CS605%20

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CS605 The Mathematics and Theory of Computer
Science

• Turing Machines

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Equivalence

• In order that we show that two models are
  equivalent we simply need to show that we can
  simulate one by the other.

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Multitape TMs

• A multitape TM is like an ordinary TM with
  several tapes each with its own head for reading
  and writing.
• Initially the input appears on one tape with all
  the others blank.
• The transition function is changed to allow for
  reading, writing and moving the heads on some or
  all of the tapes simultaneously.

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Multitape TMs

• Formally this transition is
• d Q Gk ? Q G k L,R,Sk where k is the
  number of tapes.
• d(qi, a1, , ak) (qj, b1, , bk, L, R, , L)
  means that, if the machine is in state qi and
  heads 1 through k are reading symbols a1 through
  ak, the machine goes to state qj, writes symbols
  b1 through bk, and directs each head to move
  Left, Right or Stay Put.

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Multitape TMs

• Multitape TMs are equivalent (recognise the same
  language) in power to ordinary TMs.
• Every multitape TM has an equivalent single-tape
  TM.
• Proof?

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Turing- Recognisable

• A language is Turing-Recognisable if and only if
  some multitape TM recognises it.
• We stated previously that A language is Turing
  Recognisable if some Turing Machine recognises
  it.
• This new definition does not go against this as a
  multitape TM can be simulated by an ordinary TM.

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Nondeterministic Turing Machines

• A nondeterministic TM is a generalisation of the
  standard TM for which every configuration may
  yield none, or one or more than one next
  configurations.
• In contrast to a deterministic TM, for which a
  computation is a sequence of configurations, a
  computation of a nondeterministic TM is a tree of
  configurations that can be reached from the start
  configuration.

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Nondeterministic Turing Machines

• The transition function for a nondeterministic TM
  has the form
• d Q G ? P(Q G L,R)
• where P refers to the power set.
• The computation of a nondeterministic TM is a
  tree whose branches correspond to different
  possibilities for the machine.
• If some branch of the computation leads to an
  accept state the machine accepts the input.

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Nondeterministic Turing Machines

• An input to a NTM is said to be accepted if there
  exists at least one node of the computation tree
  which is an accept-configuration.
• The path from the root to the accept-configuration
  is said to be non-deterministically selected.
• A NTM is called a decider if all branches halt on
  all inputs.

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Nondeterministic Turing Machines

• Every nondeterministic TM has an equivalent
  deterministic TM.

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Do TM always halt?

• When can a machine not halt? There are a number
  of possibilities
• The machine stops, but not in the halt state. We
  call this a crash.
• No transition is specified for the current state
  and tape symbol this is also a crash.
• The machine enters an infinite loop, and so never
  gets out of this to reach a halt state.

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Implications of Non-Halting

• The first sounds ominous, but it is ok. When
  machine crashes we know input was invalid.
• The second is to be expected when we design the
  machines we dont have to specify all the
  transitions the ones we dont specify should
  naturally lead to the error state, i.e. crash.
• The third is far more serious If the machine
  enters an infinite loop, the user is left waiting
  for an answer. As time goes on they are wondering
  Should I turn the machine odd? or If I leave
  it running maybe it will stop in the next few
  minutes. This issue is non-trivial

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Implications of Non-Halting

• This problem is referred to as the Halting
  Problem and we will look at it soon