Introduction to the Theory of Computation

1
Introduction to the Theory of Computation

• John Paxton
• Montana State University
• Summer 2003

2
Humor

• George W. Bush's Intelligence Quiz
• While visiting England, George W. Bush is invited
  to tea with the Queen. He asks her what her
  leadership philosophy is. She says that it is to
  surround herself with intelligent people. He asks
  how she knows if they're intelligent.
• "I do so by asking them the right questions,"
  says the Queen. "Allow me to demonstrate."
• She phones Tony Blair and says, "Mr. Prime
  Minister. Please answer this question Your
  mother has a child, and your father has a child,
  and this child is not your brother or sister. Who
  is it?"
• Tony Blair responds, "It's me, ma'am."
• "Correct. Thank you and good-bye, sir," says the
  Queen. She hangs up and says, "Did you get that,
  Mr. Bush?"
• "Yes ma'am. Thanks a lot. I'll definitely be
  using that!"

3
Humor

• Upon returning to Washington, he decides he'd
  better put the Chairman of the Senate Foreign
  Relations Committee to the test. He summons Jesse
  Helms to the White House and says, "Senator
  Helms, I wonder if you can answer a question for
  me."
• "Why, of course, sir. What's on your mind?"
• "Uh, your mother has a child, and your father has
  a child, and this child is not your brother or
  your sister. Who is it?"
• Helms hems and haws and finally asks, "Can I
  think about it and get back to you?" Bush agrees,
  and Helms leaves. He immediately calls a meeting
  of other senior senators, and they puzzle over
  the question for several hours, but nobody can
  come up with an answer. Finally, in desperation,
  Helms calls Colin Powell at the State Department
  and explains his problem.

4
Humor

• "Now look here Colin Powell, your mother has a
  child, and your father has a child, and this
  child is not your brother, or your sister. Who is
  it?" Powell answers immediately, "It's me, of
  course."
• Much relieved, Helms rushes back to the White
  House and exclaims, "I know the answer, sir! I
  know who it is! It's Colin Powell!" And Bush
  replies in disgust, "Wrong, It's Tony Blair!"

5
2.3 Non-Context-Free Languages

• To show that a language is not context-free, we
  can use the pumping lemma for context-free
  languages

6
Pumping Lemma

• If A is a Context-Free Language, there 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 five pieces s uvxyz satisfying
  these conditions
• for each i gt 0, uvixyiz ?A
• vy gt 0
• vxy lt p

7
Example 1

• Show that B anbncn n gt 0 is not context
  free
• Select s apbpcp
• Case 1 v and y contain only one type of alphabet
  symbol. uv2wy2z ? B since there arent enough of
  the third symbol.

8
Example 1

• Case 2 Either v or y or both contain two
  symbols. Now uv2wy2z ? B because either v2 or y2
  scrambles the order of the symbols.
• B is not context-free!

9
Example 2

• Show that C aibjck 0 lt i lt j lt k is not
  context-free.
• Select s apbpcp
• Case 1 either v or y or both contain more than
  one symbol. Then uv2xy2z ? C

10
Example 2

• Case 2 v contains as and y contains bs. Now
  uv2xy2z ? C.
• Case 3 v contains as and y contains cs. Now
  uxz ? C.
• Case 4 v contains bs and y contains cs. Now
  uxz ? C.

11
Exercise

• Show that D ww w ? 0, 1 is not context
  free.
• Hint Let s 0p1p0p1p

12
Exercises

• Prove whether or not E 0n02n n gt 0
  is context-free.
• Prove whether or not F wx w is a substring
  of x over a,b is context-free.