Transcript DCP 1172: Introduction to Artificial Intelligence

DCP 1172
Introduction to Artificial Intelligence
Chang-Sheng Chen
Topics Covered:
•Introduction to Nonmonotonic Logic
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Monotonic logic
• Standard logic is monotonic:
• once you prove something is true, it is true
forever
• Monotonic Logic is not a good fit to reality
• If the wallet is in the purse, and the purse in is the
car, we can conclude that the wallet is in the car
• But what if we take the purse out of the car?
DCP 1172, Ch. 6
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Monotonic Logic
• Given a collection of facts D that entail some sentence
s (s is a logical conclusion of D):
• for any collection of facts D’ such that DD’ , D’ also
entails s.
• in other words: s is also a logical conclusion of any
superset of D.
DCP 1172, Ch. 6
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Nonmonotonic Logic
• In a nonmonotonic system:
• the addition of new facts can reduce the set of
logical conclusions.
• S is a conclusion of D, but is not necessarily a
conclusion of D+newfact.
• Humans use nonmonotonic reasoning constantly!
DCP 1172, Ch. 6
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What is “Non-monotonic Logic” ?
• To understand what nonmonotonic logic means simple
consider a standard example:
"all birds fly",
"Tweety is a bird",
"Does Tweet fly?".
• The obvious answer is yes,
• however what if later you learned that Tweety had a broken wing,
then the answer becomes no,
• then what if you learned that tweet was an airplane pilot, or had a
jet pack, the answer can change again.
• The important point is that as new information is added
the answers change
DCP 1172, Ch. 6
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Nonmonotonic logic
• Facts and rules can be changed at any time
• such facts and rules are said to be dynamic
• Prolog uses nonmonotonic logic
• assert(...) adds a fact or rule
• retract(...) removes a fact or rule
• assert and retract are said to be extralogical
predicates
DCP 1172, Ch. 6
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Intelligent Reasoning
• One of the characteristics associated with intelligent
systems is adaptability - the ability to deal with a
changing environment.
• Adaptation requires that a system be capable of
adding and retracting beliefs as new information is
available.
• This requires nonmonotonic reasoning.
DCP 1172, Ch. 6
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Uncertainty
• Another characteristic of intelligent systems is the ability
to reason under conditions of uncertainty.
• Another way of saying this: the ability to reason
with an incomplete set of facts.
DCP 1172, Ch. 6
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Can we implement inheritance using predicate logic?
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Pat is a Bat.
Bats are Mammals.
Bats can fly.
Bats have 2 legs.
Mammals cannot fly.
Mammals have 4 legs.
How many legs does Pat have?
DCP 1172, Ch. 6
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Inheritance
• Reasoning about inheritance of properties from one
class to another:
Bird(x)  Flies(x)
• Clearly this is not a good rule, since we know there
are exceptions.
Bird(x)  Normal(x)  Flies(x)
• This provides for exceptions, although we must define
the conditions that imply Normal(x).
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Normal(x)
• Assuming we know that:
Ostrich(x)  Bird(x)  ~Flies(x)
• we can derive:
Ostrich(x)  ~Normal(x)
• So an ostrich is not a normal bird.
• But what about all the the other things that are birds?
DCP 1172, Ch. 6
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Assumptions and Defaults
• If there is no reason to believe otherwise, assume that
Normal(x) is TRUE.
• The default is that everything is normal.
• Now we only need to supply additional information for
exceptions.
DCP 1172, Ch. 6
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How to specify defaults
• A number of formal systems have been developed to handle
defaults.
• Nonmonotonic logics formalize unsound but reasonable
patterns of reasoning with uncertain, incomplete and
inconsistent information
• Default Logic: New rule of inference
• Abduction: New interpretation of implication.
DCP 1172, Ch. 6
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And More Logics To Think About!
• Modal logic is useful for modeling reasoning about
knowledge, actions, time or obligations.
• Epistemic logics apply the techniques of modal logic to
reasoning about knowledge.
• Both individual and group knowledge is studied. The study of
epistemic logic is relevant to communication protocols and
cooperation.
• Deontic logic formalizes normative modalities.
• Deontic logic can be applied to representation of
normative (e.g. legal) knowledge.
DCP 1172, Ch. 6
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Default Reasoning with Nonmonotonic Logic
• Predicate logic with an extension:
• a modal operator M which means is “consistent
with everything we know”.
• Example:
x,y: Related(x,y)  M GetAlong(x,y)  WillDefend (x,y)
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Default Logic
• New rule of inference:
A:B
C
• If A is true and it is consistent to assume B, then C is true.
• Same idea, but now used as a rule of inference. The
new rule extends the knowledge base to a set of
plausible extensions, any new statement that is true in
all extensions is added.
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Inheritance with Default Logic
• Support for inheritance using Default Logic:
Mammal(x) : Legs(x,4)
Legs(x,4)
• In the absence of contradictory information, we
can assume anything that is a mammal has 4 legs. (also
need a rule stating that nothing can have 2 different
numbers of legs!)
DCP 1172, Ch. 6
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Abduction
• Deduction (演繹):
Given A(x)  B(x) and A(x), we assume that B(x) is true.
 Similar to forward reasoning
• [Cf.] reasoning from the general to the particular (or from cause to
effect)
• Abduction:
Given A(x)  B(x) and B(x), we assume that A(x) is true.
 Similar to backward reasoning
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Inheritance Diagrams
• The book shows how we can also express default reasoning using
diagrams.
Flying Things
Normal Facts
Default
Ostriches
Birds
Fred
Tweety
Default
DCP 1172, Ch. 6
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A Problem with NML
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x: Republican(x)  M ~ Pacifist(x)  ~ Pacifist(x)
 x: Quaker(x)  M Pacifist(x)  Pacifist(x)
Republican(Dick)
Quaker(Dick)
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Not quite this easy
• Assuming we have some mechanism for representing
defaults, there can still be problems:
• Is Nixon a pacifist?
Pacifists
Republicans
Quakers
Nixon
DCP 1172, Ch. 6
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Nixon Dilemma
• In general we must be prepared to deal with multiple,
possibly conflicting consequences of a set of facts.
• One simple idea - rank all the assumptions and use
rank to determine which to believe.
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Other approaches to handling conflicting
assumptions.
• Minimalist Reasoning
• Assume that there are fewer true statements that false
statements in the world.
• Find the smallest interpretation that satisfies all the
statements we know to be true.
• Closed World Assumption: the only objects that
satisfy a predicate are those that must.
• forces positive assertions to take priority over negative
assertions
DCP 1172, Ch. 6
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Closed world assumption
• If we are told nothing about Tweety, other than Tweety is a
bird,
• we assume that Tweety's feet are not in concrete, or
Tweety's wings are not broken, this is the closed world
assumption.
• Humans regularly make assumptions and when new evidence
appears those assumptions can be changed, causing a different
answer, thus behaving nonmonotonicaly.
DCP 1172, Ch. 6
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Using Probabilities
• Probabilities can also be used determine which
defaults apply when contradictions arise.
• Label each fact with a probability of being true.
• There is a big split in the A.I. community over whether
symbolic methods or numeric methods are best for
handling these types of issues.
DCP 1172, Ch. 6
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Summary - Nonmonotonic logic vs. Probabilty
• Nonmonotonic logic systems may miss the
importance of probability.
• Probabilistic reasoners can also represent
uncertainty, and in a different (probabilistic) way.
• These systems exhibit a different set of properties,
with which nonmonotonic logic can not effectively
deal with.
DCP 1172, Ch. 6
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Nonmonotonic vs. Classical logic
• Nonmonotonic logic does not have many essential
properties of classical first order logic, specifically semidecidability.
 In classical logic, it is possible for a system to halt
(be stuck in an infinite loop) trying to prove the
negation of something for which there is
insufficient information.
 In nonmonotonic default logic, rather than return
with no answer, the process returns with a wrong
(default answer).
DCP 1172, Ch. 6
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Summary - Nonmonotonic vs. Classical logic
• First order logic although descriptively universal, is not
effective at handling large classes of problems.
 If computers are going to handle common sense we need
to be able to have some form of default reasoning.
• Nonmonotonic logic can be used in many domains where
classical logic falls short:
 such as in the areas of default diagnosis, diagnosis,
action, and temporal logic.
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