MC 306 Theory of Computation Tuesday, 111103

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MC 306 Theory of ComputationTuesday, 11/11/03

• Last time
• Closure properties of context-free languages
• Review of Pumping Lemma for Regular Languages
• Todays Class
• One more closure property
• Pumping Lemma for Context-free Languages
• Intro to next topic Turing Machines

• Reading 4.1
• Exercises
• 4.3b, 4.4
• No new hand-in problems yet

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Our final machine

• DFA/NFA Model of computer that has fixed memory.
• Simple calculators, digital watches
• PDA Model of computer with a single stack. More
  powerful than DFA/NFA
• Some scientific calculators
• Turing Machine Model of fully general computer.

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Turing Machine Basics

• Turing Machine has parts in common with DFAs,
  NFAs, PDAs
• States and transitions
• Start state
• Turing machine differs from DFAs, NFAs, PDAs
• Halting states (not accepting/rejecting)
• Much more flexible input tape
• Is also an output tape can write on it
• No need for final states can write results on
  tape!
• Is bidirectional can move left or right (or
  stay still)
• Tape is infinite never runs out of room for
  writing

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TM Computation

• We run a TM on a string w by writing it on the
  tape, and then running the machine from its start
  state
• A transition
• (1) reads the current symbol,
• (2) Either (a) writes a symbol in that spot, or
  (b) moves left or right on tape
• (3) Changes from the current state to a new state
• We can accept/reject by writing 1 for yes, or
  0 for no, and then halting (enter a halting
  state)
• Can write lots of other stuff too, so can have
  much more complex output!

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Example

• Note Special symbols used by every Turing
  machine
• ? marks the left end of the tape
• ? marks a blank square on the tape
• Let ? a, b, c, and L ?c? ? ? a, b
• Is L context-free?
• Design a Turing machine to accept L
• Design a tape-cleaning machine over a,b. This
  machine simply erases all the non-blank symbols.