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Expert System
Seyed Hashem Davarpanah
[email protected]
University of Science and Culture
Inexact Reasoning
References:
 Jackson, Chapter 19, Truth Maintenance Systems
 Giarratano and Riley, Chapters 4 and 5
 Luger and Stubblefield 'Artificial Intelligence', Addison-Wesley,
2002, Chapter 7
Knowledge & Inexact Reasoning
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inexact knowledge (truth of  not clear)
incomplete knowledge (lack of knowledge about
)
defaults, beliefs (assumption about truth of )
contradictory knowledge ( true and false)
vague knowledge (truth of  not 0/1)
Inexact Reasoning
Inexact Reasoning
 CF Theory - uncertainty
uncertainty about facts and conclusions

Fuzzy - vagueness
truth not 0 or 1 but graded (membership fct.)
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Truth Maintenance - beliefs, defaults
assumptions about facts, can be revised

Probability Theory - likelihood of events
statistical model of knowledge
Inexact Reasoning not necessary ...
NOT necessary when assuming:
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complete knowledge about the "world"
no contradictory facts or rules
everything is either true or false
This corresponds formally to a complete consistent theory in FirstOrder Logic, i.e.
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everything you have to model is contained in the theory, i.e. your
theory or domain model is complete
facts are true or false (assuming your rules are true)
your sets of facts and rules contain no contradiction (are consistent)
Exact Reasoning:
Theories in First-Order Predicate Logic
Theory (Knowledge Base) given as a set of well-formed formulae.
Formulae include facts like
mother (Mary, Peter)
and rules like
mother (x, y)  child (y, x)
Reasoning based on applying rules of inference of first-order
predicate logic, like Modus Ponens:
p, pq
q
If p and pq given then q can be inferred (proven)
Forms of Inexact Knowledge

uncertainty (truth not clear)
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incomplete knowledge (lack of knowledge)
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assume P is true, as long as there is no counter-evidence (i.e. that ¬P is
true)
assume P is true with Certainty Factor
contradictory knowledge (true and false)


P true or false not known ( defaults)
defaults, beliefs (assumptions about truth)

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probabilistic models, multi-valued logic (true, false, don't know,...),
certainty factor theory
inconsistent fact base; somehow P and ¬P true
vague knowledge (truth value not 0/1; not crisp sets)

graded truth; fuzzy sets
Inexact Knowledge - Example
Person A walks on Campus towards the bus stop. A few hundred
yards away A sees someone and is quite sure that it's his next-door
neighbor B who usually goes by car to the University. A screams
B's name.
Q: Which forms of inexact knowledge and reasoning are
involved here?
default - A wants to take a bus
belief, (un)certainty - it's the neighbor B
probability, default, uncertainty - the neighbor goes home by car
default - A wants to get a lift
default - A wants to go home
Examples of Inexact Knowledge
Person A walks on Campus towards the bus stop. A few hundred yards
away A sees someone and is quite sure that it's his next-door neighbor B
who usually goes by car to the University. A screams B's name.
Fuzzy - a few hundred yards
define a mapping from "#hundreds" to 'few', 'many', ...
not uncertain or incomplete but graded, vague
Probabilistic - the neighbor usually goes by car
probability based on measure of how often he takes car;
calculates always p(F) = 1 - p(¬F)
Belief - it's his next-door neighbor B
"reasoned assumption", assumed to be true
Default - A wants to take a bus
assumption based on commonsense knowledge
Dealing with Inexact Knowledge
Methods for representing and handling:
1. incomplete knowledge: defaults, beliefs
 Truth Maintenance Systems (TMS); non-monotonic reasoning
2. contradictory knowledge: contradictory facts or different
conclusions, based on defaults or beliefs
 TMS, Certainty Factors, ... , multi-valued logics
3. uncertain knowledge: hypotheses, statistics
 Certainty Factors, Probability Theory
4. vague knowledge: "graded" truth
 Fuzzy, rough sets
5. inexact knowledge and reasoning
 involves 1-4; clear 0/1 truth value cannot be assigned
Truth Maintenance
Systems
Truth Maintenance
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Necessary when changes in the fact-base lead to
inconsistency / incorrectness among the facts 
non-monotonic reasoning
A Truth Maintenance System tries to adjust the
Knowledge Base or Fact Base upon changes to
keep it consistent and correct.
A TMS uses dependencies among facts to keep
track of conclusions and allow revision /
retraction of facts and conclusions.
Non-monotonic Reasoning
non-monotonic reasoning
 The set of currently valid (believed) facts does NOT increase
monotonically.
 Adding a new fact might lead to an inconsistency which
requires the removal of one of the contradictory facts.
 Thus, the set of true (or: believed as true) facts can shrink
and grow with reasoning.
 This is why it’s called “non-monotonic reasoning”.
 In classical logic (first-order predicate logic) this does not
happen. Once a fact is asserted, it’s forever true.
Non-monotonic Reasoning - Example
Example: non-monotonic reasoning
Your are a student, it's 8am
, you are in bed.
You slip out of your dreams and think: Today is Sunday. No classes today.
l don't have to get up. You go back to sleep.

You wake up again. It's 9:30am
now and it is slowly coming to your
mind: Today is Tuesday. What an unpleasant surprise.
P1 = today-is-Tuesday
P3 = have-class-at-10am
P5 = have-to-get-up
P2 = today-is-Sunday
P4 = no-classes
P6 = can-stay-in-bed
Non-monotonic Reasoning - Example
P1 = today-is-Tuesday
P3 = have-class-at-10am
P5 = have-to-get-up
P1  P3  P5
P2  P4  P6
P2 = today-is-Sunday
P4 = no-classes
P6 = can-stay-in-bed
Assume: P1 and P2, P3 and P4, P5
and P6 are mutually exclusive, i.e.
P1  P2, P3  P4, P5  P6
assume P2; conclude  P1 ; P4 ;  P3 ; P6 ;  P5
assume P1; conclude  P2 ; P3 ;  P4 ; P5 ;  P6

Truth Maintenance Theories
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TMS are often based on dependency-directed backtracking to the
point in reasoning where a wrong assumption was used.
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McAllester (1978,1980)
“propositional constraint propagation”
employs a dependency network which reflects the
justification of conclusions of new facts
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Doyle (1979)
justification based Truth Maintenance System
Truth Maintenance Theories - McAllester
McAllester “propositional constraint propagation”


network representing conclusions, where
proposition-nodes are connected if one of the nodes
is a reason for concluding the other node.
Example:
pq
(pq)
If p is known to be true, q can be concluded.
Connections from p and pq to q mean that p and
pq are reasons to conclude p.
Truth Maintenance Theories - McAllester
McAllester (1980)
proposition-nodes are connected if one of the nodes is a reason for
concluding the other node (simplified version).
Example:
Connections from p and pq to combination and then to q
represent justification for q
p  q
p
p  q
p
q
Truth Maintenance Theories - Doyle
Doyle (1979)
deals with beliefs as justified assumptions.
As long as there is no contra-evidence for a fact (belief) we can assume
that it is true.
INp facts which support P; OUTp facts which prevent P.
Distinguishes:
 Premises - always true (INp = OUTp = )
 Deductions - derived (INp  ; OUTp = )
 Assumptions – depends (INp = ; OUTp  )
Truth Maintenance Theories - Doyle
Doyle (1979)
As long as there is no contra-evidence for a fact (belief) we can
assume that it is true.
Theory is based on the concept of Support-Lists (SL).
A Support-List of a Fact (Belief) P specifies Facts (Beliefs) which
support the conclusion of the Fact P or prevent its conclusion.
The TMS maintains and updates the set of current Facts/Beliefs
if changes occur. Uses justification networks, similar to
McAllester’s dependency networks.
Certainty Factor
Theory
Certainty Factor Theory
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Certainty Factor CF of Hypothesis H
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ranges between -1 (denial of H) and +1 (confirmation of H)
allows the ranking of hypotheses
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Based on measures of belief MB and disbelief MD
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MB is expressing the belief that H is true
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MD is expressing the belief that H is not true
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MB is not 1-MD - it’s not like probabilities
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Experts determine values for MB, MD of H based on given
evidence E  subjective
Stanford Certainty Factor Theory
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Certainty Factor CF of Hypothesis H is based on difference
between Measure of Belief MB and Measure of Disbelief MD
in hypothesis H, given evidence E.
Certainty Factor of hypothesis H given evidence E:
CF (H|E) = MB(H|E) – MD(H|E)


-1  CF(H)  1
Can integrate different experts’ assessments.
Basis to combine support/rejection for H within one rule and
using different rules.
Stanford Certainty Factor Theory
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Remember the base rule for Certainty Factor CF (H|E) :
CF (H|E) = MB(H|E) – MD(H|E)
-1  CF(H)  1
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Integrate Certainty Factors into reasoning.
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CF-value for H calculated using CFs of premises P in rule
CF(H) = CF(P1 and P2) = min (CF(P1),CF(P2))
CF(H) = CF(P1 or P2) = max (CF(P1),CF(P2))

CF-value for H combined from different rules, experts, ...
CF(H) = CF1 + CF2 – CF1∙ CF2if both CF1,CF2 > 0
CF(H) = CF1 + CF2 + CF1∙ CF2
if both CF1,CF2  0
CF(H) =
CF1 + CF2
else
1 – min ( |CF1|,|CF2| )
Characteristics of Certainty Factors
(Believed)
Probability
Aspect
MB MD CF
Certainly true
P(H|E) = 1
1
0
1
Certainly false
P(H|E) = 1
0
1
-1
No evidence
P(H|E) = P(H)
0
0
0
Ranges
measure of belief
measure of disbelief
certainty factor
0 ≤ MB ≤ 1
0 ≤ MD ≤ 1
-1 ≤ CF ≤ +1
Probability Theory
Basics of Probability Theory
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mathematical approach to process uncertain information
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sample space (event) set: S = {x1, x2, …, xn}
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collection of all possible events
probability p(xi) is likelihood that the event xiS occurs
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non-negative values in [0,1]
total probability of the sample space is 1,  p(xi , xiS) = 1
experimental probability
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based on the frequency of events
subjective probability (CF Theories, like Dempster-Shafer, ...)
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based on expert assessment
Compound Probabilities
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for independent events
do not affect each other in any way
 example: cards and events “hearts” and “queen”
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joint probability of independent events A and B
P(A  B) = |A  B| / |S| = P(A) * P(B)
where |S| is the number of elements in S

union probability of independent events A and B
P(A  B) = P(A) + P(B) - P(A  B)
= P(A) + P(B) - P(A) * P (B)
Situation in which either event occurs. Subtract probability of
their accidental co-occurrence - P(A  B) is already included in
P(A)+P(B) and would otherwise be counted twice.
Compound Probabilities
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For mutually exclusive events
can not occur together at the same time
 Examples: one dice and events “1” and “6”; one coin
and events “heads” and “tail”
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joint probability of two different events A and B
P(A  B) = 0
Throw dice and show both “1” and “6” cannot happen.
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union probability of two events A and B
P(A  B) = P(A) + P(B)
Throw coin and show either “heads” or “tail”.
This is also called “special addition”.
Conditional Probabilities
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describes dependent events
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affect each other in some way
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Example: Throw dice twice; second throw has to give
larger value than first throw.
conditional probability
of event A given that event B has already occurred
P(A|B) = P(A  B) / P(B)
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example: B = throw(x); A = throw(y>x)
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See next slide.
Conditional Probabilities
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Example: B = throw(x); A = throw(y>x)
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P(A|B) = P(throw x and then throw y with y>x)
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P(A|B) = P(A  B) / P(B)
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P(A  B) = P(throw x)  P(throw y, y>x) = 1/6  (1/6  (6-x))
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If x=5 then P(AB) = 1/6  1/6  (6-5) = 1/36
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If x=1 then P(AB) = 1/6  1/6  5 = 5/36
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P(B) = P(throw x) = 1/6
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P(A|B) = P(A  B) / P(B)
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If x=1 then P(A|B) = 5/36*6 = 5/6  0.8...
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If x=5 then P(A|B) = 5/36*1 = 5/36  0.14
Bayesian Approaches
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derive the probability of a cause given a
symptom
has gained importance recently due to advances
in efficiency
more computational power available
 better methods
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especially useful in diagnostic systems
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medicine, computer help systems
inverse or a posteriori probability
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inverse to conditional probability of an earlier event
given that a later one occurred
Bayes’ Rule for Single Event
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single hypothesis H, single event E
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P(H | E) = (P(E | H) * P(H)) / P(E)
or
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P(H | E) = (P(E | H) * P(H) /
(P(E | H) * P(H) + P(E | H) * P(H) )
Example
Fred and the Cookie Bowls
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Suppose there are two bowls full of cookies.
Bowl #1 has 10 chocolate chip cookies and 30 plain cookies, while
bowl #2 has 20 of each.
Fred picks a bowl at random, and then picks a cookie at random.
We may assume there is no reason to believe Fred treats one bowl
differently from another, likewise for the cookies.
The cookie turns out to be a plain one.
How probable is it that Fred picked it out of bowl #1?
From: http://en.wikipedia.org/wiki/Bayes'_theorem
The Cookie Bowl Problem
“What’s the probability that Fred picked bowl #1, given that he has a plain cookie?”
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Event A is that Fred picked bowl #1.
Event B is that Fred picked a plain cookie.
Compute P(A|B). We need:
 P(A) - the probability that Fred picked bowl #1 regardless of any other information.
Since Fred is treating both bowls equally, it is 0.5.
 P(B) is the probability of getting a plain cookie regardless of any information on the
bowls. It is computed as the sum of the probability of getting a plain cookie from a
bowl multiplied by the probability of selecting this bowl. We know that the probability
of getting a plain cookie from bowl #1 is 0.75, and the probability of getting one from
bowl #2 is 0.5. Since Fred is treating both bowls equally the probability of selecting
any one of the bowls is 0.5 (see next slide).
 Thus, the probability of getting a plain cookie overall is 0.75×0.5 + 0.5×0.5 = 0.625.
 P(B|A) is the probability of getting a plain cookie given that Fred has selected bowl
#1. From the problem statement, we know this is 0.75, since 30 out of 40 cookies in
bowl #1 are plain.
The Cookie Bowls
Number of cookies in each bowl
by type of cookie
Bowl #1
Bowl #2
Totals
Chocolate
10
20
30
Plain
30
20
Total
40
40
Relative frequency of cookies in
each bowl
by type of cookie
Bowl #1
Bowl #2
Totals
Chocolate
0.125
0.250
0.375
50
Plain
0.375
0.250
0.625
80
Total
0.500
0.500
1.000
The table on the right is derived from the table on the left by dividing each entry by the total
Fred and the Cookie Bowl

Given all this information, we can compute the probability of Fred
having selected bowl #1 (event A) given that he got a plain cookie
(event B), as such:

As we expected, it is more than half.
http://en.wikipedia.org/wiki/Bayes'_theorem
Fuzzy Set Theory
Fuzzy Set Theory (Zadeh)
Aimed to model and formalize "vague" Natural Language terms and
expressions.
Example: Peter is relatively tall.
Define a set of fuzzy sets (predicates or categories), like tall, small.
Each fuzzy subset has an associated membership function mapping
(exact) domain values into a (graded) membership value.
tall would be one fuzzy subset defined by such a function which
takes the height (e.g. in inches) as input, and determines a fuzzy
membership-value (between 0 and 1) for tall and small as output.
Fuzzy Set Membership Function
If Peter is 6' high, and the fuzzy membership value of tall for 6' is
0.9, then Peter is quite tall.
Review
Inexact Reasoning
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uncertain reasoning – uncertainty about facts and/or rules –
CF Theory
vagueness – truth not 0 or 1 - Fuzzy sets and Fuzzy logic
beliefs, defaults – assumptions about truth, can be revised –
non-monotonic reasoning, Truth Maintenance System
likelihood of event – statistical model of knowledge Probability Theory
Other Forms of Representing and
Reasoning with Inexact Knowledge

Logics
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Explicit modeling of Belief- and KnowsOperators in Modal Logic or Autoepistemic
Logic.
Probabilistic Reasoning

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Bayes’ Theory
Dempster-Shafer Theory