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Knowledge Representation (II) Paolo Turrini Department of - - PowerPoint PPT Presentation

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Intro to AI Knowledge Representation (II) Paolo Turrini Department of Computing, Imperial College London Introduction to Artificial Intelligence Paolo Turrini Intro to AI Intro to AI The main reference Stuart Russell and Peter Norvig

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Intro to AI

Knowledge Representation (II)

Paolo Turrini

Department of Computing, Imperial College London

Introduction to Artificial Intelligence

Paolo Turrini Intro to AI

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Intro to AI

The main reference

Stuart Russell and Peter Norvig Artificial Intelligence: a modern approach Chapter 9

Paolo Turrini Intro to AI

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Intro to AI

Today’s class

Start with a knowledge base and try to prove something interesting Various methods for doing so, with different computational properties

Paolo Turrini Intro to AI

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Intro to AI

Artificial Intelligence and Law

The formalization of legislation and the development of computer systems to assist with legal problem solving provide a rich domain for developing and testing artificial-intelligence technology. Marek Sergot, Fariba Sadri, Robert Kowalski, (and others) The British Nationality Act as a logic program Communications of the ACM, 1986

Paolo Turrini Intro to AI

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Intro to AI

An informal knowledge base

The law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy

  • f America, has some missiles, and all of its missiles were

sold to it by Colonel West, who is American.

Paolo Turrini Intro to AI

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Intro to AI

An informal knowledge base

The law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy

  • f 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

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

. . . it is a crime for an American to sell weapons to hostile nations:

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

. . . 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)

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

. . . 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

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

. . . 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 Owns(Nono, M1) and Missile(M1)

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

. . . 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 Owns(Nono, M1) and Missile(M1) . . . all of its missiles were sold to it by Colonel West

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

. . . 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 Owns(Nono, M1) and Missile(M1) . . . all of its missiles were sold to it by Colonel West ∀ x Missile(x) ∧ Owns(Nono, x) ⇒ Sells(West, x, Nono)

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

Missiles are weapons:

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

Missiles are weapons: Missile(x) ⇒ Weapon(x)

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

Missiles are weapons: Missile(x) ⇒ Weapon(x) An enemy of America ”counts as” hostile:

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

Missiles are weapons: Missile(x) ⇒ Weapon(x) An enemy of America ”counts as” hostile: Enemy(x, America) ⇒ Hostile(x)

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

Missiles are weapons: Missile(x) ⇒ Weapon(x) An enemy of America ”counts as” hostile: Enemy(x, America) ⇒ Hostile(x) West, who is American.

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

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)

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

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 . . .

Paolo Turrini Intro to AI

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Intro to AI

Example knowledge base contd.

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)

Paolo Turrini Intro to AI

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Intro to AI

Forward chaining proof

Paolo Turrini Intro to AI

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Intro to AI

Forward chaining proof

Paolo Turrini Intro to AI

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Intro to AI

Forward chaining proof

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Intro to AI

Footage from Tuesday

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Intro to AI

GMP: recall...

Definite clause: disjunction of literals, exactly one of which positive e.g., (p1 ∧ p2 ∧ . . . ∧ pn ⇒ q)

p1′, p2′, . . . , pn′, (p1 ∧ p2 ∧ . . . ∧ pn ⇒ q) qθ where pi ′θ = piθ for all i

Assuming all variables are universally quantified...

p1′ is King(Joffrey) p1 is King(x) p2′ is Greedy(y) p2 is Greedy(x) θ is {x/Joffrey, y/Joffrey} q is Evil(x) qθ is Evil(Joffrey)

Paolo Turrini Intro to AI

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Intro to AI

Forward chaining algorithm

Paolo Turrini Intro to AI

function FOL-FC-Ask(KB,α) returns a substitution or false inputs: KB, the knowledge base, a set of definite clauses α, the (atomic) query local variables: new, the new sentences inferred at each iteration repeat until new is empty new ← { } for each sentence r in KB do ( p1 ∧ . . . ∧ pn ⇒ q) ← Standardize-Apart(r) for each θ such that (p1 ∧ . . . ∧ pn)θ = (p′

1 ∧ . . . ∧ p′ n)θ

for some p′

1, . . . , p′ n in KB do

q′ ← qθ if q′ does not already unify in KB or new then add q′ to new φ ← Unify(q′,α) if φ is not fail then return φ add new to KB return false

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Intro to AI

First iteration

On the first iteration... American(x)∧Weapon(y)∧Sells(x, y, z)∧Hostile(z) ⇒ Criminal(x)

Paolo Turrini Intro to AI

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Intro to AI

First iteration

On the first iteration... American(x)∧Weapon(y)∧Sells(x, y, z)∧Hostile(z) ⇒ Criminal(x) has unsatisfied premises

Paolo Turrini Intro to AI

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Intro to AI

First iteration

∀ x Missile(x) ∧ Owns(Nono, x) ⇒ Sells(West, x, Nono)

Paolo Turrini Intro to AI

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Intro to AI

First iteration

∀ x Missile(x) ∧ Owns(Nono, x) ⇒ Sells(West, x, Nono) is satisfied with {x/M1} and Sells(West, M1, Nono) is added

Paolo Turrini Intro to AI

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Intro to AI

First iteration

Missile(x) ⇒ Weapon(x)

Paolo Turrini Intro to AI

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Intro to AI

First iteration

Missile(x) ⇒ Weapon(x) is satisfied with {x/M1} and Weapon(M1) is added

Paolo Turrini Intro to AI

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Intro to AI

First iteration

Enemy(x, America) ⇒ Hostile(x)

Paolo Turrini Intro to AI

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Intro to AI

First iteration

Enemy(x, America) ⇒ Hostile(x) is satisfied with {x/Nono} and Hostile(Nono) is added

Paolo Turrini Intro to AI

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Intro to AI

First iteration

So in total we have added...

Paolo Turrini Intro to AI

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Intro to AI

First iteration

So in total we have added... Sells(West, M1, Nono)

Paolo Turrini Intro to AI

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Intro to AI

First iteration

So in total we have added... Sells(West, M1, Nono) Weapon(M1)

Paolo Turrini Intro to AI

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Intro to AI

First iteration

So in total we have added... Sells(West, M1, Nono) Weapon(M1) Hostile(Nono)

Paolo Turrini Intro to AI

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Intro to AI

Second iteration

On the second iteration...

Paolo Turrini Intro to AI

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Intro to AI

Second iteration

On the second iteration... American(x)∧Weapon(y)∧Sells(x, y, z)∧Hostile(z) ⇒ Criminal(x)

Paolo Turrini Intro to AI

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Intro to AI

Second iteration

On the second iteration... American(x)∧Weapon(y)∧Sells(x, y, z)∧Hostile(z) ⇒ Criminal(x) is satisfied with {x/West, y/M1, z/Nono} and Criminal(West) is added

Paolo Turrini Intro to AI

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Intro to AI

Second iteration

On the second iteration... American(x)∧Weapon(y)∧Sells(x, y, z)∧Hostile(z) ⇒ Criminal(x) is satisfied with {x/West, y/M1, z/Nono} and Criminal(West) is added Yay! :)

Paolo Turrini Intro to AI

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Intro to AI

Forward chaining proof

Paolo Turrini Intro to AI

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Intro to AI

Forward chaining proof

Paolo Turrini Intro to AI

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Intro to AI

Forward chaining proof

Paolo Turrini Intro to AI

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Intro to AI

Properties of forward chaining

Sound (because of soundness of GMP)

Paolo Turrini Intro to AI

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Intro to AI

Properties of forward chaining

Sound (because of soundness of GMP) Complete for first-order (entailed!) definite clauses

Paolo Turrini Intro to AI

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Intro to AI

Properties of forward chaining

Sound (because of soundness of GMP) Complete for first-order (entailed!) definite clauses May not terminate in general if α is not entailed

Paolo Turrini Intro to AI

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Intro to AI

Properties of forward chaining

Sound (because of soundness of GMP) Complete for first-order (entailed!) definite clauses May not terminate in general if α is not entailed It’s inefficient:

Paolo Turrini Intro to AI

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Intro to AI

Properties of forward chaining

Sound (because of soundness of GMP) Complete for first-order (entailed!) definite clauses May not terminate in general if α is not entailed It’s inefficient:

but there can be improvements

Paolo Turrini Intro to AI

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Intro to AI

Properties of forward chaining

Sound (because of soundness of GMP) Complete for first-order (entailed!) definite clauses May not terminate in general if α is not entailed It’s inefficient:

but there can be improvements well... matching conjunctive premises against known facts is NP-hard

Paolo Turrini Intro to AI

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Intro to AI

Properties of forward chaining

Sound (because of soundness of GMP) Complete for first-order (entailed!) definite clauses May not terminate in general if α is not entailed It’s inefficient:

but there can be improvements well... matching conjunctive premises against known facts is NP-hard

Forward chaining is widely used in deductive databases

Paolo Turrini Intro to AI

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Intro to AI

Backward chaining example

Paolo Turrini Intro to AI

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Intro to AI

Backward chaining example

Paolo Turrini Intro to AI

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Intro to AI

Backward chaining example

Paolo Turrini Intro to AI

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Intro to AI

Backward chaining example

Paolo Turrini Intro to AI

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Intro to AI

Backward chaining example

Paolo Turrini Intro to AI

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Intro to AI

Backward chaining example

Paolo Turrini Intro to AI

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Intro to AI

Backward chaining example

Paolo Turrini Intro to AI

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Intro to AI

Backward chaining algorithm

Paolo Turrini Intro to AI

function FOL-BC-Ask(KB,goals,θ) returns a set of substitutions inputs: KB, a knowledge base goals, a list of conjuncts forming a query (θ already applied) θ, the current substitution, initially the empty substitution { } local variables: answers, a set of substitutions, initially empty if goals is empty then return {θ} q′ ← First(goals)θ for each sentence r in KB where Standardize-Apart(r) = ( p1 ∧ . . . ∧ pn ⇒ q) and θ′ ← Unify(q,q′) succeeds new goals ← [ p1, . . . , pn|Rest(goals)] answers ← FOL-BC-Ask(KB,new goals,Compose(θ′,θ)) ∪ answers return answers

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Intro to AI

Backward chaining example

Paolo Turrini Intro to AI

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Intro to AI

Backward chaining example

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Intro to AI

Backward chaining example

Paolo Turrini Intro to AI

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Intro to AI

Backward chaining example

Paolo Turrini Intro to AI

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Intro to AI

Backward chaining example

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Intro to AI

Backward chaining example

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Intro to AI

Backward chaining example

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Intro to AI

Properties of backward chaining

Depth-first recursive proof search

Paolo Turrini Intro to AI

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Intro to AI

Properties of backward chaining

Depth-first recursive proof search space is linear in size of proof

Paolo Turrini Intro to AI

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Intro to AI

Properties of backward chaining

Depth-first recursive proof search space is linear in size of proof Incomplete due to infinite loops

Paolo Turrini Intro to AI

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Intro to AI

Properties of backward chaining

Depth-first recursive proof search space is linear in size of proof Incomplete due to infinite loops Widely used for logic programming

Paolo Turrini Intro to AI

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Intro to AI

Resolution

Full first-order version:

ℓ1 ∨ · · · ∨ ℓk, m1 ∨ · · · ∨ mn (ℓ1 ∨ · · · ∨ ℓi−1 ∨ ℓi+1 ∨ · · · ∨ ℓk ∨ m1 ∨ · · · ∨ mj−1 ∨ mj+1 ∨ · · · ∨ mn)θ where Unify(ℓi, ¬mj) = θ.

Paolo Turrini Intro to AI

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Intro to AI

Resolution

Full first-order version:

ℓ1 ∨ · · · ∨ ℓk, m1 ∨ · · · ∨ mn (ℓ1 ∨ · · · ∨ ℓi−1 ∨ ℓi+1 ∨ · · · ∨ ℓk ∨ m1 ∨ · · · ∨ mj−1 ∨ mj+1 ∨ · · · ∨ mn)θ where Unify(ℓi, ¬mj) = θ.

For example, ¬Rich(x) ∨ Unhappy(x)

Paolo Turrini Intro to AI

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Intro to AI

Resolution

Full first-order version:

ℓ1 ∨ · · · ∨ ℓk, m1 ∨ · · · ∨ mn (ℓ1 ∨ · · · ∨ ℓi−1 ∨ ℓi+1 ∨ · · · ∨ ℓk ∨ m1 ∨ · · · ∨ mj−1 ∨ mj+1 ∨ · · · ∨ mn)θ where Unify(ℓi, ¬mj) = θ.

For example, ¬Rich(x) ∨ Unhappy(x) Rich(Berlusconi)

Paolo Turrini Intro to AI

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Intro to AI

Resolution

Full first-order version:

ℓ1 ∨ · · · ∨ ℓk, m1 ∨ · · · ∨ mn (ℓ1 ∨ · · · ∨ ℓi−1 ∨ ℓi+1 ∨ · · · ∨ ℓk ∨ m1 ∨ · · · ∨ mj−1 ∨ mj+1 ∨ · · · ∨ mn)θ where Unify(ℓi, ¬mj) = θ.

For example, ¬Rich(x) ∨ Unhappy(x) Rich(Berlusconi) Unhappy(Berlusconi)

Paolo Turrini Intro to AI

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Intro to AI

Resolution

Full first-order version:

ℓ1 ∨ · · · ∨ ℓk, m1 ∨ · · · ∨ mn (ℓ1 ∨ · · · ∨ ℓi−1 ∨ ℓi+1 ∨ · · · ∨ ℓk ∨ m1 ∨ · · · ∨ mj−1 ∨ mj+1 ∨ · · · ∨ mn)θ where Unify(ℓi, ¬mj) = θ.

For example, ¬Rich(x) ∨ Unhappy(x) Rich(Berlusconi) Unhappy(Berlusconi)

Paolo Turrini Intro to AI

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Intro to AI

Resolution

Full first-order version:

ℓ1 ∨ · · · ∨ ℓk, m1 ∨ · · · ∨ mn (ℓ1 ∨ · · · ∨ ℓi−1 ∨ ℓi+1 ∨ · · · ∨ ℓk ∨ m1 ∨ · · · ∨ mj−1 ∨ mj+1 ∨ · · · ∨ mn)θ where Unify(ℓi, ¬mj) = θ.

For example, ¬Rich(x) ∨ Unhappy(x) Rich(Berlusconi) Unhappy(Berlusconi) with θ = {x/Berlusconi}

Paolo Turrini Intro to AI

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Intro to AI

Resolution

Full first-order version:

ℓ1 ∨ · · · ∨ ℓk, m1 ∨ · · · ∨ mn (ℓ1 ∨ · · · ∨ ℓi−1 ∨ ℓi+1 ∨ · · · ∨ ℓk ∨ m1 ∨ · · · ∨ mj−1 ∨ mj+1 ∨ · · · ∨ mn)θ where Unify(ℓi, ¬mj) = θ.

For example, ¬Rich(x) ∨ Unhappy(x) Rich(Berlusconi) Unhappy(Berlusconi) with θ = {x/Berlusconi} Apply resolution steps to CNF(KB ∧ ¬α)

Paolo Turrini Intro to AI

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Intro to AI

Conversion to CNF

Everyone who dislikes pizzas is disliked by someone:

Paolo Turrini Intro to AI

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Intro to AI

Conversion to CNF

Everyone who dislikes pizzas is disliked by someone: ∀ x [∀ y Pizza(y) ⇒ Dislikes(x, y)] ⇒ [∃ y Dislikes(y, x)]

Paolo Turrini Intro to AI

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Intro to AI

Conversion to CNF

Everyone who dislikes pizzas is disliked by someone: ∀ x [∀ y Pizza(y) ⇒ Dislikes(x, y)] ⇒ [∃ y Dislikes(y, x)]

  • 1. Eliminate biconditionals and implications

Paolo Turrini Intro to AI

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Intro to AI

Conversion to CNF

Everyone who dislikes pizzas is disliked by someone: ∀ x [∀ y Pizza(y) ⇒ Dislikes(x, y)] ⇒ [∃ y Dislikes(y, x)]

  • 1. Eliminate biconditionals and implications

∀ x [¬∀ y ¬Pizza(y) ∨ Dislikes(x, y)] ∨ [∃ y Dislikes(y, x)]

Paolo Turrini Intro to AI

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Intro to AI

Conversion to CNF

Everyone who dislikes pizzas is disliked by someone: ∀ x [∀ y Pizza(y) ⇒ Dislikes(x, y)] ⇒ [∃ y Dislikes(y, x)]

  • 1. Eliminate biconditionals and implications

∀ x [¬∀ y ¬Pizza(y) ∨ Dislikes(x, y)] ∨ [∃ y Dislikes(y, x)]

  • 2. Move ¬ inwards: ¬∀ xϕ

≡ ∃ x ¬ϕ, ¬∃ xϕ ≡ ∀ x ¬ϕ:

Paolo Turrini Intro to AI

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Intro to AI

Conversion to CNF

Everyone who dislikes pizzas is disliked by someone: ∀ x [∀ y Pizza(y) ⇒ Dislikes(x, y)] ⇒ [∃ y Dislikes(y, x)]

  • 1. Eliminate biconditionals and implications

∀ x [¬∀ y ¬Pizza(y) ∨ Dislikes(x, y)] ∨ [∃ y Dislikes(y, x)]

  • 2. Move ¬ inwards: ¬∀ xϕ

≡ ∃ x ¬ϕ, ¬∃ xϕ ≡ ∀ x ¬ϕ: ∀ x [∃ y ¬(¬Pizza(y) ∨ Dislikes(x, y))] ∨ [∃ y Dislikes(y, x)]

Paolo Turrini Intro to AI

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Intro to AI

Conversion to CNF

Everyone who dislikes pizzas is disliked by someone: ∀ x [∀ y Pizza(y) ⇒ Dislikes(x, y)] ⇒ [∃ y Dislikes(y, x)]

  • 1. Eliminate biconditionals and implications

∀ x [¬∀ y ¬Pizza(y) ∨ Dislikes(x, y)] ∨ [∃ y Dislikes(y, x)]

  • 2. Move ¬ inwards: ¬∀ xϕ

≡ ∃ x ¬ϕ, ¬∃ xϕ ≡ ∀ x ¬ϕ: ∀ x [∃ y ¬(¬Pizza(y) ∨ Dislikes(x, y))] ∨ [∃ y Dislikes(y, x)] ∀ x [∃ y ¬¬Pizza(y) ∧ ¬Dislikes(x, y)] ∨ [∃ y Dislikes(y, x)]

Paolo Turrini Intro to AI

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Intro to AI

Conversion to CNF

Everyone who dislikes pizzas is disliked by someone: ∀ x [∀ y Pizza(y) ⇒ Dislikes(x, y)] ⇒ [∃ y Dislikes(y, x)]

  • 1. Eliminate biconditionals and implications

∀ x [¬∀ y ¬Pizza(y) ∨ Dislikes(x, y)] ∨ [∃ y Dislikes(y, x)]

  • 2. Move ¬ inwards: ¬∀ xϕ

≡ ∃ x ¬ϕ, ¬∃ xϕ ≡ ∀ x ¬ϕ: ∀ x [∃ y ¬(¬Pizza(y) ∨ Dislikes(x, y))] ∨ [∃ y Dislikes(y, x)] ∀ x [∃ y ¬¬Pizza(y) ∧ ¬Dislikes(x, y)] ∨ [∃ y Dislikes(y, x)] ∀ x [∃ y Pizza(y) ∧ ¬Dislikes(x, y)] ∨ [∃ y Dislikes(y, x)]

Paolo Turrini Intro to AI

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Intro to AI

Conversion to CNF

Everyone who dislikes pizzas is disliked by someone: ∀ x [∀ y Pizza(y) ⇒ Dislikes(x, y)] ⇒ [∃ y Dislikes(y, x)]

  • 1. Eliminate biconditionals and implications

∀ x [¬∀ y ¬Pizza(y) ∨ Dislikes(x, y)] ∨ [∃ y Dislikes(y, x)]

  • 2. Move ¬ inwards: ¬∀ xϕ

≡ ∃ x ¬ϕ, ¬∃ xϕ ≡ ∀ x ¬ϕ: ∀ x [∃ y ¬(¬Pizza(y) ∨ Dislikes(x, y))] ∨ [∃ y Dislikes(y, x)] ∀ x [∃ y ¬¬Pizza(y) ∧ ¬Dislikes(x, y)] ∨ [∃ y Dislikes(y, x)] ∀ x [∃ y Pizza(y) ∧ ¬Dislikes(x, y)] ∨ [∃ y Dislikes(y, x)]

Paolo Turrini Intro to AI

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Intro to AI

Conversion to CNF contd.

  • 3. Standardise variables: each quantifier should use a different one

Paolo Turrini Intro to AI

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Intro to AI

Conversion to CNF contd.

  • 3. Standardise variables: each quantifier should use a different one

∀ x [∃ y Pizza(y) ∧ ¬Dislikes(x, y)] ∨ [∃ z Dislikes(z, x)]

Paolo Turrini Intro to AI

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Intro to AI

Conversion to CNF contd.

  • 3. Standardise variables: each quantifier should use a different one

∀ x [∃ y Pizza(y) ∧ ¬Dislikes(x, y)] ∨ [∃ z Dislikes(z, x)]

  • 4. Skolemize: a more general form of existential instantiation.

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Conversion to CNF contd.

  • 3. Standardise variables: each quantifier should use a different one

∀ x [∃ y Pizza(y) ∧ ¬Dislikes(x, y)] ∨ [∃ z Dislikes(z, x)]

  • 4. Skolemize: a more general form of existential instantiation.

Each existential variable is replaced by a Skolem function

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Conversion to CNF contd.

  • 3. Standardise variables: each quantifier should use a different one

∀ x [∃ y Pizza(y) ∧ ¬Dislikes(x, y)] ∨ [∃ z Dislikes(z, x)]

  • 4. Skolemize: a more general form of existential instantiation.

Each existential variable is replaced by a Skolem function of the enclosing universally quantified variables:

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Conversion to CNF contd.

  • 3. Standardise variables: each quantifier should use a different one

∀ x [∃ y Pizza(y) ∧ ¬Dislikes(x, y)] ∨ [∃ z Dislikes(z, x)]

  • 4. Skolemize: a more general form of existential instantiation.

Each existential variable is replaced by a Skolem function of the enclosing universally quantified variables: ∀ x [Pizza(F(x)) ∧ ¬Dislikes(x, F(x))] ∨ Dislikes(G(x), x)

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Conversion to CNF contd.

  • 5. Drop universal quantifiers:

[Pizza(F(x)) ∧ ¬Dislikes(x, F(x))] ∨ Dislikes(G(x), x)

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Conversion to CNF contd.

  • 5. Drop universal quantifiers:

[Pizza(F(x)) ∧ ¬Dislikes(x, F(x))] ∨ Dislikes(G(x), x)

  • 6. Distribute ∧ over ∨:

[Pizza(F(x)) ∨ Dislikes(G(x), x)]∧ ∧[¬Dislikes(x, F(x)) ∨ Dislikes(G(x), x)]

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Resolution proof: definite clauses

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What we have seen

Inference methods with different structure and properties Forward-chaining (deductive databases) Backward-chaining (logic programming) Resolution (full-first order logic)

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Coming next

Knowledge and uncertainty: what if we don’t know exactly? How to handle uncertain information:

representation reasoning

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Appendix: Universal Instantiation

Every instantiation of a universally quantified sentence is entailed by it: ∀ v α α({v/g}) for any variable v and ground term g

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Appendix: Universal Instantiation

Every instantiation of a universally quantified sentence is entailed by it: ∀ v α α({v/g}) for any variable v and ground term g E.g., ∀ x King(x) ∧ Greedy(x) ⇒ Evil(x) yields

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Appendix: Universal Instantiation

Every instantiation of a universally quantified sentence is entailed by it: ∀ v α α({v/g}) for any variable v and ground term g E.g., ∀ x King(x) ∧ Greedy(x) ⇒ Evil(x) yields

King(Joffrey) ∧ Greedy(Joffrey) ⇒ Evil(Joffrey) King(Aerys) ∧ Greedy(Aerys) ⇒ Evil(Aerys) King(Father(Joffrey)) ∧ Greedy(Father(Joffrey)) ⇒ Evil(Father(Joffrey)) . . .

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Appendix: Existential instantiation (EI)

For any sentence α, variable v, and constant symbol k that does not appear elsewhere in the knowledge base: ∃ v α α({v/g})

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Appendix: Existential instantiation (EI)

For any sentence α, variable v, and constant symbol k that does not appear elsewhere in the knowledge base: ∃ v α α({v/g}) E.g., ∃ x Crown(x) ∧ OnHead(x, John) yields

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Appendix: Existential instantiation (EI)

For any sentence α, variable v, and constant symbol k that does not appear elsewhere in the knowledge base: ∃ v α α({v/g}) E.g., ∃ x Crown(x) ∧ OnHead(x, John) yields Crown(C1) ∧ OnHead(C1, John)

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Appendix: Existential instantiation (EI)

For any sentence α, variable v, and constant symbol k that does not appear elsewhere in the knowledge base: ∃ v α α({v/g}) E.g., ∃ x Crown(x) ∧ OnHead(x, John) yields Crown(C1) ∧ OnHead(C1, John) provided C1 is a new constant symbol, called a Skolem constant

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Appendix: Existential instantiation contd.

UI can be applied several times to add new sentences; the new KB is logically equivalent to the old

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Appendix: Existential instantiation contd.

UI can be applied several times to add new sentences; the new KB is logically equivalent to the old EI can be applied once to replace the existential sentence;

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Appendix: Existential instantiation contd.

UI can be applied several times to add new sentences; the new KB is logically equivalent to the old EI can be applied once to replace the existential sentence; the new KB is not equivalent to the old,

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Appendix: Existential instantiation contd.

UI can be applied several times to add new sentences; the new KB is logically equivalent to the old EI can be applied once to replace the existential sentence; the new KB is not equivalent to the old, but is satisfiable iff the old KB was satisfiable

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