1. 1 CMSC 250 Chapter 2, Predicate Logic

2. 2 CMSC 250 Definitions l Subject / predicate John / went to the store. The sky / is blue. l Propositional logic- uses statements l Predicate logic or calculus- uses predicates –predicates must be applied to a subject in order to be true or false l P(x) – means this predicate represented by P –applied to the object represented by x

3. 3 CMSC 250 Quantification l  x There exists an x l  xFor all x's l Domain- the set where these subjects come from l (  x) [P(x)]there exists an x for which predicate p is true l (  x) [P(x)]for all x, predicate p is true l Notation: –  x  Z there exists an x in the integers –  x  R for all x's in the reals

4. 4 CMSC 250 Predicate translation l A student of mine is wearing a blue shirt. –Domain: all people P –Quantification: There is at least one –Predicates: "wearing a blue shirt" and "is my student" Let B(x) represent "x is wearing a blue shirt" Let S(x) represent "x is my student"  x  P such that B(x) ^ S(x) l All good students are in class. –Domain: all people P –Quantification: All of them –Predicates: "are in class" and "is a good student" Let C(x) represent "x is in class" Let G(x) represent “x is a good student"  x  P such that G(x)  C(x)

5. 5 CMSC 250 Establishing truth and falsity l To show an existentially quantified statement (  ) is true, find an example. l One way to show an existentially quantified statement is false is to show it's false for every member of the domain. l To show a universally quantified statement (  ) is false, find a counterexample. l One way to show a universally quantified statement is true is to show it's true for every member of the domain.

6. 6 CMSC 250 Negation of quantified statements l  (there is a cat which can fly)  –some cats can’t fly? –only some cats can fly? –cats can only fly sometimes? –all cats can’t fly? –watch me throw this cat out the window… l Let F(x) represent the predicate "x can fly" ~ (  x  cats) [F(x)]  (  x  cats) [~F(x)]

7. 7 CMSC 250 Negation of quantified statements l  (everybody likes me)  –nobody likes me? –everybody doesn’t like me? –somebody doesn’t like me? l Let L(x) represent the predicate "x likes me" ~(  x  people) [L(x)]  (  x  people) [~L(x)]

8. 8 CMSC 250 Discrete Structures CMSC 250 Lecture 9 February 15, 2008

9. 9 CMSC 250 Vacuous cases (for universally quantified statements) l All even prime numbers which are greater than or equal to 10 are the sum of two squares. l All students in this class who are ten feet tall have green hair. l Are these true or false? How would we show they’re true or false?

10. 10 CMSC 250 Universal conditional statements and variants l A universal conditional statement: (  x  D) [P(x)  Q(x)] l Its contrapositive: (  x  D) [~Q(x)  ~P(x)] l Its converse: (  x  D) [Q(x)  P(x)] l Its inverse: (  x  D) [~P(x)  ~Q(x)] l Also applies to existentially quantified conditional statements

11. 11 CMSC 250 Multiple quantification l Let C = {all chairs} and P = {all people}, and S(c,p) represent “person p is sitting in chair c " l (  p  P) (  c  C) [S(c,p)] l (  c  C) (  p  P) [S(c,p)] l (  p  P) (  c  C) [S(c,p)] l (  c  C) (  p  P) [S(c,p)]

12. 12 CMSC 250 Order of mixed quantifiers l  and  are different! l Example- domain N- the following do not mean the same thing! l (  x) (  y) [y > x] l (  y) (  x) [y > x]

13. 13 CMSC 250 Mixed multiple quantification variations l Let C = {all chairs} and P = {all people}, and S(c,p) represent “person p is sitting in chair c" l (  c  C) (  p  P) [S(c,p)] l (  p  P) (  c  C) [S(c,p)] l (  p  P) (  c  C) [S(c,p)] l (  c  C) (  p  P) [S(c,p)]

14. 14 CMSC 250 Negations of multiply quantified statements l Let P= {all people}, and C= {all cats}, and L(p,c) represent “person p likes cat c" ~((  p  P) (  c  C) [L(p,c)])  (  p  P) (  c  C) [~L(p,c)]

15. 15 CMSC 250 Other variations l Let S be the set of all students and C be the set of all colleges, and A(s,c) represent "student s attends college c". l Exactly one student attends college: (  s  S) (  c  C) [A(s,c)  ~((  p  S) (  u  C) [p  s  A(p,u)])] (  s  S) (  c  C) [A(s,c)  ((  p  S) (  u  C) [p = s  ~A(p,u)])] l At most one student attends college: (  s,p  S) (  c,u  C) [(A(s,c)  A(p,u))  p = s] l At least two students attend college: (  s,p  S) (  c,u  C) [A(s,c)  A(p,u)  p  s]

16. 16 CMSC 250 Discrete Structures CMSC 250 Lecture 10 February 18, 2008

17. 17 CMSC 250 Establishing truth and falsity with multiple quantification l Recall: –to show  x is true, find an example. –to show  x is false, find a counterexample. l Let the domain be the natural numbers

18. 18 CMSC 250 Set (Euler) diagrams l Circles are used to indicate “truth sets" for a predicate –where the predicate applied to an object is true l A dot is used to indicate a specific instance l A completely contained circle indicates a "for all" relationship l An overlapping circle indicates "some" SBC P A U OR U P A All college students are brilliant. All brilliant people are scientists.  All college students are scientists. Some poets are unsuccessful. Some athletes are unsuccessful.  Some poets are athletes.

19. 19 CMSC 250 Interpretations l Assign meanings to predicates and domains to quantified variables l One interpretation: domain N, L is < l Another interpretation: domain Q, L is < l Another interpretation: domain N, L is / (L(x, y) means "x can be divided by y evenly") l An interpretation is analogous to a row of a truth table (except there are an infinite number of them).

20. 20 CMSC 250 Rules of inference for quantified statements Universal Modus Ponens (  x  D) [P(x)  Q(x)] a  D P(a)  Q(a) Universal Modus Tollens (  x  D) [P(x)  Q(x)] a  D ~Q(a)  ~P(a) Universal instantiation (  x  D) [P(x)] a  D  P(a) Existential generalization P(a) a  D  (  x  D) [P(x)]

21. 21 CMSC 250 Rules that don't exist or need more clarification l Existential Modus Ponens doesn't exist l Existential Modus Tollens doesn't exist l Universal generalization: P(a),  (  x  D) [P(x)] –only if a is completely arbitrary in the domain l Existential instantiation: (  x  D) [P(x)],  P(a) –only if a is completely arbitrary in the domain

22. 22 CMSC 250 Errors in deduction l Converse error: (  x  D) [P(x)  Q(x)] Q(a)  P(a) l Called asserting the consequence l Inverse error: (  x  D) [P(x)  Q(x)] ~P(a)  ~Q(a) l Called denying the hypothesis

23. 23 CMSC 250 Easy formal direct proofs by deduction P1: (  x  D) [P(x)  Q(x)] P2: ~Q(a), where a  D  (  x  D) [~P(x)] P1: (  x  D) [P(x)  Q(x)] P2: (  x  D) [R(x)  ~P(x)] P3: P(b), where b  D  Q(b) ^ ~R(b)

24. 24 CMSC 250 More formal direct proofs by deduction P1: (  x  D) [P(x)  Q(x)] P2: (  x  D) [~P(x)  R(x)] P3: P(b), where b  D  (  x  D) [Q(x) ^ R(x)] P1: (  x  D) [P(x)  Q(x)] P2: (  y  D) [~P(y)  R(y)] P3: (  z  D) [~Q(z)]  (  x  D) [R(x)]

25. 25 CMSC 250 Discrete Structures CMSC 250 Lecture 11 February 20, 2008

26. 26 CMSC 250 One more proof l Note: (  x  D) [A(x)  ~N(x)]  (  y  D) [A(y)  ~N(y)]  (  z  D) [A(z)  ~N(z)] etc. l Also, the quantified variables are completely independent in different quantified predicates like (  y  D) [A(y)] and (  y  D) [B(y)  Q(y)] P1: (  x  D) [(A(x)  B(x))  (M(x)  N(x))] P2: (  y  D) [A(y)  ~N(y)]  (  z  D) [M(z)  B(z)]

27. 27 CMSC 250 Example using predicate logic l Once we define operations or concepts we can use them in defining other operations. l Consider only , , =, and the domain N l Define DIV(x,y): (  z)[x  z = y] l Now we can define PRIME(y): (  x)[DIV(x,y)  x = 1  x = y] l Also ODD(y): (  x)[y = 2  x + 1]

28. 28 CMSC 250 Example, continued l Using these, define SUM3P(x): l Now we can state the following: (  n)(  x)[x  n  ODD(x)  SUM3P(x)]

29. 29 CMSC 250 Another application- describing properties of programs l Motivation: –finding bugs –proofs of correctness –specifying programs- clarifying goals before coding A precondition is a property which must be true before some piece of code is executed, otherwise it won’t work correctly l A postcondition is a property which will be true after some piece of code executes (assuming the precondition was true beforehand) l An invariant is a property which is true at every iteration of a loop (at the top of the loop)

30. 30 CMSC 250 Example i= 0; // precondition: true (meaning there are no // requirements or conditions which must be // satisfied in order for the code to work correctly) while (i < n && arr[i] < x) // invariant: 0  i < n  // (  j, 0  j < i)[arr[j] < x] i++; // postcondition: n < 0  // (0  i  n  // (  j, 0  j < i)[arr[j] < x]  // i < n  arr[i]  x)

31. 31 CMSC 250 Discrete Structures CMSC 250 Lecture 12 February 22, 2008

32. 32 CMSC 250 Another example i= 1; max= arr[0]; // precondition: true while (i < n) { // invariant: 1  i < n  // (  j, 0  j < i)[arr[j] = max]  // (  k, 0  k < i)[arr[k]  max] if (arr[i] > max) max= arr[i]; i++; } // postcondition: n < 1  // (i = n  // (  j, 0  j < n)[arr[j] = max]  // (  k, 0  k < n)[arr[k]  max])

33. 33 CMSC 250 A further example i= 0; found= 0; // precondition: true while (i < n && !found) // invariant: found = 0  0  i < n  // (  j, 0  j < i)[arr[j]  x] if (arr[i] == x) found= 1; else i++; // postcondition: // n < 0  // ((found = 0  i = n  // (  j, 0  j < i)[arr[j]  x]))  // (found = 1  0  i < n  arr[i] = x  // (  j, 0  j < i)[arr[j]  x]]))

34. 34 CMSC 250 One more example l If we need to refer to the initial value of a variable, the convention is to use its name with a prime symbol i= 0; // precondition: n  0 while (i < n – 1) { // invariant: i < n – 1  // (  j, 0 ≤ j < i)[arr[j] = arr'[j + 1]]  // (  j, i ≤ j < n)[arr[j] = arr'[j]] arr[i]= arr[i + 1]; i++; } // postcondition: // n < 0  // (i = n – 1  // (  j, 0 ≤ j < i)[arr[j] = arr'[j + 1]]  // [arr[n - 1] = arr'[n - 1]])