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Uncertainty and Rules
• We have already seen that expert systems can
operate within the realm of uncertainty.
• There are several sources of uncertainty in rules:
– Uncertainty related to individual rules
– Uncertainty due to conflict resolution
– Uncertainty due to incompatibility of rules
© C. Kemke
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Figure 5.1 Major Uncertainties in
Rule-Based Expert Systems
© C. Kemke
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Figure 5.2 Uncertainty Associated
with the Compatibilities of Rules
© C. Kemke
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Knowledge Engineer
• The knowledge engineer endeavors to minimize,
or eliminate, uncertainty if possible.
• Minimizing uncertainty is part of the verification
of rules.
• Verification is concerned with the correctness of
the system’s building blocks – rules.
© C. Kemke
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Verification vs. Validation
• Even if all the rules are correct, it does not
necessarily mean that the system will give the
correct answer.
• Verification refers to minimizing the local
uncertainties.
• Validation refers to minimizing the global
uncertainties of the entire expert system.
• Uncertainties are associated with creation of rules
and also with assignment of values.
© C. Kemke
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Sources of Uncertainty
• Potential contradiction of rules – the rules may
fire with contradictory consequents, possibly as a
result of antecedents not being specified properly.
• Subsumption of rules – one rules is subsumed by
another if a portion of its antecedent is a subset of
another rule.
© C. Kemke
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Cont’d…
• Information is partial
• Information is not fully reliable.
• Representation language is inherently
imprecise.
• Information comes from multiple sources
and it is conflicting.
• Information is approximate
• Non-absolute cause-effect relationships exist
© C. Kemke
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• In many cases, our knowledge of the world is incomplete
(not enough information) or uncertain (sensors are
unreliable).
• Often, rules about the domain are incomplete or even
incorrect
• We have to act in spite of this!
• Drawing conclusions under uncertainty
© C. Kemke
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Uncertainty
• When a fact is entered in the working memory, it
receives a unique timetag – indicating when it
was entered.
• The order that rules are entered may be a factor
in conflict resolution – if the inference engine
cannot prioritize rules, arbitrary choices must be
made.
• Redundant rules are accidentally entered / occur
when a rule is modified by pattern deletion.
© C. Kemke
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Uncertainty
• Deciding which redundant rule to delete is not a
trivial matter.
• Uncertainty arising from missing rules occurs if
the human expert forgets or is unaware of a rule.
• Data fusion is another cause of uncertainty –
fusing of data from different types of
information.
© C. Kemke
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State of Uncertainty
• There are two mountains – logic and uncertainty
• Expert systems are built on the mountain of logic
and must reach valid conclusions given a set of
premises – valid conclusions given that –
– The rules were written correctly
– The facts upon which the inference engine generates
valid conclusions are true facts
© C. Kemke
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Knowledge & Inexact Reasoning
• 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)
© C. Kemke
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Inexact Reasoning
Inexact Reasoning
CF Theory - uncertainty
•
•
uncertainty about facts and conclusions
 Fuzzy - vagueness
•
truth not 0 or 1 but graded (membership fct.)
 Truth Maintenance - beliefs, defaults
•
assumptions about facts, can be revised
 Probability Theory - likelihood of events
•
© C. Kemke
statistical model of knowledge
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Inexact Reasoning not necessary ...
NOT necessary when assuming:
• 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 First-Order Logic, i.e.
• 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)
© C. Kemke
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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)
© C. Kemke
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Forms of Inexact Knowledge
• uncertainty (truth not clear)
– probabilistic models, multi-valued logic (true, false, don't
know,...), certainty factor theory
• incomplete knowledge (lack of knowledge)
– P true or false not known ( defaults)
• defaults, beliefs (assumptions about truth)
– 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)
– inconsistent fact base; somehow P and ¬P true
• vague knowledge (truth value not 0/1; not crisp sets)
– graded truth; fuzzy sets
© C. Kemke
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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
© C. Kemke
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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 nextdoor 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
© C. Kemke
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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
© C. Kemke
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



In many cases, our knowledge of the world is incomplete
(not enough information) or uncertain (sensors are
unreliable).
Often, rules about the domain are incomplete or even
incorrect
We have to act in spite of this!
Drawing conclusions under uncertainty
© C. Kemke
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Example

Goal: The agent wants to drive someone to air port
to catch a flight
Let action At = leave for airport t minutes before flight
Will At get me there on time?
Problems:
1.
2.
3.
4.
partial observability (road state, other drivers' plans, etc.)
noisy sensors (traffic reports)
uncertainty in action outcomes (flat tire, etc.)
immense complexity of modeling and predicting traffic
Hence a purely logical approach either
1.
2.
risks falsehood: “A25 will get me there on time”, or
leads to conclusions that are too weak for decision making:
“A25 will get me there on time if there's no accident on the bridge and it doesn't rain and
my tires remain intact etc etc.”
(A1440 might reasonably be said to get me there on time but I'd have to stay overnight in
the airport …)
© C. Kemke
Inexact Reasoning
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Making decisions under uncertainty
Suppose I believe the following:
P(A25 gets me there on time | …)
P(A90 gets me there on time | …)
P(A120 gets me there on time | …)
P(A1440 gets me there on time | …)
= 0.04
= 0.70
= 0.95
= 0.9999
Which action to choose? Which one is rational?
Depends on my preferences for missing flight vs. time
spent waiting, etc.
Utility theory is used to represent and infer preferences
Decision theory = probability theory + utility theory
The fundamental idea of decision theory is that an agent is rational if
and only if it chooses the action that yields that highest expected
utility, averaged over all the possible outcomes of the action.
© C. Kemke
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Uncertainty in logical rules
© C. Kemke
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Probability
Imagine an urn containing 1500 red, pink, yellow, blue
and white marbles.
Take one ball from the urn. What is:
P(black) =
P(~black) =
0
1
~ = NOT
Probabilities are all greater than or equal to zero and less
than or equal to one.
© C. Kemke
Inexact Reasoning
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Same urn:
Suppose the number of balls is as follows:
Red
400
Pink
100
Yellow
400
Blue
500
White
100
Total
1500
What is:
P(Red) =
400/1500 = .267
P(Pink) =
100/1500 = .067
P(Yellow) =
400/1500 = .267
P(Blue) =
500/1500 = .333
P(White) =
100/1500 = .067
Total =
© C. Kemke
1
Inexact Reasoning
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Joint probabilities and
independence
Define A as the event “draw a red or a pink marble.”
We know 500 marbles are either red or pink.
What are:
P(A) =
P(~A) =
© C. Kemke
400  100
1500
= .33
(1 - P(A)) = .67
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Joint probabilities and
independence (we’re getting
there)
Define B as the event, “draw a pink or white marble.”
We know 200 marbles are pink or white.
What are:
© C. Kemke
P(B) =
.133
P(~B) =
.867
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Joint probabilities and
independence
Define A as the event “draw a red or a pink marble.”
Define B as the event “draw a pink or white marble.”
What is:
P(A, B) = P(A  B)
This is the joint probability of A and B.
What color is the marble?
P(A, B) = P(pink) =
© C. Kemke
Pink
100
1500
= .0667
Inexact Reasoning
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Conditional probabilities
A
~A
The probability that a particular
event will occur, given we already
know that another event has
occurred.
B
100
pink
100
white
200
900
400
~B
1300
all
others
red
We have information to bring
to bear on the base rate
1500
500
1000
probability of the event
P(A | ~B) =
© C. Kemke
400
1300
P(~B | A) =
What is
P(A | B) =
100
200
P(~A | ~B) =
900
1300
400
500
Inexact Reasoning
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© C. Kemke
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