Introduction to the Theory of Computation

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Introduction to the Theory of Computation

• John Paxton
• Montana State University
• Summer 2003

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Humor

• In a train carriage there was Bill Clinton,
  George Bush, Janet Reno and Bo Derek. After
  several minutes of the trip, the train passes
  through a dark tunnel and the unmistakable sound
  of a slap is heard.              When they
  leave the tunnel, Clinton has a big red slap mark
  on his cheek.              (1) Bo Derek thought
  - "That sleazeball Clinton wanted to touch me and
  by mistake, he must have put his hand on Janet
  Reno, who in turn must have slapped his face."
               (2) Janet Reno thought - "That
  dirty Bill Clinton laid his hands on Bo Derek and
  she smacked him."              (3) Bill Clinton
  thought - "George put his hand on Bo Derek and by
  mistake she slapped me."              (4)
  George Bush thought - "I hope there's another
  tunnel soon so I can smack Clinton again."

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1.4 Nonregular Languages

• A w w has an equal number of 0s and 1s.
  Is A regular?
• B w w has an equal number of 01s and 10s
  Is B regular?

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Pumping Lemma

• If A is a regular language, then there is a
  number p (the pumping length) where, if s is any
  string in A of length at least p, then s may be
  divided into three pieces, s xyz, satisfying
  the following conditions

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Pumping Lemma

1. for each i gt 0, xyiz ? A
2. y gt 0
3. xy lt p

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Proof Idea

• Pigeonhole Principle

y
qm
x
qn
z
q1
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Example 010

• p gt 3

1
0
0
e
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Example 010

• s 00012p0 x 000, y 1, z 12p-10
• s 0p0 x e, y 0, z 0p-10

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Example

• Prove that C 0n1n n gt 0 is not regular
• Assume C is regular, let s 0p1p
• Since xy lt p and y gt 0, then y must consist
  of 1 or more 0s, call this number k
• xyz 0j0k0p-j-k1p 0p1p ? C
• xy2z 0j02k0p-j-k1p 0pk1p ? C
• Contradiction!

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Exercises

• Prove that A w w has an equal number of 0s
  and 1s is not regular.
• Prove that D ww w ? 0,1 is not
  regular.
• Prove that E 0i1j i gt j is not regular.

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Exercise

• Show that F ak where k is a multiple of n
  is regular for all n gt 1.
• Show the the language G w w is not a
  palindrome satisifies the three conditions of
  the pumping lemma even though it is not regular.
  Explain why this fact does not contradict the
  pumping lemma.