Loop Invariants

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Loop Invariants

• CSC 171 FALL 2001
• LECTURE 6

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History Herman Hollerith

• Herman is said to have been a bright and able
  child at school, but had an inability to learn
  spelling easily. His determined teacher made his
  life miserable to the extent that he used to
  avoid school whenever possible and run away when
  his teacher showed renewed effort to improve his
  spelling.

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History Herman Tabulating

• 1890 - Herman Hollerith won the competition for
  the processing equipment to assist in the 1890
  US Census
• The Hollerith Tabulating Company, eventually
  became the Calculating-Tabulating-Recording
  (C-T-R) company in 1914, and eventually was
  renamed IBM in 1924.

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Loop invariants

• In order to verify loops we often establish an
  assertion (boolean expression) that is true each
  time we reach a specific point in the loop.
• We call this assertion, a loop invariant

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Assertions

• When ever the program reaches the top of the
  while loop, the assertion is true

INIT
BODY
INVARIANT
TEST
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Example

• Write a method to compute an
• public static int power(int a , int n)
• You have 5 minutes

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Possible solution

• public static double power(int a, int n)
• double r 1 double b a int i n
• while (igt0)
• if (i2 0) b b b i i / 2
• else r r b i--
• return r

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Does it work?
b i r
a 100 1
a2 50
a4 25
24 a4
a8 12
a16 6
a32 3
2 a36
a64 1
0 a100

• SURE! TRUST ME!
• Well, look at a100 if you dont believe me!
• Note, less loops!
• Can you prove that it works?

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What is the loop invariant?

• At the top of the while loop, it is true that
• rbi an
• It is?
• Well, at the top of the first loop
• r1
• ba
• in

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So, if its true at the start

• Even case
• rnew rold
• bnew (bold)2
• inew(iold)/2
• Therefore,
• rnew (bnew)i-new rold ((bold)2)i-old/2
• rold
  ((bold)2)i-old
• an

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So, if its true at the start II

• Odd case
• rnew roldbold
• bnew bold
• inewiold-1
• Therefore,
• rnew (bnew)i-new rold bold (bold)i-old-1
• rold
  (bold)i-old
• an

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So,

• If its true at the start
• And every time in the loop, it remains true
• Then, it is true at the end
• rbi an
• And, i 0 ( the loop ended)
• What do we know?

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Correctness Proofs

• Proof are more valuable than testing
• Tests demonstrate limited correctness
• Proofs demonstrate correctness for all inputs
• For some time, people hoped that all formal logic
  would replace programming
• The naïve idea that programming is a form of
  math proved to be an oversimplification

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Correctness Proofs

• Unfortunately, in practice, these methods never
  worked very well.
• Instead of buggy programs,
• people wrote buggy logic
• Nonetheless, the approach is useful for program
  analysis

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The take away message?

• In the end, engineering and (process) management
  are at least as important as mathematics and
  logic for the successful completion of large
  software projects