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

TOPICS Recursion Recursive and iterative algorithms Program correctness

MON 10/7 RECURSION (3.3)

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

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…

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 3.3.31: 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?

Examples III Exercise 3.3.33: 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 3.3.34: 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.

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

Concepts / Vocabulary Recursive algorithm Iterative algorithm

Examples Algorithm 7: Recursive Fibonacci Complexity of f7 (Exercise 14), fn Algorithm 8: Iterative Fibonacci Complexity f7 (Exercise 14), fn Eercise 3.4.23: Give a recursive algorithm for finding the reversal of a bit string.

FRI 10/11 PROGRAM CORRECTNESS (3.5)

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

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.

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.