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 & Norvig’s 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 WestMissile(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)

Missile(x) Owns(Nono,x) Sells(West,x,Nono)

Enemy(x,America) Hostile(x)

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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(θ1, θ2), p) = SUBST(θ2, SUBST(θ1, 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