Logic Programming Part 2: Semantics

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Logic Programming Part 2 Semantics

• James Cheney
• CS 411

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Resolution

• Resolution principle
• If A1 ... An ? B and B,C1,...,Cm ? D
• Then A1 ... An,C1,...,Cm ? D
• Proof search
• To prove B, try to prove B ? False
• Apply resolution until LHS is empty.

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Unification

• In Prolog, logic variables X,Y,Z can be
  substituted by any other term
• Unification Algorithm for finding substitution
  making two terms equal

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Examples

• f(X,Y,Z) f(g(Z),W)
• unifies with X g(Z), W Y,Z
• f(X,g(X)) f(2,g(Y))
• Doesnt iunify X 2 and X g(Y) impossible

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More Examples

• f(X) X
• Doesnt unify!
• X f(X) f(f(X)) ...
• f(X,Y) f(Y,g(X))
• Doesnt unify! (Why?)

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Example
f
f
X
Y
Y
g
X
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Example
f
f
X
Y
Y
g
X
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Example
X
Y
Y
g
X
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Example
Answer X Y
Y
g
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Example
Answer X Y
Y
g
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Example
Answer X Y Y g(Y)
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Example
Answer X Y Y g(Y)
Problem In equation Y f(Y), Y
occurs circularly. There is no finite term
satisfying this equation. Checking that this is
not the case is called the occurs check. For
efficiency reasons, PROLOG skips this check.
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Robinsons Algorithm

• Consider a general unification problem
• P t1 u1, t2 u2, ...
• Reduce the problem by replacing one equation with
  a set of equations
• Succeed if all equations reducible, else fail.

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Robinsons Algorithm

• Two constants unify if they are equal.
• c c, P ? P
• Else fail.

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Robinsons Algorithm

• Two function applications unify if the symbols
  are equal, and the corresponding arguments unify.
• f(t1,...tn) f(u1,...,un), P ?
• t1 u1,... tn un, P
• Must have equal number of arguments

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Robinsons Algorithm

• Two variables unify if they are the same.
• X X, P ? P

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Robinsons Algorithm

• Otherwise, a variable X unifies with a term t
  provided X does not occur in t.
• X t, P ? Pt/X (if X not in FV(t))
• Proceed by substituting t for X in P.

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Operational Semantics

• In general, we have
• a program P clauses,
• a goal G atomic goals,
• and a substitution q Var ? Term
• Base case If there are no goals then succeed
• (P,,q) ? Yes q

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Operational Semantics

• Inductive case Suppose state is
• (P,G,q) with G nonempty.
• Pick a p(t1,...,tn) in G.
• Assume clause p(u1,...,un) - G in P.
• Assume q unifies p(t1,...,tn) and p(u1,...,un)
• Then
• (P,G,q) ? (P,qG, qq).