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SubstitutionSystem
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SubstitutionSystem

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SubstitutionSystem[rule,init,t]

generates a list representing the evolution of the substitution system with the specified rule from initial condition init for t steps.

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SubstitutionSystem[rule,init]

gives the result of evolving init for one step.

✖

is an operator form of SubstitutionSystem that corresponds to one step of evolution.

Details

  • In SubstitutionSystem[rule,…], rule can be of the following forms:
  • {i1rhs1,i2rhs2,…}list substitution system
    {"s1"rhs1,"s2"rhs2,…}string substitution system
  • In list substitution systems, the rhsi can be lists of any length or can be rectangular arrays of any depth but all with the same dimensions. They can also be individual elements such as integers.
  • In string substitution systems, the rhsi can be strings of any length.
  • Both lists and string substitution systems can have rules that involve patterns, but every object that appears in the rhsi must have a transformation defined by the rules given.
  • List substitution systems work with SparseArray objects.
  • In string substitution systems, the initial condition init must be a string; in list substitution systems, it must be an array whose depth is equal to the depth of the rhsi.
  • In SubstitutionSystem[rule,init,tspec], the time specification tspec can have any of the following forms:
  • tall steps 0 through t
    {t}a list containing only step t
    {t1,t2}steps t1 through t2
    {t1,t2,dt}steps t1, t1+dt, …
  • SubstitutionSystem uses the first substitutions that apply at each step, in the same way as SequenceReplace and StringReplace.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Five steps of a string substitution system:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-cffwyh
Out[1]=1

Analogous substitution system with lists:

In[2]:=2
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https://wolfram.com/xid/09a8lh2zm-zana6h
Out[2]=2

Steps in a 2D substitution system:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-pv6wsv
Out[1]=1

Give the fourth step only:

In[2]:=2
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https://wolfram.com/xid/09a8lh2zm-4fsoto
Out[2]=2

Alternative form:

In[3]:=3
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https://wolfram.com/xid/09a8lh2zm-utt8m6
Out[3]=3

Generate a rule icon for a substitution system:

In[1]:=1
✖
https://wolfram.com/xid/09a8lh2zm-rme7zl
Out[1]=1

Scope  (16)Survey of the scope of standard use cases

1D List Substitution Systems  (3)

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-qv0bhd
Out[1]=1

Lists do not have to be the same length:

In[2]:=2
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https://wolfram.com/xid/09a8lh2zm-64bbbw
Out[2]=2

The initial condition can be of any length:

In[1]:=1
✖
https://wolfram.com/xid/09a8lh2zm-htphyh
Out[1]=1

Rules can involve patterns:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-n1p9y3
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/09a8lh2zm-4wvwo0
Out[2]=2

Higher-Dimensional List Substitution Systems  (5)

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-3gpdqz
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/09a8lh2zm-vlomqm
Out[2]=2

Arrays do not have to be square:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-xp13c3
Out[1]=1

Arrays can have any depth:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-fxdvpc
Out[1]=1

The right-hand side of a rule can be a SparseArray:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-ees1a
Out[1]=1

SparseArray as an initial condition:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-k21uz
Out[1]=1

String Substitution Systems  (2)

A string substitution system:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-6qxf2e
Out[1]=1

Use any characters in the strings:

In[2]:=2
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https://wolfram.com/xid/09a8lh2zm-ndr3ff
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/09a8lh2zm-6k3ruq
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/09a8lh2zm-441ejj
Out[4]=4

Arrange output in a column:

In[5]:=5
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https://wolfram.com/xid/09a8lh2zm-1bao9n
Out[5]=5

Characters that do not appear in the rules are not replaced at each step:

In[1]:=1
✖
https://wolfram.com/xid/09a8lh2zm-7c81gl
Out[1]=1

Include additional characters in rules:

In[2]:=2
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https://wolfram.com/xid/09a8lh2zm-1cfeni
Out[2]=2

Time Step Specifications  (6)

Steps 0 through 5:

In[1]:=1
✖
https://wolfram.com/xid/09a8lh2zm-kkfx7p
Out[1]=1

Step 5 only:

In[1]:=1
✖
https://wolfram.com/xid/09a8lh2zm-o6se30
Out[1]=1

Steps 3 through 5:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-09q4vt
Out[1]=1

Odd-numbered steps:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-qux3yi
Out[1]=1

Apply a single step of evolution:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-txm18t
Out[1]=1

Apply in operator form:

In[1]:=1
✖
https://wolfram.com/xid/09a8lh2zm-k1oedx
Out[1]=1

Generalizations & Extensions  (1)Generalized and extended use cases

Set up neighbor-dependent string rules; rules are applied in the order given:

In[1]:=1
✖
https://wolfram.com/xid/09a8lh2zm-upasqs
Out[1]=1

Applications  (3)Sample problems that can be solved with this function

Steps in constructing a Cantor set:

In[1]:=1
✖
https://wolfram.com/xid/09a8lh2zm-89bzs2
Out[1]=1

Create an analogous 2D nested object:

In[2]:=2
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https://wolfram.com/xid/09a8lh2zm-kijr6q
Out[2]=2

3D Menger sponge:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-f5wso5
Out[1]=1

Generate a "C curve" fractal:

In[1]:=1
✖
https://wolfram.com/xid/09a8lh2zm-5dvfz3
Out[1]=1

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

Approximate a Cantor staircase function:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-479ufe
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/09a8lh2zm-uk11kb
Out[2]=2

Generate steps in a Thue–Morse substitution system:

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-57fr5g
Out[1]=1

The output at each step k is given by ThueMorse[Range[0,2^k-1]:

In[2]:=2
✖
https://wolfram.com/xid/09a8lh2zm-7nadpv
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

SubstitutionSystem always uses only the first substitution that applies:

In[1]:=1
✖
https://wolfram.com/xid/09a8lh2zm-8bnxhf
Out[1]=1

Neat Examples  (2)Surprising or curious use cases

In[1]:=1
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https://wolfram.com/xid/09a8lh2zm-6t25sh
Out[1]=1

3D substitution system based on the Levi–Civita tensor:

In[1]:=1
✖
https://wolfram.com/xid/09a8lh2zm-dtvyic
Out[1]=1
Wolfram Research (2015), SubstitutionSystem, Wolfram Language function, https://reference.wolfram.com/language/ref/SubstitutionSystem.html.
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Wolfram Research (2015), SubstitutionSystem, Wolfram Language function, https://reference.wolfram.com/language/ref/SubstitutionSystem.html.

Text

Wolfram Research (2015), SubstitutionSystem, Wolfram Language function, https://reference.wolfram.com/language/ref/SubstitutionSystem.html.

✖
Wolfram Research (2015), SubstitutionSystem, Wolfram Language function, https://reference.wolfram.com/language/ref/SubstitutionSystem.html.

CMS

Wolfram Language. 2015. "SubstitutionSystem." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SubstitutionSystem.html.

✖
Wolfram Language. 2015. "SubstitutionSystem." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SubstitutionSystem.html.

APA

Wolfram Language. (2015). SubstitutionSystem. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SubstitutionSystem.html

✖
Wolfram Language. (2015). SubstitutionSystem. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SubstitutionSystem.html

BibTeX

@misc{reference.wolfram_2025_substitutionsystem, author="Wolfram Research", title="{SubstitutionSystem}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/SubstitutionSystem.html}", note=[Accessed: 15-April-2025 ]}

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@misc{reference.wolfram_2025_substitutionsystem, author="Wolfram Research", title="{SubstitutionSystem}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/SubstitutionSystem.html}", note=[Accessed: 15-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_substitutionsystem, organization={Wolfram Research}, title={SubstitutionSystem}, year={2015}, url={https://reference.wolfram.com/language/ref/SubstitutionSystem.html}, note=[Accessed: 15-April-2025 ]}

✖
@online{reference.wolfram_2025_substitutionsystem, organization={Wolfram Research}, title={SubstitutionSystem}, year={2015}, url={https://reference.wolfram.com/language/ref/SubstitutionSystem.html}, note=[Accessed: 15-April-2025 ]}