Intelligent Information Systems

1. Intelligent Information Systems Prof. M. Muraszkiewicz Institute of Information and Book Studies Warsaw University mietek@n-s.pl M. Muraszkiewicz

2. Non- Standard Logics Module 4 M. Muraszkiewicz

3. Prologue M. Muraszkiewicz

4. Incompleteness, Uncertainty, … M.C. Escher – Night and Day M. Muraszkiewicz

5. Incompleteness, Uncertainty, … M.C. Escher – Ascending or Descending M. Muraszkiewicz

6. Table of Contents • Prologue • Definitive and Non-Definitive Database • Note on Reiter’s Default Logic • Note on Zadeh’s Fuzzy Logic • Epilogue M. Muraszkiewicz

7. Definitive and Non-Definitive Databases M. Muraszkiewicz

8. Monotonicity A consistent theory T based on logic L is cumulative iff as a result of any valid sequence of inferences generating new axioms the total number of axioms belonging to T does not decrease and T remains consistent. If each theory based on logic L is cumulative, then L is called monotonic. Classic logics are monotonic. M. Muraszkiewicz

9. Incompleteness, Uncertainty Incompleteness- lack of “part” of information. Range of incompleteness <0, 1). Uncertainty– choice dilemma SurnameNameAddress Watson - Warsaw Samson - - Zaffir {Ian, Bill} Krakow Relational Database How such situations are represented in logic? Incompleteness - lack of formula in classic logic; non-classic logics Uncertainty - non-definitive databases; non-classic logics M. Muraszkiewicz

10. Definitive, Non-Definitive Definitive database:facts - P(arguments are constants); rules - P1  .. . Pn  R ; semantic dependency – any formula Non-Definitive facts - P1(arguments are constants)  ... databasePn(arguments are constants); rules - P1  .. . Pn  R1 ...  Rn semantic dependency – any formula CWA - Closed World Assumption; OWA - Open World Assumption M. Muraszkiewicz

11. Non-Monotonicity Non-classic deductive database (non-definitive) are non-monotonic theories. Since some axioms of a non-monotonic theory might be invalidated as a result of inference, therefore we rather use the term convictions than axioms. Truth Maintenance System (TMS, Doyle'79) are used to verify convictions. M. Muraszkiewicz

12. Note on Reiter’s Default logic? M. Muraszkiewicz

13. Default Reasoning, Reiter, 1980 M. Muraszkiewicz

14. Surprising Result Convictions obtained through both inferences are contradictory! (which the default logic allows) Theories that are generated as extensions of a given theory might be contradictory. M. Muraszkiewicz

15. Elegant Notation of CWA : M  P(x1, ..., xn) --------------------------  P(x1, ..., xn) If the default  P(x1, ..., xn) is consistent with our knowledge, then the above inference rule represents the CWA principle. M. Muraszkiewicz

16. Note on Zadeh’s Fuzzy Logic M. Muraszkiewicz

17. Origins of Doubts One of the Aristotelian laws of thought, borrowed from Parmenides (around 400 B.C.), is the Excluded Middle, which states that every proposition must either be True or False. But already Heraclitus proposed that things could be simultaneously True and not True. Plato also indicated that that there was a third region (beyond True and False) where these opposites "tumbled about." Also more modern philosophers echoed his sentiments, notably Hegel and Marx. Yet it was the Polish logician Lukasiewicz who first proposed a systematic alternative to the binary logic of Aristotle M. Muraszkiewicz

18. Fuzziness The concept of Fuzzy Logic was conceived by Prof. Lotfi Zadeh. In 1973 he proposed the concept of linguistic or "fuzzy" variables. The fuzzy variables themselves are adjectives that modify the variable (e.g. “high" temperature, “short” man. Additional ranges such as "very high" and "very short" could also be added. Lotfi Zadeh M. Muraszkiewicz

19. Fuzziness The main assumption of fuzzy logic is that truth values or membership values (in fuzzy sets) are indicated by a value on the range [0, 1], with 0 representing absolute Falseness and 1representing absolute Truth. ExampleLet us take the statement: Jane is young If Jane's age was 25, we might assign the statement the truth value of 0.80. The statement could be translated into set terminology as follows: "Jane is a member of the set of young people." This statement would be rendered symbolically with fuzzy sets as: mYoung(Jane) = 0.80 where m is the membership function, operating in this case on the fuzzy set of young people, which returns a value between 0.0 and 1.0. M. Muraszkiewicz

20. Fuzziness A basic application might characterize subranges of a continuous variable. For instance, a temperature measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled. In this image, cold, warm, and hot are functions mapping a temperature scale. A point on that scale has three "truth values" — one for each of the three functions. For the particular temperature shown, the three truth values could be interpreted as describing the temperature as, say, "fairly cold", "slightly warm", and "not hot". http://en.wikipedia.org/wiki/Fuzzy_logic M. Muraszkiewicz

21. Readings • Baldwin J.F., "Fuzzy logic and fuzzy reasoning," in Fuzzy Reasoning and Its Applications, E.H. Mamdani and B.R. Gaines (eds.), London: Academic Press, 1981. • Gensler H., “Introduction to Logic”, Routledge, 2001. • "Fuzzy Sets and Applications: Selected Papers by L.A. Zadeh", ed. R.R. Yager et al. (John Wiley, New York, 1987). • Lejewski C., "Jan Lukasiewicz," Encyclopedia of Philosophy, Vol. 5, MacMillan, NY, 1967. http://en.wikipedia.org/wiki/Fuzzy_logic M. Muraszkiewicz

22. Epilogue M. Muraszkiewicz

23. To Remember Fuzzy does not mean unlikely or uncertain. M. Muraszkiewicz

24. M. Muraszkiewicz