CS 1502 Formal Methods in Computer Science

1. CS 1502 Formal Methods in Computer Science Lecture Notes 14 Translations Mixed Quantifiers

2. 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)

3. 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))]

4. 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))]

5. 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))]

6. 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))]

7. 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)

8. 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) (sameforb,c,d) Want: all x all y ((Cube(x) ^ Cube(y) ^ x != y)  (LeftOf(x,y) v RightOf(x,y)))

9. Translation • x y P(x,y)For each x there is a y such that P(x,y). y x

10. Translation • x y P(x,y)There is a special x such that for all y, P(x,y).

11. 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.

12. 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

13. 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))]

14. 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)]]

15. 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…

16. 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

17. 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 ownsandwich: all x exist y Ate(x,y) • But perhaps it was one huge sandwich: exist y all x Ate(x,y)

18. What do these sentences mean? • exist y (Small(y) ^ all x (Small(x)  y=x)) There is exactly one small thing! • all x all y ((Small(x) ^ Small(y))  y=x) There is at most one small thing T if there are 0 or 1 small things (try it in TW)

19. 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))

20. 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))

21. 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)  y=z)) Moral: use a function when appropriate

22. 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))