1. Approximate Reasoning 1 Expert Systems Dr. Samy Abu Nasser

2. Approximate Reasoning 2 Overview Approximate Reasoning u Motivation u Objectives u Approximate Reasoning u Variation of Reasoning with Uncertainty u Commonsense Reasoning u Fuzzy Logic u Fuzzy Sets and Natural Language u Membership Functions u Linguistic Variables u Important Concepts and Terms u Chapter Summary

3. Approximate Reasoning 3 Motivation u reasoning for real-world problems involves missing knowledge, inexact knowledge, inconsistent facts or rules, and other sources of uncertainty u while traditional logic in principle is capable of capturing and expressing these aspects, it is not very intuitive or practical u explicit introduction of predicates or functions u many expert systems have mechanisms to deal with uncertainty u sometimes introduced as ad-hoc measures, lacking a sound foundation

4. Approximate Reasoning 4 Objectives u be familiar with various approaches to approximate reasoning u understand the main concepts of fuzzy logic u fuzzy sets u linguistic variables u fuzzification, defuzzification u fuzzy inference u evaluate the suitability of fuzzy logic for specific tasks u application of methods to scenarios or tasks u apply some principles to simple problems

5. Approximate Reasoning 5 Approximate Reasoning u inference of a possibly imprecise conclusion from possibly imprecise premises u useful in many real-world situations u one of the strategies used for “common sense” reasoning u frequently utilizes heuristics u especially successful in some control applications u often used synonymously with fuzzy reasoning u although formal foundations have been developed, some problems remain

6. Approximate Reasoning 6 Approaches to Approximate Reasoning u fuzzy logic u reasoning based on possibly imprecise sentences u default reasoning u in the absence of doubt, general rules (“defaults) are applied u default logic, nonmonotonic logic, circumscription u analogical reasoning u conclusions are derived according to analogies to similar situations

7. Approximate Reasoning 7 Advantages of Approximate Reasoning u common sense reasoning u allows the emulation of some reasoning strategies used by humans u concise u can cover many aspects of a problem without explicit representation of the details u quick conclusions u can sometimes avoid lengthy inference chains

8. Approximate Reasoning 8 Problems of Approximate Reasoning u nonmonotonicity u inconsistencies in the knowledge base may arise as new sentences are added u sometimes remedied by truth maintenance systems u semantic status of rules u default rules often are false technically u efficiency u although some decisions are quick, in general such systems are very slow v especially when truth maintenance is used

9. Approximate Reasoning 9 Fuzzy Logic u approach to a formal treatment of uncertainty u relies on quantifying and reasoning through natural language u linguistic variables v used to describe concepts with vague values u fuzzy qualifiers v a little, somewhat, fairly, very, really, extremely u fuzzy quantifiers v almost never, rarely, often, frequently, usually, almost always v hardly any, few, many, most, almost all

10. Approximate Reasoning 10 Fuzzy Sets u categorization of elements x i into a set S u described through a membership function  (s) :x  [0,1] v associates each element x i with a degree of membership in S: 0 means no, 1 means full membership v values in between indicate how strongly an element is affiliated with the set

11. Approximate Reasoning 11 Fuzzy Set Example membership height (cm) 0 0 50100150200250 0.5 1 shortmedium tall

12. Approximate Reasoning 12 Fuzzy vs. Crisp Set membership height (cm) 0 0 50100150200250 0.5 1 short medium tall

13. Approximate Reasoning 13 Possibility Measure u degree to which an individual element x is a potential member in the fuzzy set S Poss{x  S} u combination of multiple premises with possibilities u various rules are used u a popular one is based on minimum and maximum v Poss(A  B) = min(Poss(A),Poss(B)) v Poss(A  B) = max(Poss(A),Poss(B))

14. Approximate Reasoning 14 Possibility vs.. Probability u possibility refers to allowed values u probability expresses expected occurrences of events u Example: rolling dice u X is an integer in U = {2,3,4,5,6,7,8,9,19,11,12} u probabilities p(X = 7) = 2*3/36 = 1/6 7 = 1+6 = 2+5 = 3+4 u possibilities Poss{X = 7} = 1 the same for all numbers in U

15. Approximate Reasoning 15 Fuzzification u the extension principle defines how a value, function or set can be represented by a corresponding fuzzy membership function u extends the known membership function of a subset to a specific value, or a function, or the full set function f: X  Y membership function  A for a subset A  X extension  f(A) ( f(x) ) =  A (x) [Kasabov 1996]

16. Approximate Reasoning 16 Fuzzification Example u function f(x) = (x-1) 2 u known samples for membership function “about 2” u membership function of f(“about 2”) x 01234 f(x) 10149 1234 “about 2” 0.51 0 x 1234 f(x) 0149 f (“about 2”) 0.51 0 [Kasabov 1996]

17. Approximate Reasoning 17 Defuzzification u converts a fuzzy output variable into a single-value variable u widely used methods are u center of gravity (COG) v finds the geometrical center of the output variable u mean of maxima v calculates the mean of the maxima of the membership function [Kasabov 1996]

18. Approximate Reasoning 18 Fuzzy Logic Translation Rules u describe how complex sentences are generated from elementary ones u modification rules u introduce a linguistic variable into a simple sentence v e.g. “John is very tall” u composition rules u combination of simple sentences through logical operators v e.g. condition (if... then), conjunction (and), disjunction (or) u quantification rules u use of linguistic variables with quantifiers v e.g. most, many, almost all u qualification rules u linguistic variables applied to truth, probability, possibility v e.g. very true, very likely, almost impossible

19. Approximate Reasoning 19 Fuzzy Probability u describes probabilities that are known only imprecisely u e.g. fuzzy qualifiers like very likely, not very likely, unlikely u integrated with fuzzy logic based on the qualification translation rules v derived from Lukasiewicz logic

20. Approximate Reasoning 20 Fuzzy Inference Methods u how to combine evidence across fuzzy rules u Poss(B|A) = min(1, (1 - Poss(A)+ Poss(B))) v implication according to Max-Min inference u also Max-Product inference and other rules u formal foundation through Lukasiewicz logic v extension of binary logic to infinite-valued logic

21. Approximate Reasoning 21 Fuzzy Inference Rules u principles that allow the generation of new sentences from existing ones u the general logical inference rules (modus ponens, resolution, etc) are not directly applicable u examples u entailment principle u compositional rule X,Y are elements F, G, R are relations X is F F  G X is G X is F (X,Y) is R Y is max(F,R)

22. Approximate Reasoning 22 Example Fuzzy Reasoning 1 u bank loan decision case problem u represented as a set of two rules with tables for fuzzy set definitions  fuzzy variables CScore, CRatio, CCredit, Decision  fuzzy values high score, low score, good_cc, bad_cc, good_cr, bad_cr, approve, disapprove Rule 1: If (CScore is high) and (CRatio is good_cr) and (CCredit is good_cc) then (Decision is approve) Rule 2: If (CScore is low) and (CRatio is bad_cr) or (CCredit is bad_cc) then (Decision is disapprove ) [Kasabov 1996]

23. Approximate Reasoning 23 Example Fuzzy Reasoning 2 u tables for fuzzy set definitions [Kasabov 1996] CScore 150155160165170175180185190195200 high 0000000.20.7111 low 110.80.50.2000000 CCredit 012345678910 good_cc 1110.70.3000000 bad_cc 0000000.30.7111 CRatio 0.10.30.40.410.420.430.440.450.50.71 good_cc 110.70.30000000 bad_cc 00000000.30.711 Decision 012345678910 approve 0000000.30.7111 disapprove 1110.70.3000000

24. Approximate Reasoning 24 Advantages and Problems of Fuzzy Logic u advantages u foundation for a general theory of commonsense reasoning u many practical applications u natural use of vague and imprecise concepts u hardware implementations for simpler tasks u problems u formulation of the task can be very tedious u membership functions can be difficult to find u multiple ways for combining evidence u problems with long inference chains u efficiency for complex tasks

25. Approximate Reasoning 25 Important Concepts and Terms u approximate reasoning u common-sense reasoning u crisp set u default reasoning u defuzzification u extension principle u fuzzification u fuzzy inference u fuzzy rule u fuzzy set u fuzzy value u fuzzy variable u imprecision u inconsistency u inexact knowledge u inference u inference mechanism u knowledge u linguistic variable u membership function u non-monotonic reasoning u possibility u probability u reasoning u rule u uncertainty

26. Approximate Reasoning 26 Summary Approximate Reasoning u attempts to formalize some aspects of common- sense reasoning u fuzzy logic utilizes linguistic variables in combination with fuzzy rules and fuzzy inference in a formal approach to approximate reasoning u allows a more natural formulation of some types of problems u successfully applied to many real-world problems u some fundamental and practical limitations v semantics, usage, efficiency