Compensatory Fuzzy Logic Discovery of strategically useful knowledge

1. 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@ind.cujae.edu.cu, rafaelespin@yahoo.com

4. 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

5. Fuzzy Logic basedDecision 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

6. 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

7. Existing efforts to create fuzzy semantics standards • Using min-max logic • Using a pragmatic combination of operators

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

9. 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) • SDI’Readiness

10. Compensatory Conjunction Geometric Mean

11. Negation n(x) = 1-x.

12. Compensatory Disjunction Dual of Geometric Mean

13. Zadeh Implication i(x,y)=d(n(x),c(x,y))

14. Operators of CFL using SWRL Rule: Definition of And Rule: Definition of Negation

15. Operators of CFL using SWRL Rule Definition of Or

16. Operators of CFL using SWRL Rule Definition of Implication

17. Operators of CFL using SWRL Rule Definition of Equivalence

18. 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

19. No Associativity Level Properties

20. As higher level a basic predicatebe, 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.

21. Natural Implication i(x,y)=d(n(x),y)

22. Natural Implication

23. Zadeh Implication i(x,y)=d(n(x),c(x,y))

25. Universal and Existential Quantifiers

26. Universal and Existential Quantifiers over bounded universes of Rn

27. Compatibility with Propositional Classical Calculus

28. 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

29. 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

30. 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

31. 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 t0+t 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 t0+t 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 t0+t will be high. (sufficient condition for goodness of future PIB)

34. Membership Functions : as true as false : almost false

35. Membership function As true as false:10; Almost false:5

36. As true as false:40; Almost false:15

38. 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)

39. 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. 39

40. Function u(x)=ln(v(x) have second diferential positive (risk averse) when v es sigmoidal.

41. Relation between CFL and Utility Theory • Second outlook Opportunity: There are convenient scenarios according with their probabilities (It is equivalent to be risk prone)

42. 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.

43. 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))

44. 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).

45. 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 x<a that utility function is convex and for x>a is concave.

46. 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.

47. 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.

48. 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.

49. Prize1 Prize2 Prob1 Prob2 Equiv Universal Existential Conjunction Disjunction