1. Week #7 – 7/9/11 October 2002 Prof. Marie desJardins
CMSC 203 / 0201 Fall 2002
Week #7 – 7/9/11 October 2002
Prof. Marie desJardins

2. TOPICS Recursion Recursive and iterative algorithms
Program correctness

3. MON 10/7 RECURSION (3.3)

4. Concepts/Vocabulary Recursive (a.k.a. inductive) function definitions
Recursively defined sets
Special sequences:
Factorial F(0)=1, F(n) = F(n-1)(n) = n!
Fibonacci numbers f0 = 0, f1 = 1, fn = fn-1 + fn-2
Strings
*: Strings over alphabet 
Empty string 
String length l(s)
String concatenation

5. Examples S1 = Even positive integers S2 = Even integers
2  S1; x+2  S1 if x  S1
S2 = Even integers
2  S2; x+2  S2 if x  S2; x-2  S2 if x  S2
Prove that S1 is the set of all even positive integers
Every even positive integer is in S1…
Every member of S1 is an even positive integer…

6. Examples II
Exercise 3.3.5: Give a recursive definition of the sequence {an}, n = 1, 2, 3, … if
(a) an = 6n
(b) an = 2n + 1
(c) an = 10n
(d) an = 5
Exercise : When does a string belong to the set A of bit strings defined recursively by    0x1  A if x  A, where  is the empty string?

7. Examples III
Exercise : Use Exercise 29 [definition of wi] and mathematical induction to show that l(wi) = i  l(w), where w is a string and i is a nonnegative integer.
Exercise : Show that (wR)I = (wi)R whenever w is a string and I is a nonnegative integer; that is, show that the ith power of the reversal of a string is the reversal of the ith power of the string.

8. WED 10/9 RECURSIVE ALGORITHMS (3.4)
** HOMEWORK #4 DUE **
** UNGRADED QUIZ TODAY **

9. Concepts / Vocabulary
Recursive algorithm
Iterative algorithm

10. Examples Algorithm 7: Recursive Fibonacci
Complexity of f7 (Exercise 14), fn
Algorithm 8: Iterative Fibonacci
Complexity f7 (Exercise 14), fn
Eercise : Give a recursive algorithm for finding the reversal of a bit string.

11. FRI 10/11 PROGRAM CORRECTNESS (3.5)

12. Concepts / Vocabulary Initial assertion, final assertion
Correctness, partial correctness, termination
“Partially correct with respect to initial assertion p and final assertion q”
Rules of inference
Composition rule
Conditional rules
Loop invariants

13. Examples
Exercise 3.5.3: Verify that the program segment x := 2 z := x + y if y > 0 then z := z + 1 else z := 0 is correct with respect to the initial assertion y=3 and the final assertion z=6.

14. Examples II
Exercise 3.5.6: Use the rule of inference developed in Exercise 5 [if… else if … else …] to verify that the program if x < 0 then y := -2|x| / x else if x > 0 then y := 2|x| / x else if x = 0 then y := 2 is correct with respect to the initial assertion T and the final assertion y=2.