Variants of Turing Machines

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Variants of Turing Machines

• Lecture 26
• Section 3.2
• Mon, Oct 22, 2007

2
Increasing the Power of a Turing Machine

• It is hard to believe that something as simple as
  a Turing machine could be powerful enough for
  complicated problems.

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Increasing the Power of a Turing Machine

• We can imagine a number of improvements.
• Multiple tapes
• Two-way infinite tape
• Two-dimensional tape
• Addressable memory
• Nondeterminism
• etc.

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Multiple Tapes

• Would a Turing machine with k tapes, k gt 1, be
  more powerful than a standard Turing machine?
• Each tape could be processed independently of the
  others.

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Multiple Tapes

• In other words, each transition would read each
  tape, write to each tape, and move left or right
  independently on each tape.

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Multiple Tapes

• Theorem Any language that is accepted by a
  multitape Turing machine is also accepted by a
  standard Turing machine.

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Multiple Tapes

• Sketch of the proof
• On a single tape, we could write the contents of
  all k tapes.
• If tape i contains xi1xi2xi3, for each i, then
  write
• x11x21xk1x12x22xk2...
• on the single tape.

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Multiple Tapes

• To show the current location on each tape, put a
  special mark on one of that tapes symbols
• x11x21xk1x12x22xk2...
• Now the Turing machine scans the tape, locating
  the current symbol on each tape.

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Multiple Tapes

• It then makes the appropriate transition.
• Write a symbol over each of the current symbols.
• Move the location of the current symbol one space
  left or right for each tape.
• Of course, the devil is in the details.

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Two-way Infinite Tape

• We can use a two-tape machine to simiulate the
  two-way infinite tape.
• The right half of the two-way tape is stored on
  tape 1.
• The left half is stored on tape 2.
• Transitions are modified to handle the transition
  from tape 1 to tape 2.

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Two-way Infinite Tape

• Theorem Any language accepted by a two-way
  infinite tape is also accepted by a standard
  Turing machine.

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Other Variants

• Metatheorem Any language accepted by a Turing
  machine with any variant that anyone has ever
  thought of is also accepted by a standard Turing
  machine.

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Nondeterminism

• A nondeterministic Turing machine is defined like
  a standard Turing machine except for the
  transition function.
• ? Q? ? ? ? ?(Q ? ? ? L, R)
• where Q? Q qacc, qrej.

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Nondeterminism

• That is, ?(q, a) may result in any of a number of
  actions.
• If any sequence of transitions leads to the
  accept state, then the input is accepted.
• If all sequences of transitions lead to the
  reject state or to looping, then the input is not
  accepted.

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Nondeterminism

• Theorem Any language accepted by a
  nondeterministic Turing machine is also accepted
  by a standard Turing machine.

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Nondeterminism

• Proof
• We may use a three-tape machine to simulate a
  nondeterministic Turing machine.
• Tape 1 preserves a copy of the original input.
• Tape 2 contains a working copy of the input.
• Tape 3 keeps track of the current state in the
  nondeterministic machine.

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Nondeterminism

• Start with the input w on Tape 1 and with Tapes 2
  and 3 empty.
• Copy w from Tape 1 to Tape 2.
• For Tape 3, imagine the transitions starting from
  the start state as forming a tree.
• Each state has child states.

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Nondeterminism

• Let b be the largest number of children of any
  node.
• Number the children of each state using the
  numbers 1, 2, , b (as many as needed).

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Nondeterminism

• Now each finite string of numbers from 1, 2, ,
  b represents a particular path through the
  nondeterministic Turing machine, or else
  represents no path at all.
• Beginning by writing the empty string on Tape 3,
  representing no moves at all.

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Nondeterminism

• If that leads to acceptance, then quit.
• If not, then replace ? with its lexicographical
  successor.
• For that string, follow the sequence of
  transitions that it describes.

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Nondeterminism

• If that sequence leads to acceptance, then quit
  and accept.
• If not, then continue in the same manner.

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Nondeterminism

• If w is accepted by the nondeterministic Turing
  machine, then some sequence of transitions leads
  to the accept state.
• Eventually that sequence will be written on Tape
  3 and tried.

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Nondeterminism

• On the other hand, if no sequence of transitions
  leads to the accept state, then the deterministic
  Turing machine will loop.