Basic Examples 
(2)
 

Sample evolution of a symbolic system:
In[3]:=
[◼]
SymbolicSystem
[{e[x_][y_]x[e[y]][x]},e[e[e][e]][e][e],5]//Column
Out[3]=
e[e[e][e]][e][e]
e[e][e][e[e]][e[e][e]][e]
e[e[e]][e][e[e]][e[e[e]][e]][e]
e[e][e[e]][e[e]][e[e]][e[e][e[e]][e[e]]][e]
e[e[e[e]]][e][e[e]][e[e]][e[e[e[e]]][e][e[e]]][e]
e[e[e]][e[e]][e[e[e]]][e[e]][e[e]][e[e[e]][e[e]][e[e[e]]][e[e]]][e]
———
Define the auxiliary function
ToBrackets
:
In[9]:=
ToBrackets[rule_,init_,steps_]:=Module{e},​​StringSplit[StringReplace[ToString[#1],"e"->""],""]&/@
[◼]
SymbolicSystem
[rule,init,steps];
In[10]:=
ToBrackets[e[x_][y_]x[e[y]][x],e[e[e][e]][e][e],5]//Column
Out[10]=
{[,[,],[,],],[,],[,]}
{[,],[,],[,[,],],[,[,],[,],],[,]}
{[,[,],],[,],[,[,],],[,[,[,],],[,],],[,]}
{[,],[,[,],],[,[,],],[,[,],],[,[,],[,[,],],[,[,],],],[,]}
{[,[,[,],],],[,],[,[,],],[,[,],],[,[,[,[,],],],[,],[,[,],],],[,]}
{[,[,],],[,[,],],[,[,[,],],],[,[,],],[,[,],],[,[,[,],],[,[,],],[,[,[,],],],[,[,],],],[,]}
Steps versus step lengths:
In[11]:=
ListLinePlotLength/@ToBrackets[e[x_][y_]x[e[y]][x],e[e[e][e]][e][e],37],FillingAxis,AspectRatio
1
4

Out[11]=
Plot the evolution of the symbolic system:
In[12]:=
Module{evolved=ToBrackets[e[x_][y_]x[e[y]][x],e[e[e][e]][e][e],5]},GraphicsFlattenTableBlack,EdgeForm[Thin],evolvedji/.{"]"White,"["
,Rectangle[{i-1,1-j},{i,2-j}],evolvedji/.{"]"
,"["
,TextStyle[evolved〚j〛〚i〛,FontSize8],
1
2
(2i-1),
1
2
(3-2j),{j,Length[evolved]},{i,Length[evolved〚j〛]}
Out[12]=
In[13]:=
Module[{evolved=ToBrackets[e[x_][y_]x[e[y]][x],e[e[e][e]][e][e],37]},ArrayPlot[evolved/.{"]"0,"["1},PixelConstrained5,ColorRules{0LightGray,1Black},FrameFalse]]
Out[13]=