The Turing Machine

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The Turing Machine
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What Is Turing Machine?

• It is a conceptual machine put forward by Allan
  Turing in 1938.
• It is composed of
• A unlimitedly long tape with 0/1
• A device with a read/write head which moves one
  step at a time
• A process box with pre-stored rules.

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What A Turing Machine Can Do?

• A Turing Machine can carry out a particular task,
  such as adding one to a number, multiply two
  numbers, find the highest common factor between
  two numbers, pick the larger from two numbers.

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Church-Turings Thesis

• Any problem that has an algorithmic solution can
  be solved on a Turing Machine.

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Algorithm

• An algorithm is a step-by-step procedure, by
  following which we can get a problem solved.
• The results of an algorithm is independent of
  executor and times of executions. That is, no
  matter who is execute the algorithm, and how many
  times it is executed, the results will be, if
  input data are same, either identical or
  undistinguishable.

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Any Algorithmic Solution Can Be Realized on a
Computer

• If a problem can be solved by an algorithm, then
  a computer program can be generated to solve the
  problem.

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Therefore

• Any task that is accomplished on a computer can
  be accomplished on a Turing Machine.

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Universal Turing Machine

• If the pre-stored rules in the process box are
  put on the tape, then individual Truing Machines
  with particular tasks can be combined into one
  Turing Machine Universal Turing Machine.
• It has been proved that those pre-stored rules in
  process box can be coded onto the tape.

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Therefore

• A computer as we have now can be conceptually
  reduced to an equivalent universal Turing Machine.

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Implications

• We can predict some features of computers in the
  future, even we do not know what specifically
  computers can do and how complicated it is.
• For example, what the computer does in the future
  must be duplicable.

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Turing Machine May Set a Limit for How Far
Computer Can Go

• Since we can foresee some features of future
  computers through Turing Machine, we may figure
  out what a computer is unable to do.
• For example, if something is not duplicable, then
  it cannot be programmed.

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Theorem of Incompleteness

• The enthusiasm and fanaticism of developing a
  computing machine able to develop all theorems in
  a math system came to a sudden stop after Gödel
  published his Incompleteness Theorem in 1936
• Gödel showed its not possible to exhaust all
  theorems of a math system if staying in the
  system.

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Be Humble

• Gödels Theorem of Incompleteness served as a
  sharp warning against the blind arrogance of
  humans.
• Can we humans understand ourselves so well that
  we can make machines more intelligent than us?