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Extensions to FOL
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Extensions to FOL•When the form of the statements provides useful information
•Rule-based systems
•Frame systems
•When FOL isn’t enough
•Default reasoning and circumscription
•Reasoning with uncertainty
•Degrees of membership (fuzzy logic)
•Reasoning about belief
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What Does Really Mean?
HasChildren Mother A B definition of B
Raining Wet A B A causes B
Fever Infection A B A is a symptom of BB causes A
LikeBig GetHummer A B whenever A occurs,B usually does too
So how should we reason with these very different things?
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Rule-Based SystemsThe logic:
a b is equivalent to: a b
So, given: fever infection fever infection fever
Conclude: infection
Given: fever infection fever infection infection
Conclude: fever
But are these two inferences equally useful?
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An Example for a Design Task: XCON (1982)
From XCON (1982):If: the most current active context is distributing
massbus devices, and there is a single-port disk drive that has not been
assigned to a massbus, and there are no unassigned dual-port disk drives, and the number of devices that each massbus should support is known, and there is a massbus that has been assigned at least one disk drive that should support additional disk drives, and the type of cable needed to connect the disk drive to the previous
device on the massbus is knownThen: assign the disk drive to the massbus.
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An Example for a Diagnosis Task: Mycin (1975)
If: (1) the stain of the ogranism is gram-positive, and (2) the morphology of the organism is coccus, and (3) the growth conformation of the organism is clumps
Then: there is suggestive evidence (0.7) that the identity of the organism is staphylococcus.
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Simple Examples Today
eXpertise2Go: http://www.expertise2go.com/
AcquiredIntelligence: http://www.aiinc.ca/demos/
(whales, graduate school)
DecisionScript: http://www.vanguardsw.com/decisionscript/Examples.htm
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Implementation of Rule-Based Systems
•Prolog: The KB: reply(sampcor) :- a, b
A query: ?- reply(X)
Use backward chaining to answer the question.
•Expert system shells: Typically combine methods
If (1) the suggested technique category is correlation and regression analysis, and (2) one of the values of the desired correlation/regression result is a measure of the degree to which 2 variables move together Then the suggested analysis approach is to calculate a sample correlation coefficient
If a b reply(sampcor)
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Expert System Shells
Some rules are best used in forward chaining mode. For example, data collection and reasoning from symptoms.
Other rules (e.g., how to achieve goals) are best used in backward chaining mode.
All these rules may also want to exploit other kinds of knowledge, like default information associated with classes of objects:
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Inheritance, Again
birds canfly T
ISA ISA
robins ostriches canfly F Instance-of Instance-of
Tweety Richy
Scooter is a bird. Can Scooter fly?
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Inheritance
Objects inherit from their parents:•Scooter inherits from Bird the facts that:
•its birthmode is eggs, and•it has two wings
Should Scooter inherit from Bird the fact that it can fly?
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Default Reasoning
•The importance of default reasoning
•Default reasoning is nonmonotonic.
•Techniques for default reasoning
•Inheritance
•The closed world assumption
•Circumscription
•Maintaining consistency in nonmonotonic reasoning systems
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Default Reasoning - Examples
Inheritance from superclasses:x bird(x) canfly(x) UNLESS ostrich(x)
The “normal” case:x bird(x) canfly(x) UNLESS (broken-wing(x)
sick(x) in(oil-slick, x))
The closed world assumption:can cats fly?
Abduction:infection fever Given fever, can we
conclude infection?
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Default Reasoning in Nonmonotonic
Inference in FOL systems is monotonic:
The addition of any new assertion that is consistent with the KB will never cause a formula that was previously true to become false.
Default reasoning may be nonmonotonic:Birds can fly.Tweety is a bird. Tweety can fly.
But what if we now learn:
Tweety is an ostrich. orTweety has a broken wing.
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Implementing Inheritance
birds canfly T
ISA ISA
robins ostriches canfly F Instance-of Instance-of
Tweety Richy
If we implement inheritance procedurally, we don’t have to write the UNLESS clauses. We assume Tweety isn’t an ostrich.
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The Closed World Assumption
The CWA: Any ground atomic sentences that are not asserted to be true in the KB can be assumed to be false.
We make the closed world assumption for two reasons:•We have to. In any complex domain, there may be a huge number of possible facts and there isn’t time to mention each of them explicitly:
•A database of classes mentions the ones that are offered.•An inventory database mentions all the objects on hand.•An airline scheduling system assumes that it will be told if the power is out or the terminal has burned down or is held by terrorists or there is a storm.
•It is consistent with felicitous human communication.
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Implementing the CWA: Negation as Failure
A common way to implement the CWA: Interpret failure to prove p as a proof of p.
Example:
hasonhand(x) uses(x) mustorder(x)
How do we prove hasonhand(x)?
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Circumscription
x bird(x) canfly(x) UNLESS (broken-wing(x) sick(x) in(oil-slick, x))
Is different from:x bird(x) broken-wing(x) sick(x) in(oil-slick, x)
canfly(x)Or:x bird(x) adult(x) withmother(x)
One way to implement this is to create the predicate Abnormal:
x bird(x) canfly(x) UNLESS Abnormal(x)(broken-wing(x) sick(x) in(oil-slick, x)) Abnormal(x)
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Circumscription
Then we circumscribe Abnormal, i.e., we prefer models in which Abnormal is true of the smallest possible number of individuals consistent with the rest of the KB.
But what happens if we are told just:bird(Tweety) and then we conclude canfly(Tweety)
Then we are told:broken-wing(Tweety)
How do we undo the conclusion canfly(Tweety)?
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Abbott, Babbitt, and Cabot
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Truth Maintenance Systems
The basic idea: Associate with each assertion one or more justifications. Believe any assertion with at least one valid justification.
Each justification is composed of two parts:•An IN-list•An OUT-list
We will define the operation of a TMS that operates as a service to a separate reasoning system. The TMS doesn’t make choices. It is just a bookkeeper.
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The Structure of a Justification
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Before Alibis
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Abbott’s Situation, with Alibi
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Babbitt’s Situation
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Cabot’s Situation
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The Big Picture
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New Facts Come In
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Deciding How to Resolve the Conflict
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Abduction
Examples:
infection fever measles spotsraining wetsidewalks
If given:
fever, can we conclude infection?spots, can we conclude measles?wetsidewalks, can we conclude raining?
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Uncertainty and Fuzziness
•Degrees of truthJohn is tall.John is very tall.
•Probability of truthJohn is in Austin (p = .6)Coin is heads (p = .5)
•Certainty of beliefJohn is in Austin (c = .2) a wild guessCoin is heads (c = 1) sure it’s 50/50
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When Must We Deal with Uncertainty?•Diagnosis:
Observe: spots, fever, headache. What’s wrong with the patient?Observe: clothes are wrinkled and hot. What’s wrong with the dryer?
•Interpretation:
•Speech understanding•Language understanding•Image understanding•Data interpretation
•Planning:
If I turn the steering wheel, where will the car go?
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Probabilistic Reasoning
P(strep) = x (the probability that a random person has strep right now)
P(staph) = y (similar)
Suppose that we can use the same drug in either case, so we want to know
P(strep staph) =
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Probabilistic Reasoning
P(strep) = x (the probability that a random person has strep right now)
P(staph) = y (similar)
Suppose that we can use the same drug in either case, so we want to know
P(strep staph) = P(strep) + P(staph) - P(strep staph)
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Probabilistic ReasoningSuppose There are Three Factors
P(a b c) = P(a) + P(b) + P(c) - P(a b) - P(a c) - P(b c)+P(a b c)
P(a b c) = P(a b c)- P(a) - P(b) - P(c) + P(a b) + P(a c) + P(b c)
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Conditional Probability
P(measles spots) = P(measles | spots) P(spots) definitionP(measles spots) = P(spots | measles) P(measles)
P(measles | spots) = P(measles spots) definition P(spots)
P(measles | spots) = P(spots | measles) P(measles) Bayes Rule
P(spots)
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Examples from Diagnosis and Interpretation
P(measles | spots) = P(spots | measles) P(measles) P(spots)
P(word x | sound y) = P(sound y | word x) P(word x) P(sound y)
Word = Argmax(P(sound y | word x) P(word x)) x
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Naïve Bayes ClassifierWhat if Multiple Observations Are Available? P(measles | spots fever)
= P(spots fever | measles) P(measles) P(spots fever)
Assume spots and fever are independent:
= P(spots fever | measles) P(measles) P(spots fever)
= P(spots|measles) P(fever | measles) P(measles) P(spots) P(fever)
Disease = Argmax(P(spots|x) P(fever |x) P(x) ) x
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Not Quite So Naïve Bayes ClassifierComparing chicken pox (pox) to measles:
P(measles | spots fever)
= P(spots fever | measles) P(measles) P(spots fever)
Assume spots and fever are independent given measles or pox and measles and pox are independent:
= P(spots|measles) P(fever | measles) P(measles) P(spots fever)
= P(spots|measles) P(fever | measles) P(measles) P(spots|measles)*P(fever|measles)*P(measles) + P(spots|pox)* P(fever|pox)*P(pox)
Disease = Argmax(P(spots|x) P(fever |x) P(x) ) x
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Learning Naïve Bayes Classification
An important aspect of naïve Bayes classification is that a classifier can be learned.
Where do numbers like p(spots | measles) come from?
Answer:
patient diagnosis fever cough spots sore throat sneezes achy
1 Measles Yes Yes Yes No No Yes
2 Measles Yes No Yes No No Yes
3 Measles Yes No No No No No
4 Chickenpox Yes No Yes Yes No No
5 Chickenpox Yes No Yes No No No
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Various ad hoc Approaches
Unfortunately, it often happens that we don’t have all the joint probabilities required to compute true probabilities for our conclusions. So a variety of approximate methods are used.
http://www.expertise2go.com/webesie/tutorials/Inference/Confidence1.htm