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TuringMachine—Wolfram Language Documentation

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• See Also
  • CellularAutomaton
  • RulePlot
  • SubstitutionSystem
  • BooleanFunction
  • AnglePath
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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.

• 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 a_i, the head is at position x relative to the a_i, 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 {s_i,a_i}->{sp_i,ap_i,off_i}, with elements as follows:
• s_i state of the head a_i value of cell under the head sp_i new state of the head ap_i new value of cell under the head off_i offset 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 off_i 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 and 0 to respectively, the following alternative forms can be given for rule:
• n 2‐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 off_i in the range to (excluding 0) {n,s,k,{r₁,r₂,…,r_d}} ‐dimensional machine with , , … offsets {n,s,k,{{off₁},{off₂},…}} machine allowing the specified explicit offsets {rule,s,k} machine with explicit rule given rule machine with explicit rule given (and s, k inferred)
• The number of possible Turing machine rules is as follows:
• 2-state, 2-color machines 4096 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 (8 s k)^(s k)
• If the machine has no rule for the configuration it is in, its configuration will not be changed.
• If a rule has multiple specifications for a given configuration, TuringMachine will use the first one listed.
• The form {rule,s,k} can be used to specify multiway, non-deterministic and other Turing machines in which there is not necessarily exactly one case in the rule for a given configuration.
• 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 a_i 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 b_i {{s,…},{a₁,a₂,…}} finite tape, assumed cyclic
• TuringMachine[rule,init,t] generates an evolution list of length .
• TuringMachine[rule][init] is equivalent to TuringMachine[rule,init].

2-state, 2-color machine 2506 with an initial tape of four 0s, evolving for 3 steps:

2-state, 2-color machine 2506 with an infinite tape of 0s, evolving for 4 steps:

Plot the successive configurations of the tape:

Show the rule icon for a Turing machine:

Plot the evolution, including the state of the head:

Show the rule icon for a Turing machine specified by explicit transitions:

Plot the evolution, including the state of the head:

A Turing machine specified by pattern-based transition rules:

2-state, 2-color machine 2506:

Plot the evolution:

Plot the evolution, including the state of the head:

3-state, 2-color machine 2139050:

Generate a rule icon:

2-state, 2-color machine 16220, with range 2:

3-state, 2-color machine 2139050, with jump offsets and 2:

Give explicit transition rules:

Explicitly specify values of the number of states s and the number of colors k for the same transition rules:

2-state, 2-color machine 2506 with head initially in state 1:

Evolution:

2-state, 2-color machine 2506 with head initially in state 2:

Evolution:

Place the head at position 3 on the initial tape:

Place the head at position 5 on the initial tape:

Start with a finite tape of four 0s, assumed cyclic:

The left neighbor of the leftmost cell is the rightmost cell, and vice versa:

Start with an infinite tape of 0s:

Start with a tape of 1 on an infinite background of 0s:

Start with a tape consisting of the block 211 on a background of 0s:

Start with the block 211 on a background of repeated 02 blocks:

2D 2-state, 2-color Turing machine 977401:

2D Turing machine specified by explicit transitions:

Evolution of Wolfram's simplest universal Turing machine from an infinite tape of 0s:

Alternative form using explicit rules:

Show the evolutions of a sequence of 2-state, 2-color machines:

Trajectory of the machine head from successive initial conditions:

Path traced by the head of a 2D machine:

Averaging tape of a 2D machine over many steps:

Successive states sequences from successive initial conditions:

Sequence of left or right movements for successive initial conditions:

Halting on a one-sided tape:

Computed function on a one-sided tape:

Show only steps on which the head reaches a new cell:

Show only steps on which the head returns to its initial location:

Causal network from random initial tape:

For rules of the form {n,s,k,…}, head states and cell values can be integers in the range 1 to s and 0 to k-1, respectively:

For rules of the form {n,s,k,…}, if the head reaches a cell whose value is not in the range 0 to k-1, the evolution of the machine halts:

Another Turing machine whose evolution halts:

Use an explicit set of rules to define a halting state:

Plot the evolution:

Generate a Turing machine evolution:

"Inject" the state information into a representation of the tape:

Show the position of the head as a red square:

Use RulePlot to generate a complete evolution picture:

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