Machine Learning Lecture 9: Progol

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Machine Learning Lecture 9 Progol

• Bob McKay
• School of Information Technology and Electrical
  Engineering
• University of New South Wales /
• Australian Defence Force Academy

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Resolution and Clauses

• Recall that the resolvent with respect to L of
  clauses L, M1, , Mm and L, N1, , Nn is
  the clause M1, , Mm , N1, , Nn
• Resolution of two clauses consists of finding
  complementary literals in them, and joining them
  together while deleting the complementary
  literals
• The resolvent of two clauses is implied by the
  combination of the original two clauses

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Least General Generalisation

• One way for one clause to imply another is for
  the first to contain all the literals of
  (subsume) the second
• Another way is for the second to be a
  substitution instance (via a substitution ?) of
  the first
• Combining these, we get the definition of
  ?-subsumption C ?-subsumes D iff there is a
  substitution ? such that ? applied to C gives a
  subset of D
• ?-subsumption generates a generalisation
  hierarchy
• The Least General Generalisation (LGG) of a set
  of clauses S is the least location in the
  ?-subsumption hierarchy that ?-subsumes each of S

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?-subsumption and Implication

• Recall the definitioin of ? -subsumption
• C ?-subsumes D iff there is a substitution ? such
  that ? applied to C gives a subset of D
• It is not an exact inverse of implication let
• C be the rule
• nat(X) ? nat(X1)
• D be the rule
• nat(X) ? nat((X1)1)
• then C implies D, but C does not ? -subsume D
• However D is just C resolved with itself twice

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?-subsumption and Implication (cont)

• Muggleton (1992) showed that this was always the
  case
• If C implies D, and D is not a tautology
• Then there is a clause E, which can be got by
  resolving C with itself zero or more times, and E
  ?-subsumes D
• There have been a number of attempts to use this
  for a generalisation algorithm
• Searching for an appropriate E and then C
• But there are too many possible choices for this
  to be a practical algorithm
• You can think of this approach as trying to find
  a generalisation by
• First undoing all of the substitutions which
  create the data from the generalisation,
• Then undoing all the resolution steps

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Self-saturation

• Closely related to this is the idea of
  self-saturation
• The self-saturation D of D has the property that
  any clause C which implies D also ?-subsumes D
• Unfortunately D is generally large, and may be
  infinite
• However there have been a number of attempts to
  build algorithms based on finding good enough
  approximations D to D, notably Muggletons
  Progol (1994)
• You can think of this approach as trying to find
  a generalisation by
• First undoing all the resolution steps
• Then undoing all of the substitutions which
  create the data from the generalisation
• Ie its the converse of the previous approach