CS1502 Formal Methods in Computer Science

1
CS1502 Formal Methods in Computer Science

• Notes 15
• Problem Sessions

2
Preliminaries

• 3 proofs we will be able replace with Taut Con
• 1 proof we will be able to replace with FO Con
• First 4 proofs in
• http//www.cs.pitt.edu/wiebe/courses/CS1502/lectu
  res/lec15solutions.pdf
• Why?
• Review
• Illustrate that you dont need any of the con
  rules

3
6 Fitch Proofs

• Well do them in Fitch in lecture
• Next 6 proofs in
• http//www.cs.pitt.edu/wiebe/courses/CS1502
  /lectures/lec15solutions.pdf
• Problems 1-3 use only Intro/Elim rules
• Problem 4 may use Taut Con on at most two
  support sentences
• Problems 5-6 May use FO Con on at most one
  support sentence, and Taut Con for the resolution
  step

4
Problem 7

• Prove the argument below is valid using a
  Fitch-style proof. Some teachers are
  scholars. No scholar has time for either
  football or basketball.? Some teachers
  do not have time for basketball.

5
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)

6
Review Questions around 10.13, 10.17 (see next
slide)

• Recall the circles from lecture
• inner tautological consequence
• middle FO but not tautological cons
• Outer logical but not FO cons
• Outside the circle not a logical cons
• Here are answers10.10 2 10.13 1 10.14 3
  10.15 2 10.16 1 10.17 3
• Varations in lecture

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Necessary S is always true Possible Satisfiable S could be true Equivalence S and S always have the same truth values Consequence Whenever P1Pn are true, Q is also true
Tautological Translate sentences into propositional logic using TFF algorithm S is a tautology S is Tautologically possible S and S are Tautologically equivalent Q is a tautological consequence of P1Pn
First Order (FO) Replace predicates with nonsense names S is an FO validity S is FO possible (FO satisfiable) S and S are FO equivalent Q is a FO consequence of P1Pn
Logical S is logically necessary (a logical truth) (logically valid) S is logically possible (satisfiable) S and S are logically equivalent Q is a logical consequence of P1Pn
8
Problem 8

• Does ?x ? y P(x, y) follow from
• ?x ? y P(x, y)?
• Hint does ?x ? y SameRow(x, y) follow from ?x
  ? y SameRow(x, y)?

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Problem 9

• Does ?x ? y P(x, y) ? Q(x) follow from
• ?x ?y P(x, y) ? Q(x)?
• Hint does ?x ? y LeftOf(x,y) ?
  Large(x)follow from
• ?x ?y LeftOf(x,y) ? Large(x)?

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Problem 10
all x (P(x) ? Q(x)) all x (Q(x) ? P(x)) ----- All
x (P(x) ?? Q(x))