Reasoning with Conflicting Knowledge

1
Reasoning with Conflicting Knowledge

• Bob McKay
• School of Computer Science and Engineering
• College of Engineering
• Seoul National University
• Partly based on
• Russell Norvig, Edn 2, Ch 10
• M Pagnucco Introduction to Belief Revision
  www.cse.unsw.edu.au/morri/ LSS/LSS99/belief_revis
  ion.pdf

2
Outline

• Non-Monotonic Logics
• Modal non-monotonic logics
• Default Logics
• Plausible Defaults
• Abduction
• Minimalist Reasoning
• The Closed World Assumption
• Circumscription

3
Conflicting Knowledge

• Classical logic handles conflicting knowledge
  poorly
• It is a theorem of classical logic that
• p . p ? q
• From an inconsistency, we can derive everything
• Conflicting information occurs in most parts of
  ordinary life
• there are very few propositions that we can be
  absolutely certain will never be falsified
• Full inconsistency is difficult to deal with
• if we are certain that both p and p are true,
  there is clearly a problem
• Most often, the situation is more like
• in the absence of evidence to the contrary, p is
  a reasonable assumption
• if p were certainly derivable, we would happily
  withdraw p
• Some extension of classical logic is required to
  deal with this situation
• We want a rational approach to inconsistency

4
Non-Monotonic Logics

• In traditional logic, deducibility is monotonic
• As you add new axioms, the set of truths
  increases
• if you add a new axiom to a theory, the set of
  theorems now derivable contains the set of
  theorems previously derivable
• as you increase the axioms, you also increase the
  theorems
• We have already encountered one form of
  non-monotonic logic
• the default inheritance discussed in connection
  with semantic nets and frames
• if the default habitat of kangaroos is
  grassland, and if Skippy is a kangaroo, one
  consequence is that the habitat of Skippy is
  grassland
• If you later add the explicit fact that Skippy's
  habitat is backyards, then you must retract the
  conclusion that Skippy's habitat is grassland
• In general, non-monotonic logics allow for
  decreasing truth

5
Modal Non-Monotonic Logic

• Modal logics extend traditional logic by adding a
  modal operator' to the logic
• Typically, the new operator represents some form
  of knowledge of the system
• Thus Kp represents the proposition
• the system knows that p'
• To be useful, such an operator must satisfy
• K(p ? q) ? (Kp ? Kq)
• It is also reasonable to assume that if p is
  provable, then Kp
• On the other hand, we would not want to assume in
  general that
• p ? Kp

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Modal Non-Monotonic Logic

• Within this range, there are a wide range of
  possible modal logics, with minor differences in
  their axioms
• Some of the possible axioms are
• Kp ? p
• ? X, Kp ? K (? X p)
• Kp ? KKp
• Kp ? KKp
• NB Depending on the text you use, you may instead
  come across the M operator, meaning it is
  believable that. They are related by
• Mp Kp

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Modal Logic and Conflicting Knowledge

• Modal non-monotonic logics handle conflicting
  knowledge
• in the sense that Mp and Mp are consistent
• The various modal logics have been studied in
  traditional logic for some considerable time
• the extension to non-monotonic reasoning is
  relatively recent
• Read More
• Modal Logics and Philosophy
• http//plato.stanford.edu/entries/logic-modal/
• A more technical introduction
• http//cs.wwc.edu/aabyan/Logic/Modal.html
• Wikipedia
• http//en.wikipedia.org/wiki/Modal_logic
• John McCarthy on modal logics
• http//www-formal.stanford.edu/jmc/mcchay69/node22
  .html

8
Default Logics

• Default logics add to traditional logic specific
  extra rules for inferring consequences
• P Q
• R
• Bird(x) Flies(x)
• Flies(x)
• interpreted as
• If one believes P, and it is consistent that Q,
  then one can also believe R
• The consistency of Q is taken to be failure to
  prove Q
• Default logics are non-monotonic
• adding a new fact may make a previously
  consistent Q inconsistent
• Penguin(x)
• and therefore remove the ability to conclude R
• Flies(x)

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Default Logics Semantics

• An extension of a default theory is formed by
  taking the underlying certain theory, and adding
  defaults to it while they are consistent
• The process stops when no more defaults can be
  added without creating inconsistency
• A theory may have a number of different
  extensions
• For example, the Nixon diamond has two
  extensions
• One in which Nixon is a pacifist
• Another in which Nixon is not a pacifist
• A theory may also have no extensions
• Entailment
• A credulous system accepts any conclusion which
  is true in some extension
• A skeptical system accepts only conclusions true
  in all extensions

10
Default Logics Semantic Variants

• An extension of a default theory is formed by
  taking the underlying certain theory, and adding
  defaults to it while they are consistent
• If we restrict the defaults which may be added,
  we get new default logics
• Justified
• the theory has to be consistent with the defaults
  added so far
• Not just their conclusions
• Constrained
• All added defaults must be consistent with each
  other, and with any consequences
• Cautious
• No default that is inconsistent with any other is
  ever applied

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Default Logics Efficiency

• The efficiency of default logics directly relates
  to the cost of computing Q
• Ie of computing the consistency of Q
• In general, this may be very expensive, but for
  particular restricted logics it may be feasible
• A number of systems in the mid-late 1990s based
  on prolog technologies for closed Worlds (see
  later)
• Closed World everything that cant be proven
  false is true
• F
• F

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Default Logics References

• Wikipedia
• http//en.wikipedia.org/wiki/Default_logic
• Neat default logic simulator
• http//www.kr.tuwien.ac.at/students/dls/english/
• Stanford Encyclopedia
• http//plato.stanford.edu/entries/logic-nonmonoton
  ic/

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Modal vs Default Logics

• Modal and Default Logics appear very similar
• But consider the case where we have the axioms
• A MB ? B
• A MB ? B
• Applying standard logic, we can conclude
• MB ? B
• It might seem equivalent to a default logic
• A B and A B
• ____ _____
• B B
• we cannot reach a conclusion about B unless we
  know about the status of A

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Deduction and Abduction

• Deduction
• Given two axioms
• forall x measles(x) ? spots(x)
• measles(fred)
• We can conclude
• spots(fred)
• This is a logical inference
• Abduction
• From
• forall x measles(x) ? spots(x)
• spots(fred)
• We conclude
• measles(fred)
• This is a plausible inference

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Non-monotonic reasoning and Abduction

• Abduction gives us a powerful source of
  non-monotonic reasoning
• Abduction is permitted so long as the conclusion
  is consistent with our other knowledge
• ((? x P(x) ? Q(x)) Q(x) MP(x)) ? P(x)
• (Modal)
• ((? x P(x) ? Q(x)) Q(x) P(x))
• P(x)
• Default Logic
• A search for plausible causes

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Default Inheritance in Default Logic

• Recall the skippy' inheritance example
• We can express it as an inference rule
• kangaroo(x) habitat(x,grasslands)
• habitat(x,grasslands)
• If we have an axiom asserting that each animal
  has only one habitat
• ? x,y,z (habitat(x,y) habitat(x,z)) ? (x z)
• then in the absence of other knowledge about a
  kangaroo leapy', the logic would conclude
• habitat(leapy,grasslands)
• but in the presence of the assertion
• habitat(skippy,backyards)
• the logic would not draw the equivalent
  conclusion about skippy

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Default Inheritance in Modal Logic

• We can also express the skippy' inheritance
  example in modal logic
• (kangaroo(x) M(habitat(x,grasslands))) ?
  habitat(x,grasslands)

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Minimalist Reasoning and the Closed World
Assumption

• Many rule-based expert systems incorporate a
  simple form of the Closed World Assumption
• P is equated with a failure to prove P
• In Default logic
• F
• F
• The general form of the CWA says that the only
  objects that satisfy a predicate P are those that
  must

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Semantic Problems with theClosed World Assumption

• This simple form has two types of problems
• Semantic problems
• The CWA applies equally to all predicates, and
  does not allow us to distinguish between
  predicates
• In some situations (routes in an airline
  database, for example) it is appropriate to make
  the CWA assumption
• Many government databases are of this kind
• Often by definition
• If youre not recorded as having a licence, you
  dont have one
• In other cases (has_bought_russell__norvig, for
  example) it would not be reasonable to assume
  that the CWA applies
• a system is unlikely to contain all the valid
  assertions of this type
• Many commercial databases are of this kind

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Syntactic Problems with theClosed World
Assumption

• Inconsistent theories
• Given the knowledge base
• A(joe) v B(joe)
• the CWA forces the conclusions
• A(joe)
• B(joe)
• which is inconsistent

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Syntactic Problems with theClosed World
Assumption

• Asymmetric conclusions
• Given a knowledge base
• single(john)
• single(mary)
• and the query
• single(jane)?
• The CWA results in the answer no
• But given a knowledge base
• married(john)
• married(mary)
• and the query
• married(jane)
• The CWA still results in the answer no

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Circumscription

• In a way, circumscriptive theories are an attempt
  to answer the problems with the CWA
• by restricting its application to particular
  predicates
• In circumscription, the predicates of a theory T
  are divided into two parts
• Some predicates express properties of the objects
  of the theory
• Other predicates are intended to express that
  particular objects are abnormal in some way
• a form of CWA applies to them
• The theory is augmented with second-order axioms
• which effectively say (for each abnormal'
  predicate) that the only abnormal objects are
  those which are abnormal as a direct consequence
  of the theory

23
Default Reasoning and Circumscription

• default reasoning can be expressed along the
  lines of
• Birds that are not abnormal can fly
• An ostrich is a bird
• An ostrich cannot fly
• under circumscription, the system can conclude
• Ostriches are abnormal birds
• And in fact, ostriches are the only abnormal
  birds
• if we then add axioms
• A penguin is a bird
• A penguin cannot fly
• the system will conclude
• Penguins are abnormal birds
• Ostriches are not the only abnormal birds

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Summary

• Non-Monotonic Logics
• Modal non-monotonic logics
• Default Logics
• Plausible Defaults
• Abduction
• Minimalist Reasoning
• The Closed World Assumption
• Circumscription