PAI Recitation 3 – Logic

Yuxin Chen12-10-2012

Inference

• Truth tables• Modus ponens• Resolution• …

Resolution

Transform KB into CNF

Resolution: Example

H: Heads upT: Tails upIW: I winUL : You lose

Heads, I winTails, you loseEither heads or tailsI win or you loseI don’t win

H => IWT => ULH <=> ¬TIW <=> UL¬IW

Resolution

• Why can’t we just use resolution to derive the sentence we are interested in (instead of adding its negation and derive false)?

Use resolution to derive from KB?

Resolution

• Why can’t we just use modus ponens instead of resolution to derive the sentence we are interested in?

Modus ponens

Example: P implies Q; Not P implies Q. Show that Q is ture.

Resolution

• Resolution is sound and complete• Recall:

First order logic

• A superset of propositional logic• Don’t treat propositions as “atomic” entities. • Sentences refer to objects, their relationships,

and their properties.Types ExampleObjects Bob, October 12

Relations/predicates MotherOf, ColorOf

Functions IsMotherOf, IsBlue

Connectives ∧∨¬ ⇔⇒

Quantifiers Universal: x:(isMan(x)) isMortal(x)) ∀ ⇒Existential: y: ( isFather (y, Fred) ) ∃

Example: Convert to First-Order Logic

• ∀x isSibling (x, Bob) isKind(x)⇒• ∀x isSibling (x, Bob) isKind(x) ∧• ∃x At(x, ETH) isSmart(x) ⇒• ∃x At(x, ETH) isSmart(x)∧

Example: Convert to English

• There is only one correct answer (do not use !) ∃

• First cousin (child of your parent’s siblings) • All grad students watch the same TV shows• Any person can fool some of the people all of

the time, all of the people some of the time but not all of the people all of the time

Example: Convert to English and FOL

Example (Cont..)

• In the sentences below, a symbol is missing. What symbol is it most likely to be? – ∀x Something(x) ??? SomethingElse(x) – ∃x Something(x) ??? SomethingElse(x)

Example (Cont..)

• Is this correctly written? – Peter has at least two children. – ∃x,y ParentOf(Peter, x) ParentOf(Peter, y) ∧

Resolution in FOL: Example

• Jack owns a dog. Every dog owner is an animal lover, and no animal lover kills an animal. Either Jack or Curiosity killed the cat, who is named Tuna.

• Did Curiosity kill the cat?

Resolution in ROL: Conjunctive Normal Form

• Jack owns a dog.

• Every dog owner is an animal lover.

• No animal lover kills an animal.

• Either Jack or Curiosity killed the cat, who is named Tuna.

• Did Curiosity kill the cat?

Resolution in ROL

Summary