Not All Functions Are Turing Computable: Universal Turing Machines and the Halting Problem

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Not All Functions Are Turing Computable
Universal Turing Machines and the Halting Problem
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Functions

• Function each input determines a unique output
• Domain of a function set of input values
• Partial function of the natural numbers
• domain is a strict subset of N
• Total function of N domain N

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Turing Computable Functions

• A function f is Turing computable if and only if
  it can be implemented by a Turing machine
  behaving as follows
• If f(x_1,,x_n) is not defined, then the machine
  will either never halt or will halt in a
  non-standard final position.
• If f(x_1,,x_n) is defined, then the machine
  will eventually halt in standard final position.

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Partial Recursive Functions

• A function is partial recursive if it can be
  obtained from a finite number of basic functions
  via a finite number of applications of
  composition, primitive recursion, and no more
  than one application of minimization.
• Turing computable partial recursive
• Churchs Thesis
• computable Turing computable

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Standard Description ltMgt of a Turing Machine M

• A Turing machine is defined by a finite sequence
  of quadruples
• Each quadruple lists, in order, the current
  state, the current symbol, the action to be
  taken, and the next state.
• The machine uses a finite number of states,
  symbols, and actions each of these can be
  encoded as finite strings of s.
• Single blanks separate encoding strings.

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Example ltSuccessorgt
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Decidable Questions

• Consider questions Q which can be written in some
  finite alphabet.
• A class of such questions is decidable if there
  is a Turing machine M which, when applied to any
  question in Q in the class, stops on if the
  answer to Q is YES and stops on _ if the answer
  to Q is NO.

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The Halting Problem

• Consider the class of questions
• Q_M Does the Turing machine M applied to ltMgt
  eventually stop on _?
• If Turing machine S answers such questions
• If S applied to ltSgt stops on , then S applied
  to ltSgt stops on _ is true!
• Is S applied to ltSgt stops on _, then S applied
  to ltSgt stops on is true!

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Universal Turing Machines

• Is the problem that no one Turing machine can
  simulate every Turing machine? NO!
• A universal Turing machine U exists
• Given machine M, encoded as ltMgt
• and input tape P for M, encoded as ltPgt
• U replaces ltPgt with the encoding of the output of
  M applied to P, if M is defined for P, and either
  halts in a nonstandard position or fails to halt,
  otherwise.

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Are Functions That Are Not Turing Computable
Unusual?

• The set of all possible Turing machines is
  denumerable. (Why?)
• The set of all functions from the natural numbers
  to the natural numbers is not denumerable. (Why?
  Diagonal argument.)
• Thus MOST functions are NOT Turing computable!