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CS1502 Formal Methods in
Computer Science
Notes 15
Problem Sessions
1
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/lectures/lec15solutions.pdf
Why?
– Review
– Illustrate that you don’t *need* any of the con rules
2
6 Fitch Proofs
We’ll 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
3
Problem 7
Prove the argument below is valid using a Fitchstyle proof.
Some teachers are scholars.
No scholar has time for either football or
basketball.
 Some teachers do not have time for
basketball.
4
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 isn’t 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 n*n = 2(2m*m + 2m) +
1, showing that n*n is odd. Thus, we have
shown ~Even(n)  ~Even(n*n)
5
Review Questions around 10.13, 10.17; (see
next slide)
1.
2.
3.
4.
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 answers:10.10: 2; 10.13: 1;
10.14: 3; 10.15: 2; 10.16: 1; 10.17: 3;
Varations: in lecture
6
Necessary
S is always true
Possible
Satisfiable
S could be true
Equivalence
Consequence
S and S’ always
have the same
truth values
Whenever
P1…Pn 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
P1…Pn
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
P1…Pn
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
P1…Pn
7
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)?
8
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)]?
9
Problem 10
all x (P(x)  Q(x))
all x (Q(x)  P(x))
----All x (P(x)  Q(x))
10