixed inference way using
  statistical and logical inference
• Its properties allows a better way to deal with
  modeling from natural and professional languages

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Some Scientific Perspectives

• Development of a new fuzzy framework of
  Cooperative Games Theory CFL-Based
• CFL-based Ontologies using OWL-SWRL-Protégé
  Ontologies
• Creation of mathematically formal hybrid
  frameworks mixing Neural networks, Evolutionary
  algorithms, Trees and CFL
• Experimental research line about judgment,
  election and reasoning from CFL