Function Repository Resource:

# IteratedFiniteAutomaton

Iteratively apply a transducer finite automaton

Contributed by: Stephen Wolfram
 ResourceFunction["IteratedFiniteAutomaton"][rule,init,t] generates a list representing the evolution of the iterated finite automaton with the specified rule from initial condition init for t steps. ResourceFunction["IteratedFiniteAutomaton"][rule,s0,init,t] starts the iterated finite automaton with state s0. ResourceFunction["IteratedFiniteAutomaton"][rule,init] gives the result of evolving init for one step. ResourceFunction["IteratedFiniteAutomaton"][rule,s0] is an operator form of IteratedFiniteAutomaton that represents one step of evolution.

## Details and Options

The rules for the iterated finite automaton can be given as a list of elements of the form {s,i}{f,o}, where s is the initial state of the finite automaton, f is the final state, i is the symbol taken as input and o is the symbol given as output.
The rules can also be given as a numerical code in the form {code,{s,k}}, where s is the number of states of the finite automaton, and k is the number of possible symbols used. There are (sk)sk possible rules.
ResourceFunction["IteratedFiniteAutomaton"][rule, init] implements a transducer finite automaton that starts in state 1 at the first symbol in init, then successively reads a symbol and writes a symbol, according to the rule.

## Examples

### Basic Examples (2)

Apply a simple iterated finite automaton to a single list:

 In[1]:=
 Out[1]=

Iterate the same rule for 100 steps:

 In[2]:=
 Out[2]=

### Scope (3)

Evolve an iterated finite automaton specified as a code for 2 steps:

 In[3]:=
 Out[3]=

Evolve an iterated finite automaton specified as replacement rules for 2 steps:

 In[4]:=
 Out[4]=

Give the result of one step of evolving an iterated finite automaton:

 In[5]:=
 Out[5]=
 In[6]:=
 Out[6]=

The operator form equivalent:

 In[7]:=
 Out[7]=
 In[8]:=
 Out[8]=

Two different ways to specify the same iterated finite automaton:

 In[9]:=
 Out[9]=

### Applications (1)

Rostislav Grigorchuk's automaton:

 In[10]:=
 Out[10]=