CS 3240: Languages and Computation

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CS 3240 Languages and Computation

• Multitape Turing Machines

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Review Formal definition of TM

• Definition A Turing machine is7-tuple
  (Q,?,?,?,q0,qaccept,qreject), where Q, ?, and ?
  are finite sets and
• Q is the set of states,
• ? is input alphabet where blank symbol ??
• ? is tape alphabet, where ????,
• ? Q???Q???L,R is transition function,
• q0?Q is start state,
• qaccept?Q is accept state, and
• qreject?Q is reject state, where qreject?qaccept

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Example

• Write a TM that accepts all strings of the form
  101001000100001
• Start with a 1
• End with a 1
• Progressively more 0s between consecutive 1s
• Possible Solution
• Check first symbol is a 1
• If not, then reject
• Move right and check if second symbol is a 0
• If not, then reject
• If so, replace with X and begin recursion

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Recursion (high level)

• Go back and forth on either side of each 1
• Replace 0s on right side of 1 with an X
• Replace Xs on left side of 1 with a Y
• After all Xs on left side of 1 are replaced with
  Ys, there should be exactly one on the right
  side that has not been Xed
• If not, then reject
• If so, repeat process (recursion step)
• If you begin to look for the next group of 0s
  and reach a then accept

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Example

• Ex1x2x3xn xi? 0,1 and xi?xj for i ?
  j
• Basic idea Compare each pair of (xi , xj) by
  zig-zagging. Use special marks to replace to
  indicate which pair of (xi , xj) is being
  compared
• Must take care special cases such as n0, n1,
  missing , etc.

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Question

• How to detect whether we have reached left-most
  cell?
• Alternative 1 If leftmost cell marked by special
  symbol, then easy
• Alternative 2 Use special property that TM
  stays put at left-most cell by replacing cell
  with special symbol and then recover

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Variants of Turing machines

• Robustness of model
• Varying the model does not change the power
• Simple variant of TM model
• Add stay put direction
• Other variants
• More tapes
• Nondeterministic

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Multitape Turing machines

• Same as standard Turing machine, but have several
  tapes
• Initially, input is on tape 1
• TM definition changes only in definition of d
• d Q ?k ? Q ?k L,Rk

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Equivalence of machines

• Theorem Every multitape Turing machine has an
  equivalent single tape Turing machine
• Proof method By construction.

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Simulating k-tape behavior

• Single tape start string is
• w...
• Each move proceeds as follows
• Start at leftmost slot
• Scan right to (k1)st to find symbol at each
  virtual tape head
• Make second pass making updates indicated by
  k-tape transition function
• When a virtual head moves onto a , shift string
  to right

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Corollary

• Corollary A language is Turing-recognizable if
  and only if some multitape Turing machine
  recognizes it.