15381: Artificial Intelligence Behrang Mohit and Davide Fossati
Inference in First Order Logic
Some slides, graphics and ideas are borrowed or adapted from courses offered by Stuart Russell, Hwee Tou Ng, Rebecca Hwa and Milos Hauskrecht.
Inference in First Order Logic • We learned about inference in proposiAonal logic • Modus Ponen • And eliminaAon • …
• We learned about proposiAonalizing the FOL statements • Doing inference in proposiAonal logic
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Unifica4on • UnificaAon: Finding the appropriate subsAtuAons for variables • Eliminates the instanAaAon step ∀x King(x) ∧ Greedy(x) ⇒ Evil(x) King(John) ∀y Greedy(y) • Here we don’t explicitly know that John is greedy.
• We can get the inference immediately if we can find a subsAtuAon θ such that King(x) and Greedy(x) match King(John) and Greedy(y) θ = {x/John,y/John} works
Unifica4on Unify(α,β) = θ if αθ = βθ •
Takes two sentences and returns a unifier for them
Let’s say we have query knows(john, x): whom does John know? p q θ Knows(John,x) Knows(John,Jane) Knows(John,x) Knows(y,OJ) Knows(John,x) Knows(y,Mother(y)) Knows(John,x) Knows(x,OJ)
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Unifica4on
Unify(α,β) = θ if αθ = βθ p q Knows(John,x) Knows(John,Jane) Knows(John,x) Knows(y,OJ) Knows(John,x) Knows(y,Mother(y)) Knows(John,x) Knows(x,OJ)
θ {x/Jane}
Unifica4on
p Knows(John,x) Knows(John,x) Knows(John,x) Knows(John,x)
q Knows(John,Jane) Knows(y,OJ) Knows(y,Mother(y)) Knows(x,OJ)
θ {x/Jane} {x/O J, y/John}
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Unifica4on
p Knows(John,x) Knows(John,x) Knows(John,x) Knows(John,x)
q Knows(John,Jane) Knows(y,OJ) Knows(y,Mother(y)) Knows(x,OJ)
θ {x/Jane} {x/O J, y/John} {y/John, x/Mother(John)}
Unifica4on
p q θ Knows(John,x) Knows(John,Jane) {x/Jane} Knows(John,x) Knows(y,OJ) {x/O J, y/John} Knows(John,x) Knows(y,Mother(y)) {y/John, x/Mother(John)} Knows(John,x) Knows(x,OJ) fail • Standardizing apart eliminates overlap of variables, e.g., Knows(z17,OJ)
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Modus Ponens in proposi4onal logic • Modus ponens • If both sentences in the premise are True, the conclusion is True
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Forward and backward chaining • Modus Ponens (for Horn Form): complete for Horn KBs α1, … ,αn,
α1 ∧ … ∧ αn ⇒ β β
• Can be used with forward chaining or backward chaining. • These algorithms are very natural and run in linear Ame
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Forward chaining • Idea: fire any rule whose premises are saAsfied in the KB, • add its conclusion to the KB, unAl query is found
Forward chaining example
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Forward chaining example
Forward chaining example
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Forward chaining example
Forward chaining example
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Forward chaining example
Forward chaining example
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Forward chaining example
Forward chaining • Idea: fire any rule whose premises are saAsfied in the KB, • add its conclusion to the KB, unAl query is found
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From PL to FOL • Modus Ponens in proposiAonal logic • Modus Ponens (for Horn Form) α1, … ,αn,
α1 ∧ … ∧ αn ⇒ β β
• Forward chaining in proposiAonal logic • Modus Ponens in forward direcAon and add atomic sentences
First order definite clauses • Definite clauses: SituaAon Response • Atomic • ImplicaAon with conjuncAve premise King (x) AND Greedy(x) Evil (x) King (John) Greedy (x)
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Generalized Modus Ponens (GMP) p1', p2', … , pn', ( p1 ∧ p2 ∧ … ∧ pn ⇒q) where qθ p1' is King(John) p1 is King(x) p2' is Greedy(y) p2 is Greedy(x) θ is {x/John,y/John} q is Evil(x) qθ is Evil(John)
pi'θ = pi θ for all i
• All variables assumed universally quanAfied • GMP is lihed version of modus ponens • Lihing from the ground (variable-‐free) to FOL
Example knowledge base • The law says that it is a crime for an American to sell weapons to hosAle naAons. 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 Col. West is a criminal
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Example knowledge base contd. ... it is a crime for an American to sell weapons to hosAle naAons: Nono … has some missiles, i.e., ∃x Owns(Nono,x) ∧ Missile(x): … all of its missiles were sold to it by Colonel West Missiles are weapons: An enemy of America counts as "hosAle“: West, who is American … The country Nono, an enemy of America …
Example knowledge base contd. ... it is a crime for an American to sell weapons to hosAle naAons: Nono … has some missiles, i.e., ∃x Owns(Nono,x) ∧ Missile(x): … all of its missiles were sold to it by Colonel West Missiles are weapons: An enemy of America counts as "hosAle“: West, who is American … American(West)
The country Nono, an enemy of America … Enemy(Nono,America)
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Example knowledge base contd. ... it is a crime for an American to sell weapons to hosAle naAons: Nono … has some missiles, i.e., ∃x Owns(Nono,x) ∧ Missile(x): … all of its missiles were sold to it by Colonel West Missiles are weapons:
Missile(x) ⇒ Weapon(x)
An enemy of America counts as "hosAle“: Enemy(x,America) ⇒ HosDle(x)
West, who is American … American(West)
The country Nono, an enemy of America … Enemy(Nono,America)
Example knowledge base contd. ... it is a crime for an American to sell weapons to hosAle naAons: 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 "hosAle“: Enemy(x,America) ⇒ HosDle(x)
West, who is American … American(West)
The country Nono, an enemy of America … Enemy(Nono,America)
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Example knowledge base contd. ... it is a crime for an American to sell weapons to hosAle naAons: American(x) ∧ Weapon(y) ∧ Sells(x,y,z) ∧ HosDle(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 "hosAle“: Enemy(x,America) ⇒ HosDle(x)
West, who is American … American(West)
The country Nono, an enemy of America … Enemy(Nono,America)
Example knowledge base contd. ... it is a crime for an American to sell weapons to hosAle naAons: American(x) ∧ Weapon(y) ∧ Sells(x,y,z) ∧ HosDle(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 "hosAle“: Enemy(x,America) ⇒ HosDle(x)
West, who is American … American(West)
The country Nono, an enemy of America … Enemy(Nono,America)
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Forward chaining algorithm
Forward chaining proof
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Forward chaining proof
Forward chaining proof
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Proper4es of forward chaining • Sound and complete for first-‐order definite clauses • May not terminate in general if α is not entailed • This is unavoidable: entailment with definite clauses is semidecidable
Backward chaining algorithm
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Backward chaining example
Backward chaining example
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Backward chaining example
Backward chaining example
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Backward chaining example
Backward chaining example
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Backward chaining example
Backward chaining example
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Proper4es of backward chaining • Depth-‐first recursive proof search: space is linear in size of proof • Widely used for logic programming
Communica4on Homework
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