Approximate Reasoning 1 Expert Systems Dr. Samy Abu Nasser

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

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

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

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

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

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

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

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

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

Approximate Reasoning 11 Fuzzy Set Example membership height (cm) shortmedium tall

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

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))

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

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]

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 f(x) “about 2” x 1234 f(x) 0149 f (“about 2”) [Kasabov 1996]

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]

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

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

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

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)

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]

Approximate Reasoning 23 Example Fuzzy Reasoning 2 u tables for fuzzy set definitions [Kasabov 1996] CScore high low CCredit good_cc bad_cc CRatio good_cc bad_cc Decision approve disapprove

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

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

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