Formal Methods in Computer Science CS1502 Quantifiers and Proofs

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Formal Methods in Computer ScienceCS1502Quantif
iers and Proofs

• Patchrawat Uthaisombut
• University of Pittsburgh

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Goals

• To gain skills in creating FOL sentences from a
  given situation.
• To learn proof rules involving quantifiers
• ? Elim, ? Intro
• ? Intro, ? Elim
• To gain skills in doing proofs using these rules.

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Exercise

• Create an argument with the shortest proof
  containing ? Elim and ? Intro.
• Create an argument with the shortest proof
  containing ? Intro.
• Create an argument with the shortest proof
  containing ? Elim.

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? Elim

• 1 ?x (Cube(x) /\ Large(x))
• 2 c Cube(c) /\ Large(c)
• 3 Cube(c) /\ Elim 2
• 4 ?x Cube(x) ? Intro 3
• 5 ?x Cube(x) ? Elim 1,2-4

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? Intro (Universal Intro)

• 1 ?x (Cube(x) /\ Large(x))
• 2 c
• 3 Cube(c) /\ Large(c) ? Elim 1
• 4 Cube(c) /\ Elim 3
• 5 ?x Cube(x) ? Intro 2-4

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? Intro

• Universal Generalization andGeneral Conditional
  Proof

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Two forms of ? Intro

• Universal Generalization
• c
• Q(c)
• ?x Q(x)

• General conditional proof
• c P(c)
• Q(c)
• ?x (P(x) ? Q(x))

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General conditional proof

• ?x (Cube(x) ? Large(x))
• ?x (Cube(x) ? Small(x))
• ?x (Cube(x) ? Large(x))
• c Cube(c)
• Cube(c) ? Large(c) ? Elim 1
• Large(c) ? Elim 3,4
• Small(c) AnaCon 6
• ?x (Cube(x) ? Small(x)) ? Intro 3-7

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General conditional proof

• ?x (P(x) ? Q(x))
• ?x (Q(x) ? R(x))
• ?x (P(x) ? R(x))

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General conditional proof

• ?x (P(x) ? Q(x))
• ?x (Q(x) ? R(x))
• c P(c)
• P(c) ? Q(c) ? Elim 1
• Q(c) ? Elim 3,4
• Q(c) ? R(c) ? Elim 2
• R(c) ? Elim 5,6
• ?x (P(x) ? R(x)) ? Intro 3-7

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? Intro (General Conditional)

• 1 ?x Large(x)
• 2 c Cube(c)
• 3 Large(c) ? Elim 1
• 4 ?x (Cube(x) ? Large(x)) ? Intro 2-4

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Translation Proofswith multiple quantifiers

• Boys and Girls
• Euclids Theorem
• The Barber Paradox

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Boys and Girls

• There is a girl that every boy likes.
• Every boy likes some girl.
• Are they equivalent?
• How to phrase them in logic?
• Does one imply another?
• How to support your answer?
• ?y Girl(y) /\ ?x(Boy(x) ? Likes(x,y))
• ?x Boy(x) ? ?y(Girl(y) /\ Likes(x,y))

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Euclids Theorem

• Theorem There are infinitely many primes.
• How to phrase this in logic?
• Predicates
• Prime, Largest, Larger,
• Function
• x y
• AnaCon
• If x is an integer, then x1 is an integer.
• How to prove it?
• ?x (Integer(x) ? ?y (Integer(y) /\ ygtx))

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The Barber Paradox

• There was once a small town where there was a
  barber who shaved all and only the men of the
  town who did not shave themselves.
• Is there such a town?
• How to support the answer?
• ?x ?y (S(x,y) ? S(y,y))

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Examples (proof)

• Boys and Girls
• Euclids Theorem
• The Barber Paradox

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Boys and Girls

• There is a girl that every boy likes.
• Every boy likes some girl.
• Are they equivalent?
• How to phrase them in logic?
• Does one imply another?
• How to support your answer?

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• ?y Girl(y) /\ ?x(Boy(x) ? Likes(x,y))
• ?x Boy(x) ? ?y(Girl(y) /\ Likes(x,y))
• Let c be one of these popular girls in the
  kindergarten class.
• Assume that d is any boy in the kindergarten
  class.
• Every boy likes c.
• Thus, d likes c.
• Thus, d likes some girl. (? intro)
• Since d was an arbitrarily chosen boy, thus every
  boy likes some girl.

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Boys and Girls
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• ?x Boy(x) ? ?y(Girl(y) /\ Likes(x,y))
• ?y Girl(y) /\ ?x(Boy(x) ? Likes(x,y))
• Pseudo-proof
• Every boy likes some girl or other.
• Let e be any boy in the domain.
• Thus, e likes some girl or other.
• Lets use f to name the girl that e likes.
• Since e was an arbitrarily chosen boy, then every
  boy likes f.
• Thus, there is some girl that every boy likes.

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Boys and Girls
,6
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Euclids Theorem

• Theorem There are infinitely many primes.
• How to phrase this in logic?
• Predicates
• Prime, Largest, Larger,
• Function
• x y
• AnaCon
• If x is an integer, then x1 is an integer.
• How to prove it?

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There are infinitely many integers
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The Barber Paradox

• There was once a small town where there was a
  barber who shaved all and only the men of the
  town who did not shave themselves.
• Is there such a town?
• How to support the answer?

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The Barber Paradox