CS 1502 Formal Methods in Computer Science

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CS 1502 Formal Methods in Computer Science

• Lecture Notes 14
• Translations
• Mixed Quantifiers

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Informal Proof

• Prove that if the square of an integer is even,
  then so is that integer.
• Proving the contrapositive is easier If an
  integer is not even, then its square isnt even
  either.
• Let n be an integer. Assume Even(n), i.e.,
  Odd(n). Then we can express n as 2m 1 for some
  m. But we see that nn 2(2mm 2m) 1,
  showing that nn is odd. Thus, we have shown
  Even(n) ? Even(nn)

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Translations

• Every Steeler is taller than Jim.
• ?x (Steeler(x) ? Taller(x, jim))
• Every Steeler is taller than anyone who is the
  same size as Jim.
• ?x Steeler(x) ? ?y (SameSize(y, jim) ? Taller(x,
  y))

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Translations

• Some Steeler is taller than Jim.
• ?x (Steeler(x) ? Taller(x, jim))
• Some Steeler is taller than anyone who is the
  same size as Jim.
• ?x Steeler(x) ? ?y (SameSize(y, jim) ? Taller(x,
  y))

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Translations

• Some Steeler is not taller than Jim.
• ?x (Steeler(x) ? ?Taller(x, jim))
• Some Steeler is not taller than someone who is
  the same size as Jim.
• ?x Steeler(x) ? ?y (SameSize(y, jim) ?
  ?Taller(x, y))

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Translations

• No Steeler is taller than Jim.
• ?x (Steeler(x) ? ?Taller(x, jim))
• No Steeler is taller than somebody who is the
  same size as Jim.
• ?x Steeler(x) ? ??y(SameSize(y, jim) ? Taller(x,
  y))

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Orders of Quantifiers

• ?x ?y P(x,y) is logically equivalent to ?y ?x
  P(x,y)
• ?x ?y P(x,y) is logically equivalent to
• ?y ?x P(x,y)
• ?x ?y P(x,y) is not logically equivalent to ?y ?x
  P(x,y)

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A Gotcha (before moving on)

• Suppose a world has 4 cubes, all in the same row
  (a,b,c,d)
• Is the following true of that world?
• all x all y ((Cube(x) Cube(y)) ?
• (LeftOf(x,y) v RightOf(x,y)))
• No! Can infer LeftOf(a,a) v RightOf(a,a) (same
  for b,c,d)
• Want all x all y ((Cube(x) Cube(y) x ! y) ?
  
• (LeftOf(x,y) v RightOf(x,y)))

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Translation

• ?x ?y P(x,y)For each x there is a y such that
  P(x,y).

y
x
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Translation

• ?x ?y P(x,y)There is a special x such that for
  all y, P(x,y).

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Prenex Normal Form

• Sentence containing no quantifiers at all, or
• A sentence of the form Q1x1
  Q2x2 Qnxn P
• where Qi are either the universal or
  existential quantifier, xi are variables and wff
  P is free of quantifiers.

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Conversion to Prenex Normal Form

• Replace implications (whose left- or right-hand
  sides include quantifiers) (A ? B) by ?A ? B
• Move ? inwards until there are no quantifiers
  in the scope of a negation
• Rename variables so each separate variable has
  its own name
• Move quantifiers to the front of the sentence,
  without changing their order

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Example
?x(C(x) ? ?y(T(y) ? L(x,y))) ? ?y(D(y) ?
B(x,y)) ?x?(C(x) ? ?y(T(y) ? L(x,y))) ?
?y(D(y) ? B(x,y)) ?x? ?y(C(x) ? T(y) ? L(x,y))
? ?y(D(y) ? B(x,y)) ?x?y?(C(x) ? T(y) ?
L(x,y)) ? ?z(D(z) ? B(x,z)) ?x?y?z?(C(x) ?
T(y) ? L(x,y)) ? (D(z) ? B(x,z)) If you want to
restore the conditional ?x?y?z(C(x) ? T(y) ?
L(x,y)) ? (D(z) ? B(x,z))
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Translation
All cubes are to the left of something large.
?x Cube(x) ? x is to the left of something
large
?x Cube(x) ? ?y (Large(y) ? LeftOf(x,y))
Some cube is to the left of everything large.
?x Cube(x) ? x is to the left of everything
large

• ?x Cube(x) ? ?y Large(y) ? LeftOf(x,y)

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Translation

• Taken(x,y) means x has taken class y
• Domain of discourse for x is all (Pitt) students
• Domain of discourse for y is all (Pitt) CS
  classes
• Continued

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• exist x exist y Taken(x,y)
• A student has taken a CS class
• exist x all y Taken(x,y)
• A student has taken all the CS classes
• all x exist y Taken(x,y)
• Each student has taken some CS class
• exist y all x Taken(x,y)
• There is a CS class that all students have taken
• all y exist x Taken(x,y)
• Each CS class has been taken by at least one
  student

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Translation

• Everyone ate a sandwich
• Ate(x,y) DoD of x is all people DoD of y is all
  sandwiches
• Most natural everyone ate their own sandwich
  all x exist y Ate(x,y)
• But perhaps it was one huge sandwich exist y all
  x Ate(x,y)

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What do these sentences mean?

• exist y (Small(y) all x (Small(x) ? yx))
• There is exactly one small thing!
• all x all y ((Small(x) Small(y)) ? yx)
• There is at most one small thing
• T if there are 0 or 1 small things (try it in TW)

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Valid Arguments?
YES! All Are Valid

• exists y (Tet(y) all x (Cube(x) ?
  SameSize(y,x)))
• ---
• all x (Cube(x) ? exists y (Tet(y)
  SameSize(y,x)))
• exists y (Girl(y) all x (Boy(x)? Likes(x,y)))
• ---
• all x (Boy(x) ? exists y (Girl(y) Likes(x,y)))
• exists y all x (x ! y ? Adjoins(x,y))
• ---
• all x exists y (x ! y ? Adjoins(x,y))

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Valid Arguments?
No! All Are Invalid!!

• all x (Cube(x) ? exists y (Tet(y)
  SameSize(y,x)))
• ---
• exists y (Tet(y) all x (Cube(x) ?
  SameSize(y,x)))
• all x (Boy(x) ? exists y (Girl(y) Likes(x,y)))
• ---
• exists y (Girl(y) all x (Boy(x)? Likes(x,y)))
• all x exists y (x ! y ? Adjoins(x,y))
• ---
• exists y all x (x ! y ? Adjoins(x,y))

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Translations with Function Symbols

• Domain of discourse people
• Every person has exactly one mother, who is older
  than he or she. With a function
• all x OlderThan(mother(x),x)
• With a predicate?
• all x exist y (MotherOf(y,x) OlderThan(y,x)
  all z (MotherOf(z,x) ? yz))
• Moral use a function when appropriate

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Proofs with Multiple Quantifiers
?x (P(x) ??y F(y,x))?x ?y (F(y,x) ?L(y,x))?x
(P(x) ? ?y L(y,x))
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