Advantages of Approximate Reasoning
KBS : Approximate Reasoning Motivation
reasoning for real-world problems involves missing Motivation Fuzzy Logic
knowledge, inexact knowledge, inconsistent facts or
Fuzzy Sets and Natural
Objectives
rules, and other sources of uncertainty Language
Approximate Reasoning Membership Functions while traditional logic in principle is capable of
Variation of Reasoning with Linguistic Variables capturing and expressing these aspects, it is not
Uncertainty very intuitive or practical
Important Concepts and Commonsense Reasoning
Terms explicit introduction of predicates or functions
Chapter Summary
many expert systems have mechanisms to deal with uncertainty
sometimes introduced as ad-hoc measures, lacking a sound foundation Approximate Reasoning 1 Approximate Reasoning 2
Objectives Approximate Reasoning
be familiar with various approaches to approximate reasoning
inference of a possibly imprecise conclusion from
understand the main concepts of fuzzy logic
possibly imprecise premises fuzzy sets
useful in many real-world situations linguistic variables
one of the strategies used for “common sense” reasoning
fuzzification, defuzzification
fuzzy inference
frequently utilizes heuristics
evaluate the suitability of fuzzy logic for specific tasks especially successful in some control applications
application of methods to scenarios or tasks often used synonymously with fuzzy reasoning
apply some principles to simple problems
although formal foundations have been developed, some problems remain
Approximate Reasoning 3 Approximate Reasoning 4 Approaches to Approximate Advantages of Approximate Reasoning Reasoning
fuzzy logic
reasoning based on possibly imprecise sentences
common sense reasoning default reasoning allows the emulation of some reasoning strategies used by humans
in the absence of doubt, general rules (“defaults) are applied concise
default logic, nonmonotonic logic, circumscription
can cover many aspects of a problem without explicit representation of the details analogical reasoning
conclusions are derived according to analogies to similar quick conclusions situations
can sometimes avoid lengthy inference chains Approximate Reasoning 5 Approximate Reasoning 6
Problems of Approximate Reasoning Fuzzy Logic nonmonotonicity approach to a formal treatment of uncertainty
inconsistencies in the knowledge base may arise as new
relies on quantifying and reasoning through natural sentences are added language
sometimes remedied by truth maintenance systems
linguistic variables
used to describe concepts with vague values semantic status of rules
default rules often are false technically fuzzy qualifiers
efficiency a little, somewhat, fairly, very, really, extremely
although some decisions are quick, such systems can be
fuzzy quantifiers
very slow almost never, rarely, often, frequently, usually, almost always especially when truth maintenance is used hardly any, few, many, most, almost all
Approximate Reasoning 7 Approximate Reasoning 8
Approximate Reasoning 9 Fuzzy Sets and Example
Possibility Measure degree to which an individual element x is a potential member in the fuzzy set S Poss{x
probabilities p(X = 7) = 2*3/36 = 1/6 7 = 1+6 = 2+5 = 3+4
X is an integer in U = {2,3,4,5,6,7,8,9,19,11,12}
Example: rolling dice
possibility refers to allowed values
probability expresses expected occurrences of eventsApproximate Reasoning 12 Possibility vs.. Probability
Poss(A ∧ B) = min(Poss(A),Poss(B)) Poss(A ∨ B) = max(Poss(A),Poss(B))
a popular one is based on minimum and maximum
various rules are used
∈S} combination of multiple premises with possibilities
short medium tall Approximate Reasoning 11
membership height (cm) 50 100 150 200 250
1
0.5
membership height (cm) 50 100 150 200 250
Approximate Reasoning 10 Fuzzy vs. Crisp Set
values in between indicate how strongly an element is affiliated with the set
0 means no, 1 means full membership
associates each element xi with a degree of membership in S:
described through a membership function µ(s) : x → [0,1]
short medium tall categorization of elements xi into a set S
1
0.5
possibilities Poss{X = 7} = 1 the same for all numbers in U Fuzzification Fuzzification Example
2 x
1
2
3
4
function f(x) = (x-1)
the extension principle defines how a value, function
1
1
4
9 f(x) or set can be represented by a corresponding fuzzy membership function
known samples for
1
2
3
4
extends the known membership function of a subset to a
membership function specific value, or a function, or the full set
0.5
1
0.5 “about 2”
“about 2” function f: X → Y membership function µ for a subset A ⊆ X
A membership function of
1
2
3
4 x f(“about 2”) extension µ ( f(x) ) = µ (x)
f(A) A
1
4
9 f(x)
0.5
1
0.5 f (“about 2”)
[Kasabov 1996] Approximate Reasoning 13 [Kasabov 1996] Approximate Reasoning 14
Defuzzification Fuzzy Logic Translation Rules
describe how complex sentences are generated from
converts a fuzzy output variable into a single-value elementary ones variable
modification rules
widely used methods are introduce a linguistic variable into a simple sentence
e.g. “John is very tall” center of gravity (COG)
composition rules
finds the geometrical center of the output variable combination of simple sentences through logical operators
mean of maxima
e.g. condition (if ... then), conjunction (and), disjunction (or)
calculates the mean of the maxima of the membership function
quantification rules use of linguistic variables with quantifiers
e.g. most, many, almost all
qualification rules linguistic variables applied to truth, probability, possibility
e.g. very true, very likely, almost impossible
[Kasabov 1996] Approximate Reasoning 15 Approximate Reasoning 16
Approximate Reasoning 17 Fuzzy Inference Methods
0.3
1
0.45
0.3
0.44
0.43
0.42
0.41
0.3
0.4
0.7
1
0.5
1 1 bad_cc good_cc
1
1
8 C Credit
9
10
7
0.7
6
0.3
0.7
0.1 C Ratio
4
5
1
1
1
8 Decision
9
10
7
0.7
6
0.3
4
1
0.3
3
0.7
2
1
1
1
0.7 bad_cc good_cc
1
1
5
0.3
how to combine evidence across fuzzy rules
principles that allow the generation of new sentences from
represented as a set of two rules with tables for fuzzy set
Example Fuzzy Reasoning 1 bank loan decision case problem
(X,Y) is R Y is max(F,R) Approximate Reasoning 19
X is F F ⊂ G X is G X is F
F, G, R are relations
X,Y are elements
compositional rule
entailment principle
examples
existing ones the general logical inference rules (modus ponens, resolution, etc) are not directly applicable
Approximate Reasoning 18 Fuzzy Inference Rules
C Score, 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)
derived from Lukasiewicz logic
integrated with fuzzy logic based on the qualification translation rules
e.g. fuzzy qualifiers like very likely, not very likely, unlikely
describes probabilities that are known only imprecisely
extension of binary logic to infinite-valued logic Fuzzy Probability
formal foundation through Lukasiewicz logic
also Max-Product inference and other rules
implication according to Max-Min inference
Poss(B|A) = min(1, (1 - Poss(A)+ Poss(B)))
definitions fuzzy variables
Rule 2: If (CScore is low) and (CRatio is bad_cr) or (CCredit is bad_cc) then (Decision is disapprove )
3
0.7
0.7
2
1
1
1
1 low high
1
1
1
150 185 200 195 190 C Score
180
[Kasabov 1996] Approximate Reasoning 20
0.2
170 175
0.2
165
0.5
160
0.8
155
1
[Kasabov 1996]
Example Fuzzy Reasoning 2
tables for fuzzy set definitions1 disapprove approve
Approximate Reasoning 21 Advantages and Problems of Fuzzy Logic
inconsistency
inexact knowledge
inference
inference mechanism
knowledge
linguistic variable
membership function
imprecision
non-monotonic reasoning
possibility
probability
reasoning
rule
advantages
fuzzy logic utilizes linguistic variables in combination with fuzzy rules and fuzzy inference in a formal approach to approximate reasoning foundation for a general theory of commonsense reasoning
many practical applications natural use of vague and imprecise concepts
hardware implementations for simpler tasks
problems
formulation of the task can be very tedious
membership functions can be difficult to find multiple ways for combining evidence
problems with long inference chains efficiency for complex tasks
Approximate Reasoning 22 Summary Approximate Reasoning
attempts to formalize some aspects of common- sense reasoning allows a more natural formulation of some types of
fuzzy value fuzzy variable
problems
successfully applied to many real-world problems some fundamental and practical limitations
semantics, usage, efficiency Approximate Reasoning 23
Important Concepts and Terms approximate reasoning
common-sense reasoning crisp set
default reasoning defuzzification
extension principle
fuzzification fuzzy inference
fuzzy rule fuzzy set
uncertainty