TECHNICAL SOLUTIONS

    Model Theory of ILP

Model Theory – Normal Semantics

  • The problem of inductive inference:
    • Given is background (prior) knowledge B and evidence E
    • The evidence E = E+ ∪ E- consists of positive evidence E+ and negative evidence E-
    • The aim is then to find a hypothesis H such that the following conditions hold:
        Prior Satisfiability: B ∪ E- ⊭ □
        Posterior Satisfiability: B ∪ H ∪ E- ⊭ □
        Prior Necessity: B ⊭ E+
        Posterior Sufficiency: B ∪ H ⊧ E+
  • The Sufficiency criterion is sometimes named completeness with regard to positive evidence
  • The Posterior Satisfiability criterion is also known as consistency with the negative evidence
  • In this general setting, background-theory, examples, and hypotheses can be any (well-formed) formula

Model Theory – Definite Semantics

  • In most ILP practical systems background theory and hypotheses are restricted to being definite clauses
    • Clause: A disjunction of literals
    • Horn Clause: A clause with at most one positive literal
    • Definite Clause: A Horn clause with exactly one positive literal
  • This setting has the advantage that definite clause theory T has a unique minimal Herbrand model M +( T )
    • Any logical formulae is either true or false in this minimal model (all formulae are decidable and the Closed World Assumption holds)

Model Theory – Definite Semantics

  • The definite semantics again require a set of conditions to hold
  • We can now refer to every formula in E since they are guaranteed to have a truth value in the minimal model

 

  • Consistency:
      Prior Satisfiability: all e in E - are false in M +( B )
    • Negative evidence should not be part of the minimal model
      Posterior Satisfiability: all e in E - are false in M +( B H )
    • Negative evidence should not be supported by our hypotheses
  • Completeness
      Prior Necessity: some e in E + are false in M +( B )
    • If all positive examples are already true in the minimal model of the background knowledge, then no hypothesis we derive will add useful information
      Posterior Sufficiency: all e in E + are true in M +( B H )
    • All positive examples are true (explained by the hypothesis) in the minimal model of the background theory and the hypothesis

Model Theory – Definite Semantics

  • An additional restriction in addition to those of the definite semantics is to only allow true and false ground facts as examples (evidence)
  • This is called the example setting
    • The example setting is the main setting employed by ILP systems
    • Only allows factual and not causal evidence (which usually captures more knowledge)
  • Example:
    • B: grandfather(X, Y) ← father(X, Z), parent(Z, Y)
          father(henry, jane) ←
          etc.
    • E: grandfather(henry, john) ←
          grandfather(henry, alice) ←
  • Not allowed in example setting:
    • ← grandfather(X, X)     [Not allowed in definite semantics]
      grandfather(henry, john) ← father(henry, jane), mother(jane, john)

Model Theory – Non-monotonic Semantics

  • In the non­monotonic setting:
    • The background theory is a set of definite clauses

    • The evidence is empty
      • The positive evidence is considered part of the background theory
      • The negative evidence is derived implicitly, by making the closed world assumption (realized by the minimal Herbrand model)

    • The hypotheses are sets of general clauses expressible using the same alphabet as the background theory

Model Theory – Non-monotonic Semantics (2)

  • Since only positive evidence is present, it is assumed to be part of the background theory:
            B’ = B ∪ E
  • The following conditions should hold for H and B’:
    • Validity: all h in H are true in M +( B’ )
      • All clauses belonging to a hypothesis hold in the database B, i.e. that they are true properties of the data
    • Completeness: if general clause g is true in M +( B’ ) then H g
      • All information that is valid in the minimal model of B’ should follow from the hypothesis
  • Additionally the following can be a requirement:
    • Minimality: there is no proper subset G of H which is valid and complete
      • The hypothesis should not contain redundant clauses

Model Theory – Non-monotonic Semantics (3)

  • Example for B (definite clauses):
        male(luc) ←
        female(lieve) ←
        human(lieve) ←
        human(luc) ←
  • A possible solution is then H (a set of general clauses):
        ← female(X), male(X)
        human(X) ← male(X)
        human(X) ← female(X)
        female(X), male(X) ← human(X)

Model Theory – Non-monotonic Semantics (4)

  • One more example to illustrate the difference between the example setting and the non-monotonic setting
  • Consider:
    • Background theory B
        bird(tweety) ←
        bird(oliver) ←
    • Examples E+:
        flies(tweety)

 

    • For the non-monotonic setting B’ = B ∪ E+ because positive examples are considered part of the background knowledge

Model Theory – Non-monotonic Semantics (5)

  • Example setting:
    • An acceptable hypothesis H1 would be
          flies(X) ← bird(X)
    • It is acceptable because if fulfills the completeness and consistency criteria of the definite semantics
    • This realizes can inductive leap because flies(oliver) is true in
          M + ( B ∪ H ) = { bird(tweety), bird(oliver), flies(tweety), flies(oliver) }
  • Non-monotonic setting:
    • H1 is not a solution since there exists a substitution {X oliver} which makes the clause false in M +( B’ ) (the validity criteria is violated:
          M + ( B’ ) = { bird(tweety), bird(oliver), flies(tweety) }
          {X ← oliver}: flies(oliver) ← bird(oliver)
          {X ← tweety}: flies(tweety) ← bird(tweety)