Wolfram Language Paclet Repository

Community-contributed installable additions to the Wolfram Language

WolframInstitute/ TuringMachine

Tools for exploring and analyzing Turing machines

Contributed by: Nik Murzin and Willem Nielsen

The paclet identifies a Turing machine by its enumeration number and its {s, k} state/color counts. OneSidedTuringMachineFunction runs a one-sided machine on an integer input and returns its halting value, step count, or tape width; OneSidedTuringMachinePlot draws the space-time evolution; and OneSidedTuringMachineFind searches for machines reproducing given outputs. The MultiwayTuringMachineFunction family explores nondeterministic (multiway) machines, while TuringMachineRuleCount and the TuringMachineOutput primitives tabulate behavior across whole rule spaces.

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall["WolframInstitute/TuringMachine"]


To load the code after installation, evaluate this code:
Needs["WolframInstitute`TuringMachine`"]

Details

A Turing machine is specified by its Wolfram enumeration number together with its state and color counts, written {number, s, k} — for example {600720, 3, 2} is rule 600720 with 3 states and 2 colors.
The one-sided functions read a non-negative integer input off the tape, run the machine, and report the halting value, the number of steps taken, or the maximum tape width reached.
A machine that does not halt within the given step bound reports Infinity / Undefined for the parts it could not resolve.
The multiway functions explore nondeterministic machines, where a {state, color} pair may carry several transitions, and return all reachable configurations.
A Rust backend (loaded from the paclet's Binaries) accelerates the bulk enumeration and search functions over whole rule spaces.

Paclet Guide

Examples

Basic Examples (5) 

Run the one-sided Turing machine defined by the s=3, k=2 rule 600720 on input 1 for at most 32 steps:

In[1]:=
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Out[1]=

Return the number of steps the machine took to terminate:

In[2]:=
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Out[2]=

Return the maximum width the head spanned on the tape:

In[3]:=
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Out[3]=

Return the steps, value, and width together:

In[4]:=
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Out[4]=

Plot the space-time evolution of the machine on input 1:

In[5]:=
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Out[5]=

Scope (2) 

Evaluate a whole family of machines at once — every s=1, k=2 one-sided machine on inputs 1 through 8, each for up to 32 steps:

In[6]:=
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Out[6]=

Count how many distinct Turing machine rules exist for a given number of states and colors:

In[7]:=
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Out[7]=

Options (2) 

Split a long evolution into several columns with the "Columns" option:

In[8]:=
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Out[8]=

Suppress the output label to show the bare space-time pattern:

In[9]:=
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Out[9]=

Applications (2) 

Search the 2-state, 2-color rule space for the machines that reproduce a target behavior, given as {input, steps, value} triples:

In[10]:=
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Out[10]=

Compare how a single machine transforms several different inputs:

In[11]:=
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Out[11]=

Properties and Relations (2) 

The All property is exactly the "Steps", "Value", and "Width" properties assembled into a triple:

In[12]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/f8d94821-e14d-4054-9bdf-0ed4cfc69647"]
Out[12]=

The number of rows tabulated by the rule-space functions equals TuringMachineRuleCount:

In[13]:=
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Out[13]=

Possible Issues (2) 

A machine that does not halt within maxSteps reports Undefined for its value:

In[14]:=
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Out[14]=

With the All property the unresolved parts come back as Infinity:

In[15]:=
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Out[15]=

Neat Examples (1) 

The space-time patterns of three different 2-state, 2-color machines on the same input 1, halting at values 3, 2, and 7:

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Publisher

Wolfram Institute

Disclosures

Compatibility

Wolfram Language Version 14.3

Version History

  • 1.1.1 – 02 June 2026
  • 1.1.0 – 01 June 2026
  • 1.0.9 – 01 June 2026
  • 1.0.8 – 01 June 2026
  • 1.0.5 – 24 January 2026
  • 1.0.4 – 23 January 2026
  • 1.0.3 – 23 January 2026
  • 1.0.2 – 21 January 2026
  • 1.0.1 – 21 January 2026
  • 1.0 – 21 January 2026

License Information

MIT License

Paclet Source

Source Metadata

  • Citation:
    • Stephen Wolfram, A New Kind of Science (Wolfram Media, 2002), Notes for Chapter 12, Section 8 (One-Sided Turing Machines), p. 1143

See Also