Halting Problem

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Halting Problem

• Introduction to Computing Science and Programming
  I

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Alan Turing

• Alan Turing 1912-1954
• Father of modern computing science
• 1936
• Turing Machine
• Church-Turing thesis
• Halting Problem
• 1950 Turing Test

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Turing Machine

• A Turing machine is a very simple theoretical
  computer with a couple basic elements.
• An infinitely long tape broken up into cells that
  can each store a single symbol from a finite set
  of symbols
• A head that can read or write one symbol at a
  time from the tape
• A state diagram that tells the head what to do,
  move left/right, print a symbol

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Church-Turing Thesis

• This thesis proves has to do with comparing what
  problems can be solved by different types of
  computer.
• It proves that a Turing Machine could
  theoretically be created that can do anything any
  digital computer can do.

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Halting Problem

• An important question of computing science is
  Are there problems that cannot be solved?
• There are, and probably the most famous of these
  is the halting problem described by Turing.
• He was thinking in terms of Turing machines
  (there were no computers), but it is easy to
  extend the idea.

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Halting Problem

• Halting problem
• Can we write a program that will look at any
  computer program and its input and decide if the
  program will halt (not run infinitely)?
• A practical solution might be to run the program
  and if it halts you have your answer. If after a
  given amount of time it doesnt halt, guess that
  it wont halt. However, you wouldnt know if the
  program would eventually halt.

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Halting Problem

• As it turns out the problem is undecidable.
• For a problem to be undecidable you just have to
  prove that there is one case it cant produce an
  answer for.
• The case that Turing came up with that can never
  be solved involves giving a program itself as
  input.

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Halting Problem

• Lets assume were given a function
  halts(prog,input) that is supposed to tell us if
  the program prog will halt if given the input.
  The function will return True if it halts, False
  otherwise.

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Halting Problem

• Using the halts function the following program
  can be written that reads in a program, given as
  a filename) from the user. It then calls the
  halts function to decide if the program will halt
  given itself as input.
• prog raw_input("Program file name ")
• if halts(prog, prog)
• while True
• print "looping
• else
• print "done"

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Halting Problem

• It may not make sense to give a program itself as
  input, but that isnt important here.
• What could happen?
• If halts(prog,prog) returns True, that means the
  program will halt when given itself as input.
  However, in this case the program would go into
  an infinite loop. Therefore the program doesnt
  halt.
• If halts(prog,prog) returns False, that means
  that it wouldnt halt, but in that case the
  program does halt.
• This contradiction is unavoidable thus proving
  that the halting problem is undecidable. No
  function can always correctly decide if a program
  will halt.

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Turing Test

• The Turing test was formulated as a way to answer
  the question, Can machines think?
• The basic idea is that there is a sort of judge
  who converses with a computer and a person.
  However, this is indirect. The judge writes down
  questions and the computer and the person, who
  are hidden from the judge, return typewritten
  answers. If the judge couldnt consistently
  identify which set of answers were from the
  person, than the computer could think.
• Turing predicted that computers could pass this
  test with 30 of humans as the judge by the year
  2000.