Formal Methods in Computer Science CS1502 Review for Exam 2

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Formal Methods in Computer ScienceCS1502Review
for Exam 2

• Patchrawat Uthaisombut
• University of Pittsburgh

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Outline

• Translation
• well-formed formula
• Aristotelian forms
• tricky points
• Proofs
• Proofs with ?, ?
• FO counterexample

• Interpretation
• Tautology,truth-functional form
• Tautological consequence
• FO validity
• FO consequence

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Well-Formed Formula (wff)
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Well-Formed Formula

• Definition of wff.
• Satisfaction of P(x)
• Object c satisfies S(x) if and only if S(c) is
  true.
• Truth value of P(c)
• Truth value of ?x P(x)
• ?x S(x) is true if and only if every object in
  the domain satisfies S(x).
• Truth value of ?x P(x)
• ?x S(x) is true if and only if at least one
  object in the domain satisfies S(x).

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Wffs vs Sentences

• Differences between wffs and sentences.
• Sentences are wffs with no free variables.
• All sentences are wffs
• Not all wffs are sentences.
• Translate this into logic.

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Translation
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Aristotelian Forms

• All Ps are Qs
• ?x (P(x) ? Q(x))
• Some Ps are Qs
• ?x (P(x) /\ Q(x))

• No Ps are Qs
• All Ps are not Qs.
• ?x (P(x) ? Q(x))
• ?x (P(x) /\ Q(x))
• Some Ps are not Qs
• Not all Ps are Qs.
• ?x (P(x) /\ Q(x))
• ?x (P(x) ? Q(x))

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Tricky point

• No tets are in the same column.
• ?x ?y (Tet(x) /\ Tet(y)) ? SameCol(x,y)
• Consider the case xb and yb.
• Incorrect translation.

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Identity and Variables

• For every pair of objects, it is the case that
• ?x ?y (x ! y ? )
• No tets are in the same column.
• ?x ?y (x ! y /\ Tet(x) /\ Tet(y))
  ?SameCol(x,y)

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Tricky point

• There are two Cubes that are Small
• ?x ?y ( (Cube(x) /\ Cube(y) ) /\ (Small(x) /\
  Small(y))
• Incorrect translation

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Identity and Variables

• There exist two objects which are
• ?x ?y (x ! y /\ )
• There are two Cubes that are Small
• ?x ?y ( (x ! y /\ Cube(x) /\ Cube(y) ) /\
  (Small(x) /\ Small(y))

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• There is exactly 1 cube.
• There is at least one cube and for any object in
  the domain, if it is a cube, it is that same one.
• ?x (Cube(x) /\ ?y (Cube(y) ? yx))
• ?x ( /\ ?y ( ? yx))

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Exercises

• Object c is not in the same row as any cube.
• No cubes are the same size.
• Every tet has exactly one object behind it.
• There are two cubes that are in front of all
  tets.

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Different types of truths
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Different Kinds of Truths

• Tautologies bool conn.
• First-order validity bool conn. quan.
  .
• Logical truths bool conn. quan.
  pred.
• Model-specific truths bool conn. quan. pred.
  restriction of the model/worlds.
• ?x Tet(x) ? ?x Tet(x)
• ?x Tet(x) ? ?x (Cube(x) \/ Dodec(x))

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Truth-Functional Form
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Finding the truth-functional form

• Tet(a) /\ ((Tet(b) \/ ?x Cube(x)) ? Tet(b))
• A /\ ((B \/ C) ? B)
• ?x Large(x) /\ ?y(Tet(y) ? ?z LeftOf(y,z))
• A /\ B

B
C
B
A
A
B
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First-Order Counterexample
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FO counterexample

• ?x SameShape(x,x)
• Replace the predicate with nonsensical one.
• ?x SameShape(x,x) becomes ?x P(x,x)
• Come up with a interpretation of the predicate
• P(x,y) means x is taller than y.
• Come up with a world (not necessarily Tarskis
  world).
• A world with 3 people John, James, Jack with
  height 4, 5, 6 respectively.
• Verify that the sentence is false under this
  interpretation in this world.
• P(John, John) is false. Thus, ?x P(x,x) is
  false.
• State the conslution
• Not an FO validity.

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FO counterexample

• To demonstrate that a sentence is not an FO
  validity.
• To demonstrate that in an argument, the
  conclusion is not an FO consequence of the
  premises.

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Proofs