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

2. 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. 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. 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. 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. 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. 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. 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) (same for b,c,d) Want: all x all y ((Cube(x) ^ Cube(y) ^ x != y)  (LeftOf(x,y) v RightOf(x,y)))

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

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

11. 11 Prenex Normal Form Sentence containing no quantifiers at all, or A sentence of the form Q 1 x 1 Q 2 x 2 … Q n x n P where Q i are either the universal or existential quantifier, x i are variables and wff P is free of quantifiers. where Q i are either the universal or existential quantifier, x i are variables and wff P is free of quantifiers.

12. 12 Conversion to Prenex Normal Form 1. Replace implications (whose left- or right-hand sides include quantifiers) (A  B) by  A  B 2. Move  “inwards” until there are no quantifiers in the scope of a negation 3. Rename variables so each separate variable has its own name 4. Move quantifiers to the front of the sentence, without changing their order

13. 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. 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. 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. 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. 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 own sandwich: all x exist y Ate(x,y) But perhaps it was one huge sandwich: exist y all x Ate(x,y)

18. 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. 19 Valid Arguments? 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)) YES! All Are Valid

20. 20 Valid Arguments? 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)) No! All Are Invalid!!

21. 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. 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))

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