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TuringMachine

TuringMachine[rule,init,t]
generates a list representing the evolution of the Turing machine with the specified rule from initial condition init for t steps.

TuringMachine[rule,init]
gives the result of evolving init for one step.

TuringMachine[rule]
is an operator form of TuringMachine that corresponds to one step of evolution.

DetailsDetails

  • For a 1D Turing machine, each step in the evolution generated by TuringMachine is given in the form {{s,x,dx},{a1,a2,…}}, where the head is in state s, the cells on the tape have values ai, the head is at position x relative to the ai, and has moved dx relative to its starting position.
  • If dx is omitted in the initial condition for a Turing machine, it is taken to be 0.
  • For a d-dimensional Turing machine, the tape is specified as a d-dimensional array, and the position x and relative position dx are length-d lists.
  • The rule for a Turing machine can be given as a list of replacements of the form {si,ai}->{spi,api,offi}, with elements as follows:
  • sistate of the head
    aivalue of cell under the head
    spinew state of the head
    apinew value of cell under the head
    offioffset by which the head moves
  • The states and cell values can be integers, patterns, or any other expressions. Individual cell values cannot be lists.
  • In one dimension, each offset offi is a single integer; in higher dimensions a list of integers.
  • When the states and cell values are taken to be integers in the range 1 to s and 0 to k-1 respectively, the following alternative forms can be given for rule:
  • n2‐state, 2‐color machine with number n
    {n,s}s‐state, 2‐color machine with number n
    {n,s,k}s‐state, k‐color machine with number n
    {n,s,k,r}allow offi in the range -r to +r (excluding 0)
    {n,s,k,{r1,r2,…,rd}}d‐dimensional machine with +/-r_(1), +/-r_(2), … offsets
    {n,s,k,{{off1},{off2},…}}machine allowing the specified explicit offsets
    rulemachine with explicit rule given
  • The number of possible Turing machine rules is as follows:
  • 2-state 2-color machines4096
    s-state k-color machines(2 s k)^(s k)
    s-state k-color range-r machines(2 r s k)^(s k)
    2D s-state k-color machines(4 s k)^(s k)
  • If the machine has no rule for the configuration it is in, its configuration will not be changed.
  • Typical forms for the initial conditions init are as follows:
  • {s,{{},0}}head in state s, on a 1D tape filled with 0s
    {s,{{a1,a2,…},0}}bounded region of values ai on an infinite tape
    {{s,x},{{a1,a2,…},0}}bounded region with the head initially at position x
    {{s,…},{{a1,…},{b1,…}}}repetitive background of value bi
    {{s,…},{a1,a2,…}}finite tape, assumed cyclic
  • TuringMachine[rule,init,t] generates an evolution list of length t+1.
  • TuringMachine[rule][init] is equivalent to TuringMachine[rule,init].

ExamplesExamplesopen allclose all

Basic Examples  (4)Basic Examples  (4)

2-state 2-color machine 2506 starting with a tape consisting of four 0s:

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2-state 2-color machine 2506 with an infinite blank tape:

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Plot the successive configurations of the tape:

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Generate a rule icon for a Turing machine:

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Plot the evolution, including head position:

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A machine given by a set of transition rules:

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Scope  (8)Scope  (8)

Applications  (12)Applications  (12)

Properties & Relations  (1)Properties & Relations  (1)

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Introduced in 2007
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