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RulePlot
✖
RulePlot

✖
RulePlot[sys]

generates a plot representing the rule for the computational system sys.

✖
RulePlot[sys,init,t]

generates a plot of the evolution of the system sys from initial condition init for t steps.

✖
RulePlot[sys,evol]

generates a plot of the evolution evol assuming it is derived from a system of the form sys.

Details and Options

  • Possible computational systems sys include:
  • CellularAutomaton[spec]cellular automaton
    TuringMachine[spec]Turing machine
    SubstitutionSystem[spec]substitution system
    BooleanFunction[spec]Boolean function
  • TuringMachine[{rule,s,k}] can be used to specify a Turing machine in which rule does not uniquely specify a transformation for each of the s k possible configurations, but that nevertheless will be rendered assuming s head states and k tape colors.
  • RulePlot works with 1D and 2D systems.
  • Options include:
  • Appearance Automaticoverall appearance
    ColorRules Automaticrules for determining colors from values
    Frame Automaticwhether to draw a frame around rule cases
    FrameStyle Automaticthe style to use for a frame
    Mesh Automaticwhether to draw a mesh
    MeshStyle Automaticthe style to use for a mesh
    PlotLegends Nonelegends to include
    PlotMarkers Automaticmarkers for Turing machines, etc.
    PlotTheme $PlotThemeoverall theme for the rule plot
    Spacings Automatichorizontal and vertical spacings
  • Possible appearance elements include:
  • "Contiguous"no space between states
    "Stacked"small space between states
    "Arrow"draw an arrow between states
    "Squares"squares for cells
    "Bricks"bricks for cells
    "Hexagons"hexagons for cells
    "Simplified"simplified rule cases
    "Short"show shortened list of rules
  • Possible plot legend forms include:
  • "Icon"use an icon as the legend
    "Text"use text as the legend
    exprarbitrary plot legend
  • In the form RulePlot[sys,evol], specific rules are typically ignored; only the implicit or explicit specification of number of colors, states, etc. is used.

Examples

open allclose all

Basic Examples  (5)Summary of the most common use cases

Return the rule icon for elementary cellular automaton rule 30:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-bboxcc
Out[1]=1

Plot the evolution of a cellular automaton for 10 steps:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-vd0h1l
Out[1]=1

Include the rule icon:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-gs19dp
Out[2]=2

Plot the evolution of a Turing machine:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-eae549
Out[1]=1

Return its rule icon:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-hvoap
Out[2]=2

Generate the evolution of a Turing machine:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-e7kh9s
Out[1]=1

Use RulePlot to display the history, using information on the rule to determine the form:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-tywlb
Out[2]=2

Show the rule icon for a substitution system:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-s0fom6
Out[1]=1

Scope  (42)Survey of the scope of standard use cases

System Rules  (31)

CellularAutomaton  (11)

Elementary cellular automaton rule 90:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-bwj6ic
Out[1]=1

Cellular automaton with more than two colors:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-nmtdbh
Out[1]=1

Cellular automaton with range greater than 1:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-fdcf7e
Out[1]=1

Cellular automaton with fractional range:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-clmqhk
Out[1]=1

Totalistic 3-color cellular automaton:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-olwbba
Out[1]=1

Outer totalistic 3-color cellular automaton:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-b4vlf4
Out[1]=1

General two-dimensional cellular automaton:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-b7qms1
Out[1]=1

Totalistic two-dimensional 5-neighbor rule:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-qyf924
Out[1]=1

Totalistic two-dimensional 9-neighbor rule:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-1j1m6
Out[1]=1

Outer totalistic two-dimensional rule:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-dmk01r
Out[1]=1

Rules specified using associations:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-i2hj6i
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-io2bpu
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e7pzwte-pb3iik
Out[3]=3

TuringMachine  (8)

Turing machine rule 2506:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-by46wo
Out[1]=1

Turing machine with more than two states:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-ygcin
Out[1]=1

Turing machine with more than two colors:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-c395jq
Out[1]=1

Turing machines specified by rules:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-cq5uzo
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-ulibwn
In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-sskr7s
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e7pzwte-iqqziz
Out[3]=3

Single-state Turing machine heads are represented with a dot:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-6dhczx
Out[1]=1

Increase the number of states to change the representation:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-xnwew7
Out[2]=2

An underspecified Turing machine does not have rules for all s k input configurations:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-yzh218
Out[1]=1

A nondeterministic Turing machine specifies different rules for the same input configuration:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-sys1hd
Out[1]=1

SubstitutionSystem  (8)

String substitution system for two characters:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-dkxtna
Out[1]=1

String substitution system for more than two characters:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-e583km
Out[1]=1

String substitution system for length-2 strings:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-9uape
Out[1]=1

String substitution system for strings of different lengths:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-cv05t6
Out[1]=1

List substitution system for two integer values:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-wmki
Out[1]=1

List substitution system for more than two integer values:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-l3uls8
Out[1]=1

List substitution system with lists of different lengths:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-b655gn
Out[1]=1

Two-dimensional substitution systems:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-c1cyfh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-dx6jko
Out[2]=2

BooleanFunction  (4)

Boolean function in two variables:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-e5mvn6
Out[1]=1

Boolean function in more than two variables:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-n3cvad
Out[1]=1

Boolean function based on a vector of truth values:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-d4m46l
Out[1]=1

Boolean function based on a table of truth rules:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-eqojhi
Out[1]=1

System Evolution  (11)

CellularAutomaton  (5)

Evolution of rule 30 for 10 steps:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-gwwue8
Out[1]=1

Evolution for 50 steps from a single 1 on a background of 0s:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-l9hx4d
Out[2]=2

Evolution of a 3-color cellular automaton:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-cl45b5
Out[1]=1

Evolution from a single 1 on a background of repeated 0110 blocks:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-bk3u7z
Out[2]=2

Evolution of a 3-color totalistic code:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-dmxgq6
Out[1]=1

Evolution of a 3-color outer totalistic code:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-dszfdg
Out[1]=1

Evolution of rules specified by associations:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-b3s57g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-dvg74
Out[2]=2

TuringMachine  (6)

Evolution of a 2-state, 2-color Turing machine, starting with a tape of four 0s:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-bcidn
Out[1]=1

Evolution starting with an infinite tape of 0s:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-d6kr1v
Out[2]=2

Evolution of a 3-state, 2-color Turing machine:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-ip1edc
Out[1]=1

Evolution starting with a tape of 1 on a background of 0s:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-cczn89
Out[2]=2

Evolution of a 2-state, 3-color Turing machine:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-mswnob
Out[1]=1

Evolution with the head initially in state 2:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-bp7oid
Out[2]=2

Evolution of a Turing machine specified by rules, starting from a background of repeated 021 blocks:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-k05ftq
Out[1]=1

A nondeterministic Turing machine can give different evolutions depending on the order of its rules:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-o7hx9w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-vsybfs
Out[2]=2

An underspecified Turing machine does not have rules for all s k input configurations:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-g7rkvj
Out[1]=1

Specifying a complete set of rules eliminates the fixed point:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-v2r1v5
Out[2]=2

Options  (19)Common values & functionality for each option

Appearance  (4)

Use hexagons:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-hr4x59
Out[1]=1

Use bricks:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-koknu6
Out[1]=1

Sometimes you can get a simplified result that shows cells whose value does not matter:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-3avw
Out[1]=1

Shorten the display of the rule set:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-nvh7d1
Out[1]=1

ColorRules  (1)

Specify color rules for explicit values or patterns:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-dpqhbi
Out[1]=1

Frame  (1)

Do not use a frame:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-c50uv9
Out[1]=1

FrameStyle  (2)

Make the frame blue:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-djvqp5
Out[1]=1

Give a composite directive for the frame:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-kwh37x
Out[1]=1

Mesh  (1)

By default, a mesh is drawn around individual cells:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-f0fz47
Out[1]=1

Do not use a mesh:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-bjziw
Out[2]=2

MeshStyle  (2)

Make the mesh blue:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-fc772q
Out[1]=1

Give a composite directive for the mesh:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-f7hd0u
Out[1]=1

PlotLegends  (3)

Give a plot legend describing the rule in text:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-cnlka6
Out[1]=1

Give a plot legend describing the rule as an icon:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-cbcsy7
Out[1]=1

Give an arbitrary plot legend:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-baza69
Out[1]=1

PlotMarkers  (2)

Make the marker of a Turing machine gray:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-neqegy
Out[1]=1

Draw the outline of the default markers:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-nwhpv9
Out[1]=1

PlotTheme  (2)

Use the web plot theme:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-cdykcx
Out[1]=1

Use this theme without a mesh:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-cm1jgf
Out[2]=2

Use the scientific plot theme:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-dgno3a
Out[1]=1

Spacings  (1)

Change the spacing inside the frame:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-b7loz8
Out[1]=1

Change the spacing on each side of the icon:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-89dg9
Out[2]=2

Applications  (2)Sample problems that can be solved with this function

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-f9i5o2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-9nusui
Out[2]=2

Use RulePlot to legend the evolution of a CellularAutomaton:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-lmg99
Out[1]=1

Properties & Relations  (2)Properties of the function, and connections to other functions

Plot the evolution of a cellular automaton with ArrayPlot:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-izicyr
Out[1]=1

Use RulePlot to plot the evolution and the rule icon:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-c968yz
Out[2]=2

Plot the evolution of a Turing machine with ArrayPlot:

In[1]:=1
✖
https://wolfram.com/xid/0e7pzwte-cytccx
Out[1]=1

Use RulePlot to plot the evolution and the head of the machine:

In[2]:=2
✖
https://wolfram.com/xid/0e7pzwte-b1ivvq
Out[2]=2
Wolfram Research (2016), RulePlot, Wolfram Language function, https://reference.wolfram.com/language/ref/RulePlot.html (updated 2021).
✖
Wolfram Research (2016), RulePlot, Wolfram Language function, https://reference.wolfram.com/language/ref/RulePlot.html (updated 2021).

Text

Wolfram Research (2016), RulePlot, Wolfram Language function, https://reference.wolfram.com/language/ref/RulePlot.html (updated 2021).

✖
Wolfram Research (2016), RulePlot, Wolfram Language function, https://reference.wolfram.com/language/ref/RulePlot.html (updated 2021).

CMS

Wolfram Language. 2016. "RulePlot." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/RulePlot.html.

✖
Wolfram Language. 2016. "RulePlot." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/RulePlot.html.

APA

Wolfram Language. (2016). RulePlot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RulePlot.html

✖
Wolfram Language. (2016). RulePlot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RulePlot.html

BibTeX

@misc{reference.wolfram_2025_ruleplot, author="Wolfram Research", title="{RulePlot}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/RulePlot.html}", note=[Accessed: 23-March-2025 ]}

✖
@misc{reference.wolfram_2025_ruleplot, author="Wolfram Research", title="{RulePlot}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/RulePlot.html}", note=[Accessed: 23-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_ruleplot, organization={Wolfram Research}, title={RulePlot}, year={2021}, url={https://reference.wolfram.com/language/ref/RulePlot.html}, note=[Accessed: 23-March-2025 ]}

✖
@online{reference.wolfram_2025_ruleplot, organization={Wolfram Research}, title={RulePlot}, year={2021}, url={https://reference.wolfram.com/language/ref/RulePlot.html}, note=[Accessed: 23-March-2025 ]}