CMSC 203 / 0201 Fall 2002

Week #7 – 7/9/11 October 2002 Prof. Marie desJardins

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TOPICS

 Recursion  Recursive and iterative algorithms  Program correctness

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MON 10/7 RECURSION (3.3)

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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 f 0 = 0, f 1 = 1, f n = f n-1 Strings      *: Strings over alphabet Empty string  String length l(s) String concatenation  + f n-2

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Examples

 S 1  = Even positive integers 2  S 1 ; x+2  S 1 if x  S 1  S 2  = Even integers 2  S 2 ; x+2  S 2 if x  S 2 ; x-2  S 2 if x  S 2  Prove that S 1  is the set of all even positive integers Every even positive integer is in S 1 …  Every member of S 1 is an even positive integer…

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Examples II

 Exercise 3.3.5: Give a recursive definition of the sequence {a n }, n = 1, 2, 3, … if  (a) a n = 6n  (b) a n = 2n + 1  (c) a n = 10 n  (d) a n = 5  Exercise 3.3.31: When does a string belong to the set A of bit strings defined recursively by    0

x

1  A if x  A, where  is the empty string?

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Examples III

 Exercise 3.3.33: Use Exercise 29 [definition of w i ] and mathematical induction to show that l(w i ) = i  l(w), where w is a string and i is a nonnegative integer.

 Exercise 3.3.34: Show that (w R ) I = (w i ) 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.

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WED 10/9 RECURSIVE ALGORITHMS (3.4)

** HOMEWORK #4 DUE ** ** UNGRADED QUIZ TODAY **

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Concepts / Vocabulary

 Recursive algorithm  Iterative algorithm

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Examples

 Algorithm 7: Recursive Fibonacci  Complexity of f 7 (Exercise 14), f n  Algorithm 8: Iterative Fibonacci  Complexity f 7 (Exercise 14), f n  Eercise 3.4.23: Give a recursive algorithm for finding the reversal of a bit string.

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FRI 10/11 PROGRAM CORRECTNESS (3.5)

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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

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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.

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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

the final assertion y=2.

and

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