Introduction to the Theory of Computation

1
Introduction to the Theory of Computation

• John Paxton
• Montana State University
• Summer 2003

2
Humor

• Bush, Einstein and Picasso at the Pearly Gates
• Einstein dies and goes to heaven. At the Pearly
  Gates, Saint Peter tells him, "You look like
  Einstein, but you have NO idea the lengths that
  some people will go to sneak into Heaven. Canyou
  prove who you really are?"
• Einstein ponders for a few seconds and asks,
  "Could I have a blackboard and some chalk?"
• Saint Peter snaps his fingers and a blackboard
  and chalk instantly appear. Einstein proceeds to
  describe with arcane mathematics and symbols his
  theory of relativity.
• Saint Peter is suitably impressed. "You really
  ARE Einstein!" he says. "Welcome to heaven!"

3
Humor

• The next to arrive is Picasso. Once again, Saint
  Peter asks for credentials.
• Picasso asks, "Mind if I use that blackboard and
  chalk?"
• Saint Peter says, "Go ahead."
• Picasso erases Einstein's equations and sketches
  a truly stunning mural with just a few strokes of
  chalk.
• Saint Peter claps. "Surely you are the great
  artist you claim to be!" he says. "Come on in!"

4
Humor

• Then Saint Peter looks up and sees George W.
  Bush. Saint Peter scratches his head and says,
  "Einstein and Picasso both managed to prove their
  identity. How can you proveyours?"
• George W. looks bewildered and says, "Who are
  Einstein and Picasso?"
• Saint Peter sighs and says, "Come on in, George."

5
Turing Machine

• A Turing maching is a 7-tuple (Q, S, G, d, q0,
  qaccept, qreject) where
• Q is the set of states
• S is the input alphabet not containing the
  special blank symbol
• G is the tape alphabet that contains both the
  blank symbol and S
• d Q x G -gt Q x G x L, R

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Turing Decidable

• A language is Turing decidable (or simply
  decidable) if some Turing machine decides it.
• A Turing machine can decide a language if it
  halts on all inputs.

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Equivalent Turing Machine Variants

• There may be multiple tapes.
• A Turing machine may be nondeterministic.

8
Church-Turing Thesis

• Our intuitive notion of an algorithm is
  equivalent to the algorithms that can be
  performed on a Turing Machine.
• In other words, a Turing Machine is equivalent to
  the most powerful model of computation!

9
The Halting Problem

• Does a particular Turing Machine accept the input
  w?
• This problem is undecidable and can be proven
  using a diagonalization proof.

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Classes of Languages
all problems
decidable
context-free
regular