Reasoning with Changing Knowledge

1
Reasoning with Changing 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

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Outline

• Belief Revision
• Foundationalist Approaches
• Reason Maintenance
• Truth Maintenance
• Assumption-Based Truth Maintenance
• Coherentist Approaches
• AGM Framework for Belief Revision

3
Belief Revision

• Aims to characterise the ways in which a rational
  agent can update its beliefs on receiving new
  information
• There are two main streams
• Foundational
• Demarcates a special set of beliefs (axioms)
  requiring no justification
• Coherentist
• All beliefs are equal the aim is to find a
  maximum coherent subset
• For the foundationalist every piece of knowledge
  stands at the apex of a pyramid that rests on
  stable and secure foundations whose stability and
  security does not derive from the upper stories
  or sections. For the coherentist a body of
  knowledge is a free-floating raft, every plank of
  which helps directly or indirectly to keep all
  the others in place and no plank of which would
  retain its status with no help from the others
• Sosa

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Foundational Approaches Reason Maintenance

• Abbot, Babbitt and Cabot are suspects in a murder
  case
• Abbott has an alibi, the register of a
  respectable hotel in Albany
• Babbitt also has an alibi, for his brother-in-law
  testified that Babbitt was visiting him in
  Brooklyn at the time
• Cabot pleads alibi too, claiming to have been
  watching a ski meet in the Catskills, but we have
  only his word for that
• So we believe
• That Abbott did not commit the crime
• That Babbitt did not commit the crime
• That Abbott or Babbitt or Cabot did commit the
  crime
• But then Cabot documents his alibi
• By chance, a television camera photographed him
  at the ski meet
• We have to accept a new belief
• That Cabot did not commit the crime

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Foundational Approaches Reason Maintenance

• The detective in charge of the case decides that
  Abbott is a primary suspect
• because he was a beneficiary
• He remains a suspect until he establishes an
  alibi
• In default logic, this could be represented as
• Beneficiary(X) Alibi(X)
• Suspect(X)
• For the moment, Abbott is a suspect because we
  assume there is no alibi
• as soon as an alibi is provided, Abbott should no
  longer be viewed as a suspect
• But how can we handle this efficiently in a
  rule-based system such as CLIPS?
• One possible way is to have rules reflecting each
  possible change in defaults
• It is impossibly complex

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Justification-based Truth Maintenance

• JTMS (or TMS) Doyle
• Another way is to keep a separate database of
  justifications
• The system represents justifications in terms of
  an IN-list and an OUT-list for a proposition
• Note that the form is independent of the
  particular domain
• It is also independent of the underlying
  reasoning system
• JTMS does not care where its justifications come
  from

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Labelling Nodes

• How does the JTMS decide which nodes are IN and
  which are OUT?
• A node is labelled IN if it has a valid
  justification
• A node is labelled OUT if it has no valid
  justification
• Premisses
• There is one exception some nodes (Called
  premisses) are to be treated as given
• eg that Abbott is a beneficiary
• To avoid special cases, this is handled by giving
  them a justification from the empty node, which
  is always labelled IN

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Propagating Justification

• As the investigation progresses, the detective
  discovers that Abbott was registered at a hotel
  in Albany
• This is far from the scene of the crime

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Avoiding Circularity Well Foundedness

• What about Cabot?
• (Initially) the only support for his alibi is
  that Cabot is telling the truth
• But the only support for Cabot telling the truth
  is that he was at the ski show
• A JTMS must disallow such ill-founded reasoning
• If the support for a node consists of a chain of
  positive links back to itself, the node must be
  labelled OUT
• Even worse is a circular chain with an odd number
  of negative links in that case, there can be no
  consistent labelling
• A JTMS needs to be able to detect both these cases

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Contradictions

• The detective believes that there are no other
  suspects than Abbott, Babbitt and Cabot
• This negative knowledge is recorded as a
  justification for a special node, contradiction,
  which is never allowed to become IN
• At the moment, this isn't a problem
• Suspect Abbott is OUT because of the
  justification above
• Suspect Others is OUT because it has no
  justifications
• Suspect Babbitt is OUT because of the following

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Contradictions

• Fortunately, Suspect Cabot is still in, as above
• But what happens when the television cameras pick
  Cabot out at the ski meet?
• The CONTRADICTION node is (impermissibly) IN

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Contradictions

• The JTMS has no way of knowing that this
  particular node is a contradiction
• this must be detected by the underlying reasoning
  system
• But once the contradiction has been detected, the
  JTMS can trace back over the justifications to
  detect which of the underlying assumptions have
  caused the contradiction
• the set of OUT nodes which, if they were IN,
  would remove the contradiction

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Contradictions

• In our case, either
• Abbott's register was forged
• or Babbitt's brother-in-law lied
• or Cabot's TV tape was faked
• or there is another suspect
• Having found the candidates, the contradiction
  may be removed by making one of them IN
• But how to choose which? There are two possible
  approaches
• Leave the choice to the underlying reasoning
  system, which created the dependencies in the
  first place
• Apply simple heuristics in the JTMS

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Contradictions

• Once we have chosen the node, a justification
  must be supplied
• The justification should be minimal
• in the sense that it will be invalidated if the
  system later comes to believe any other
  justifications which would resolve the
  contradiction

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Assumption-Based Truth Maintenance Systems

• (Reiter de Kleer)
• You can think of a JTMS as doing a depth-first
  search of the space of truth assignments
  (contexts), in order to find a consistent one
• From this perspective, an ATMS is simply doing a
  breadth-first (or parallel) search of the same
  space
• The ATMS starts off, in effect, with a list of
  all possible contexts
• As reasoning proceeds, it prunes this list,
  deleting contexts as contradictions are
  discovered

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Assumption-Based Truth Maintenance Systems

• As with JTMS, the ATMS sits on top of a separate
  reasoner, which
• Creates the nodes corresponding to assertions
• Provides justifications for each node
• Informs the ATMS if any context is inconsistent
• The ATMS' role is to
• Propagate inconsistencies, ruling out other
  contexts
• Label each node with contexts where it has a
  valid justification

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Context Lattices

• A context lattice looks somewhat like a
  generalisation hierarchy
• The set A1, A2 represents the context in which
  A1 and A2 are both true
• The context lattice is very large - if there are
  n assumptions, the lattice has 2n nodes - so we
  don't want to actually build it
• The lattice provides a simple method for
  propagating inconsistency
• If a node is inconsistent, then so is every node
  above it in the lattice
• The lattice also provides a simple method for
  labelling nodes with the contexts in which it has
  a justification
• For example, suppose a node's justification
  depends on assumption A1
• Then the label A1 implies that the node is
  justified not only in context A1, but also in
  any other context which contains A1

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Context Lattice Example

• Assume that the underlying reasoner generates a
  sequence of nodes and justifications as below

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Example Continued
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Example (continued)

• We assume that the reasoner can label some
  contexts as contradictory (nogoods)
• ie not permitted as contexts
• For example, nogood A7,A8
• The ATMS first computes the labels in the third
  column, one for each justification
• If the justification is an assumption
• the label is that assumption
• If the justification is a rule
• the justification is the product of
  justifications for the antecedents of the rule
• For node 8, this would give the label
• A7,A4,A6, A7,A4,A8, A7,A4,A7,A4,A6,
  A7,A4,A7,A2,A6, A7,A8,A6, etc
• But we can simplify by
• Eliminating duplicates
• Eliminating contexts containing a nogood
  (A7,A8)
• Taking the lower bound of the contexts
• ending up with just
• A7,A4,A6

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Example (continued)

• The labels for a node are the union of the labels
  for its justifications, again taking lower bounds
  
• Updating
• Suppose the reasoner now supplies additional
  information
• namely that context A2 is nogood
• The labels for nodes 1, 2, 9 and 10 immediately
  disappear, as does one of the labels for node 12
• The only remaining suspect node is Abbott but
  node 12 still has a justification, corresponding
  to the case that A, B C are not the only
  suspects
• The ATMS can report this simplification, but an
  external source is still needed to resolve
  between them

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ATMS as Explanation Generators

• ATMS have been described above as systems for
  maintaining consistency
• But there is another perspective
• We can think of an ATMS as handling a database
  consisting of two sorts of knowledge
• Common knowledge C (the nogoods above)
• Assumptions A
• An explanation for a proposition p is
• a subset E of A such that (E U C) ? p
• E is required to be minimal
• in the sense that no subset of E has the same
  property
• Minimal subsets are what an ATMS computes
• So an ATMS can be viewed as a way of generating
  explanations

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ATMS for Synthesis Planning

• So far, we have assumed that an ATMS computes all
  minimal contexts (or explanations)
• But in many contexts, only some minimal
  explanations are required
• Consider the problem of designing logic circuits
  from basic components
• We can describe the operation of each of the
  components as part of our common knowledge C
• type(c,or-gate) value(input(i,c),1) ?
  value(output(c),1)
• Similarly, rules about connecting components can
  be axiomatised
• connected(i,j) value(i,v) gt value(j,v)
• The goal would be a proposition describing the
  input-output relations required of the desired
  circuit
• Then an explanation generated by the ATMS is a
  minimal set of assumptions which will guarantee
  the input-output relations - in other words, a
  circuit design

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ATMS for Diagnosis

• Diagnosis with ATMS works in a reverse way
• The design of the circuit is held as common
  knowledge, while the intended operation of the
  components are treated as assumptions
• (ie it is now an assumption that a particular AND
  gate actually operates as an AND gate - it might
  be faulty)
• Some basic circuit knowledge is also treated as
  an assumption
• assumption that there may be a short or open
  circuit
• The proposition to be explained is the observed
  behaviour
• Then an explanation generated by the ATMS is a
  minimal set of assumptions which will generate
  that observed behaviour
• a diagnosis
• this approach assumes complete knowledge of the
  underlying rules of operation of the system
• applies to engine diagnosis, but not medical
  diagnosis

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ATMS for Database Consistency

• ATMS provide a mechanism for maintaining the
  consistency of deductive databases
• Note that this is more than just a validity check
  
• The ATMS treats the consistency rules of the
  database as common knowledge C
• The assumptions A are the database tables
• The proposition to be explained is the particular
  constraint which has been violated
• An explanation is a minimal list of changes to
  the database which are required to restore
  consistency

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Coherentist Approaches

• Positive Coherence
• The agent must possess reasons for maintaining a
  belief
• Each belief must have positive support
• Negative Coherence
• The agent is justified in holding a belief so
  long as there is no reason to think otherwise
• Innocent until proven guilty
• Linear Coherence
• The agent adopts a foundational view of reasons
  except that if we look at a reason, the reasons
  for holding reasons etc, we would never stop
• Either we have an infinite sequence of reasons,
  or there is some circularity in the reason
  structure
• Holistic Coherence
• The agent is justified in holding a belief due to
  some relationship between the belief and all
  other beliefs held
• Pollock, 1986

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The AGM Approach

• Alchourron, Gardenfors, Makinson 1985
• An example of (and the best known) coherentist
  approach
• Aims to axiomatise the behaviour of rational
  agents in reformulating beliefs
• Where possible, belief systems should remain
  consistent
• A belief system should contain all beliefs
  logically implied by beliefs in the system
• When changing belief systems, loss of information
  should be minimised
• Beliefs held in higher regard should be retained
  in preference to those held in lower regard
• AGM systems specify axioms for three types of
  belief revision operators
• Expansion
• Contraction
• Revision
• The axioms assume a belief system (knowledge
  state) K, and a new sentence ?
• There is a special (absurd) belief state K? in
  which everything is believed

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Expansion Operator

• Expansion () is applied when a proposed new
  belief ? is consistent with K, creating a new
  belief system K?
• Axioms
• Closure
• For any sentence ? and belief system K, K? is a
  belief system
• Success
• ? ? K?
• Inclusion
• K ? K?
• Vacuity
• If ? ? K, then K? K
• Monotonicity
• If H ? K, then H? ? K?
• Minimality
• K? is the smallest belief system satisfying the
  above, and containing both ? and K

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Contraction Operator

• Contraction (-) is applied when the agent wishes
  to retract a belief ?, creating a new belief
  system K-?
• Axioms
• Closure
• For any sentence ? and belief system K, K-? is a
  belief system
• Inclusion
• K-? ? K
• Vacuity
• If ? ? K, then K-? K
• Success
• Unless ? is necessarily true, ? ? K-?
• Recovery
• If ? ? K, then K ? (K-?)?
• Extensionality
• If ? is logically equivalent to ?, then K-? K-?
  
• Intersection
• (K-? ? K- ?) ? K-(???)
• Conjunction
• If ? ? K-(???) then K-(???) ? K-?

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Revision Operator

• Revision () is applied when belief ? is
  inconsistent with K, and the agent wishes to
  create a new belief system K? incorporating ?
• Axioms
• Closure
• For any sentence ? and belief system K, K? is a
  belief system
• Success
• ? ? K?
• Inclusion
• K? ? K?
• Preservation
• If ? ? K, then K? K?
• Vacuity
• K? K? If and only if ? is a logical necessity
• Extensionality
• If ? is logically equivalent to ?, then K? K?
  
• Superexpansion
• K(???) ? ( K? )?
• Subexpansion
• If ? ? K? then ( K? )? ? K(???)

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Revision Operators

• If and - are AGM expansion and contraction
  operators, the operator
• K? (K- ? ) K?
• Is an AGM revision operator

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Epistemic Entrenchment

• Remember our requirements for a reasonable update
  mechanism
• Where possible, belief systems should remain
  consistent
• A belief system should contain all beliefs
  logically implied by beliefs in the system
• When changing belief systems, loss of information
  should be minimised
• Beliefs held in higher regard should be retained
  in preference to those held in lower regard
• So far, we havent taken care of the fourth
  requirement
• Axioms of epistemic entrenchment are aimed at
  this

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Epistemic Entrenchment

• Beliefs held in higher regard should be retained
  in preference to those held in lower regard
• A preference order (or epistemic entrenchment)
  is a relationship over sentences which
  satisfies
• Axioms
• For any sentences ?, ? and ?
• Transitivity
• if ? ? and ? ? then ? ?
• Dominance
• if ? is a logical consequence of ? then ? ?
• Conjunctiveness
• Either ? ??? or ? ???
• Minimality
• When K ? K? , ? ? K iff ? ? for all possible ?
• Maximality
• If for all ?, ? ?, then ? is logically necessary

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AGM Belief Revision

• AGM belief revision then attempts to find , -
  and operators which satisfy the AGM axioms
• There are a number of different known ways to do
  this
• AGM and similar systems have very strong
  theoretical underpinnings
• Today, they are perhaps more research approaches
  than heavily used in real-world applications

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Summary

• Foundationalist Approaches
• Reason Maintenance
• Truth Maintenance
• Assumption-Based Truth Maintenance
• Coherentist Approaches
• AGM Framework for Belief Revision