Turing machine

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The Turing machine is an abstract model of computer execution and storage introduced in 1936 by Alan Turing in On Computable Numbers, with an Application to the Entscheidungsproblem to give a mathematically precise definition of algorithm or 'mechanical procedure'. As such it is still widely used in theoretical computer science, especially in complexity theory and the theory of computation. The thesis that states that Turing machines indeed capture the informal notion of effective or mechanical method in logic and mathematics is known as Turing's thesis.

Definition

A Turing machine consists of:

  1. A head that can read and write symbols on a tape and move left and right. The tape is assumed to be divided into cells that contain a symbol from some finite alphabet consisting of an blank symbol (here written as '0') and one or more other symbols. The tape is assumed to be arbitrarily extendible, i.e., the Turing machine is always supplied with as much tape as it needs for its computation. Tape that has not been written before is assumed to be filled with the empty symbol.
  2. A state register that stores the state of the Turing machine. The number of different states is always finite and there is one special start state with which the state register is initialized.
  3. An action table that tells the machine what symbol to write, how to move the head ('L' for one step left, and 'R' for one step right) and what its new state will be, given the symbol it has just read on the tape and the state it is currently in. If there is no entry in the table for the current combination of symbol and state then the machine will halt.

Note that every part of the machine is finite, but it is the potentially unlimited amount of tape that gives it an unbounded amount of storage space.

Example:

The following Turing machine has an alphabet {'0', '1'}, with 0 being the blank symbol. It expects a series of 1's on the tape, with the head initially on the leftmost 1, and doubles the 1's with a 0 in between, i.e., "111" becomes "1110111". The set of states is {s1, s2, s3, s4, s5} and the start state is s1. The action table is as follows.

 Old Read   Wr.      New     Old Read   Wr.      New
 St. Sym.   Sym. Mv. St.     St. Sym.   Sym. Mv. St.
 - - - - - - - - - - - -     - - - - - - - - - - - -
  s1  1  ->  0    R   s2      s4  1  ->  1    L   s4
  s2  1  ->  1    R   s2      s4  0  ->  0    L   s5
  s2  0  ->  0    R   s3      s5  1  ->  1    L   s5
  s3  0  ->  1    L   s4      s5  0  ->  1    R   s1
  s3  1  ->  1    R   s3

A computation of this Turing machine might for example be: (the position of the head is indicated by displaying the cell in bold face)

 Step State Tape   Step State Tape
 - - - - - - - -   - - - - - - - - -
    1  s1   11        9  s2   1001 
    2  s2   01       10  s3   1001
    3  s2   010      11  s3   10010
    4  s3   0100     12  s4   10011
    5  s4   0101     13  s4   10011
    6  s5   0101     14  s5   10011
    7  s5   0101     15  s1   11011
    8  s1   1101       -- halt --

The behavior of this machine can be described as a loop: it starts out in s1, replaces the first 1 with a 0, then uses s2 to move to the right, skipping over 1's and the first 0 encountered. S3 then skips over the next sequence of 1's (initially there are none) and replaces the first 0 it finds with a 1. S4 moves back to the left, skipping over 1's until it finds a 0 and switches to s5. s5 then moves to the left, skipping over 1's until it finds the 0 that was originally written by s1. It replaces that 0 with a 1, moves one position to the right and enters s1 again for another round of the loop. This continues until s1 finds a 0 (this is the 0 right in the middle between the two strings of 1's) at which time the machine halts.

The Universal Turing Machine

Every Turing machine computes a certain fixed function over the strings over its alphabet. In that sense it behaves like a computer with a fixed program. However, as Alan Turing already described, we can encode the action table of every Turing machine in a string. Thus we might try to construct a Turing machine that expects on its tape a string describing an action table followed by a string describing the input tape, and then computes the tape that the encoded Turing machine would have computed. As Turing showed, such a Turing machine is indeed possible and since it is able to simulate any other Turing machine it is called a universal Turing machine.

With this encoding of action tables as strings, it becomes in principle possible for Turing machines to answer questions about the behavior of other Turing machines. Most of these questions however are undecidable, see for instance the Halting problem, which was already shown to be undecidable in Turing's original paper, and Rice's theorem.

References:

  • Turing, A., On Computable Numbers, With an Application to the Entscheidungsproblem, Proceedings of the London Mathematical Soceity, Series 2, Volume 42, 1936; reprinted in M. David (ed.), The Undecidable, Hewlett, NY: Raven Press, 1965; (online version)
  • Boolos, G. and Jeffrey, R., Computability and Logic, 2nd ed., Cambridge: Cambridge University Press, 1980.

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