CMSC 203 / 0201 Fall 2002

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CMSC 203 / 0201Fall 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 10/7RECURSION (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 f0 0, f1 1, fn fn-1
  fn-2
• Strings
• ? Strings over alphabet ?
• Empty string ?
• String length l(s)
• String concatenation

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Examples

• S1 Even positive integers
• 2 ? S1 x2 ? S1 if x ? S1
• S2 Even integers
• 2 ? S2 x2 ? 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

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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?

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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.

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WED 10/9RECURSIVE 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 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.

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FRI 10/11PROGRAM 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 gt 0 then z
  z 1 else z 0is correct with respect to
  the initial assertion y3 and the final assertion
  z6.

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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 lt 0 then y
  -2x / x else if x gt 0 then y 2x /
  x else if x 0 then y 2is correct with
  respect to the initial assertion T and the final
  assertion y2.