Chapter 6 Recursive

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Chapter 6Recursive

• ???????? ????????????????
• K. H. Rosen, Discrete Mathematics and Its
  Applications, 5th Edition, McGraw-Hill.
• A. V. Aho and J. D. Ullman, Foundations of
  Computer Science, C Edition, W.H. Freeman.
• J. Martin, Introduction to Languages and the
  Theory of Computation, 3rd Edition, McGraw-Hill.

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Outline

• Concept of Recursive
• Recursive Functions
• Simple Recursive Procedure

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6.1 Concept of Recursive

• A recursive (or inductive) definition involves
• One or more basis rules,
  in which some simple objects are
  defined, and
• One or more inductive rules, whereby
  larger objects are defined in terms of
  smaller ones in the collection.

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6.1 Concept of Recursive

• Factorial function is defined by 123n to get
  n!, we can also defined the value of n! as
  follows
• Basis step 1! 1
• Induction step n! n(n-1)!
• n1, 1! 1
• n2, 21! 21 2
• n3, 32! 32 6
• n4, 43! 46 24
• and so on.

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6.1 Concept of Recursive

• 1 if n0
• n(n-1)! if n gt 0
• Factorial function is defined on the set of
  natural number, first by defining the value at 0,
  and then
• by defining the value at any larger natural
  number in terms of its value at the previous one.
• There is an obvious analogy here to the basis
  step and the induction step in a proof by
  mathematic induction.

n!
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6.2 Recursive Functions

• We use two steps to defined a function with the
  set of nonnegative integer as its domain
• Basis step Specify the value of the function at
  zero.
• Recursive step Give a rule for finding its value
  at an integer from its values at smaller
  integers.
• Such a definition is called a recursive or
  inductive definition.

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6.2 Recursive Functions

• Ex. Suppose that f is defined recursively by
• f(0) 3
• f(n1) 2f(n) 3
• Find f(1), f(2), f(3), and f(4).
• From recursive definition it follows that
• f(1) 2f(0) 3 2(3)3 63 9,
• f(2) 2f(1) 3 2(9)3 183 21,
• f(3) 2f(2) 3 2(21)3 423 45,
• f(4) 2f(3) 3 2(45)3 903 93.

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6.2 Recursive Functions

• Definition
• The Fibonacci numbers, f0, f1, f2, are defined
  by the equations f0 0, f1 1, and
• fn fn-1 fn-2 for n 2, 3, 4,
• Find the Fibonacci numbers f2, f3, f4, f5, f6
• f2 f1 f0 10 1
• f3 f2 f1 11 2
• f4 f3 f2 21 3
• f5 f4 f3 32 5
• f6 f5 f4 53 8

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6.2 Exercise

• Prove that for every ngt0, f(n) lt (5/3)n
• Define recursive definition
• Find f(0), f(1), f(2), f(3), f(4)

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6.3 Simple Recursive Procedure

• Ex.1 ????????????????????? recursive
  ??????????????????????????????? an
• ???????
• ?? Basis step
• ????? n 0, a0 1
• ?? Recursive step
• ????? n k, ak a ak-1

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6.3 Simple Recursive Procedure

• procedure power ( a, n)
• if (n 0) then
• return 1 / basis /
• else
• return a power(a, n-1) / recursive /
• end procedure

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6.3 Simple Recursive Procedure

• Ex.2 a recursive function of factorial is defined
  as
• We can write pseudo-code for this problem as
• main procedure
• read n
• result factorial(n)
• print Factorial of n is result
• end main procedure

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6.3 Simple Recursive Procedure

• procedure fact (n)
• if (n 0)
• return 1 / basis /
• else
• return n fact(n-1) /
  induction /
• end procedure

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6.3 Execise

• ???????? pseudocode ????????????? recursive
  ???????????????????????
• f(0) 3
• f(n1) 2f(n) 3
• ???????? pseudocode ????????????? Fibonacci
  number
• ???????? 1 ??? 2 ???????? n ???????????? 5
  ???????????????? pseudocode ????????
• ???????? pseudocode ??????????????????????????????
  ? (Greatest Common Divisor GCD)
  ?????????????????? 2 ????? a ??? b ?????? a
  ???????? b
• ?????????????????? ??? ?????????????????????????
  ????????????????????? ???? 16, 40
  ???????????????? 8

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