Chaining in First Order Logic

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Chaining in First Order Logic

• CS 171/271
• (Chapter 9, continued)
• Some text and images in these slides were drawn
  fromRussel Norvigs published material

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Forward Chaining

• Forward Chaining in PL extends to FOL
• A similar restriction applies to the KB
• Implications
• Atomic sentences
• KB can contain quantifiers
• Apply EI for existentially quantified sentences
• Allow variables for universally quantified
  sentences (omit quantifier in such sentences)

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Knowledge Base Example

• The law says that it is a crime for an American
  to sell weapons to hostile nations. The country
  Nono, an enemy of America, has some missiles, and
  all of its missiles were sold to it by Colonel
  West, who is American.
• Prove that Colonel West is a criminal

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Knowledge Base Example

• ... it is a crime for an American to sell weapons
  to hostile nations
• American(x) ? Weapon(y) ? Sells(x,y,z) ?
  Hostile(z) ? Criminal(x)
• Nono has some missiles, i.e., ?x Owns(Nono,x) ?
  Missile(x)
• Owns(Nono,M1) and Missile(M1)
• all of its missiles were sold to it by Colonel
  West
• Missile(x) ? Owns(Nono,x) ? Sells(West,x,Nono)
• Missiles are weapons
• Missile(x) ? Weapon(x)
• An enemy of America counts as "hostile
• Enemy(x,America) ? Hostile(x)
• West, who is American
• American(West)
• The country Nono, an enemy of America
• Enemy(Nono,America)

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Forward Chaining Algorithm
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Forward Chaining Example
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Forward Chaining Example
Missile(x) ? Weapon(x)
Enemy(x,America) ? Hostile(x)
Missile(x) ? Owns(Nono,x) ? Sells(West,x,Nono)
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Forward Chaining Example

• American(x) ? Weapon(y) ? Sells(x,y,z) ?
  Hostile(z) ? Criminal(x)

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About Forward Chaining

• Matching rules against known facts
• Incremental forward chaining
• Irrelevant facts

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Backward Chaining

• Begin with goal (query)
• Recursively build a set of substitutions that
  satisfy the premises necessary to conclude goal

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Backward Chaining Algorithm

• SUBST(COMPOSE(?1, ?2), p) SUBST(?2, SUBST(?1,
  p))

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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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About Backward Chaining

• Uses composition of substitutions
• A DFS algorithm
• Linear time
• Repeated states and incompleteness