In[]:=
TSAllInits[2]
Out[]=
{{1,0,0,1},{1,0,0,1,0},{1,0,0,1,0,0},{1,0,0,0},{1,0,0,0,0},{1,0,0,0,0,0},{0,0,0,1},{0,0,0,1,0},{0,0,0,1,0,0},{0,0,0,0},{0,0,0,0,0},{0,0,0,0,0,0}}
In[]:=
Length/@%
Out[]=
{4,5,6,4,5,6,4,5,6,4,5,6}
In[]:=
TSAllInits[UpTo[3]]
Out[]=
{{0},{0,0},{0},{0,0},{0,0,0},{1},{1,0},{1,0,0},{0,0,0,0},{0,0,0,0,0},{0,0,0,0,0,0},{0,0,0,1},{0,0,0,1,0},{0,0,0,1,0,0},{1,0,0,0},{1,0,0,0,0},{1,0,0,0,0,0},{1,0,0,1},{1,0,0,1,0},{1,0,0,1,0,0},{0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,1},{0,0,0,0,0,0,1,0},{0,0,0,0,0,0,1,0,0},{0,0,0,1,0,0,0},{0,0,0,1,0,0,0,0},{0,0,0,1,0,0,0,0,0},{0,0,0,1,0,0,1},{0,0,0,1,0,0,1,0},{0,0,0,1,0,0,1,0,0},{1,0,0,0,0,0,0},{1,0,0,0,0,0,0,0},{1,0,0,0,0,0,0,0,0},{1,0,0,0,0,0,1},{1,0,0,0,0,0,1,0},{1,0,0,0,0,0,1,0,0},{1,0,0,1,0,0,0},{1,0,0,1,0,0,0,0},{1,0,0,1,0,0,0,0,0},{1,0,0,1,0,0,1},{1,0,0,1,0,0,1,0},{1,0,0,1,0,0,1,0,0}}
In[]:=
Length[%53]
Out[]=
44
In[]:=
Union[%]
Out[]=
{{0},{1},{0,0},{1,0},{0,0,0},{1,0,0},{0,0,0,0},{0,0,0,1},{1,0,0,0},{1,0,0,1},{0,0,0,0,0},{0,0,0,1,0},{1,0,0,0,0},{1,0,0,1,0},{0,0,0,0,0,0},{0,0,0,1,0,0},{1,0,0,0,0,0},{1,0,0,1,0,0},{0,0,0,0,0,0,0},{0,0,0,0,0,0,1},{0,0,0,1,0,0,0},{0,0,0,1,0,0,1},{1,0,0,0,0,0,0},{1,0,0,0,0,0,1},{1,0,0,1,0,0,0},{1,0,0,1,0,0,1},{0,0,0,0,0,0,0,0},{0,0,0,0,0,0,1,0},{0,0,0,1,0,0,0,0},{0,0,0,1,0,0,1,0},{1,0,0,0,0,0,0,0},{1,0,0,0,0,0,1,0},{1,0,0,1,0,0,0,0},{1,0,0,1,0,0,1,0},{0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,1,0,0},{0,0,0,1,0,0,0,0,0},{0,0,0,1,0,0,1,0,0},{1,0,0,0,0,0,0,0,0},{1,0,0,0,0,0,1,0,0},{1,0,0,1,0,0,0,0,0},{1,0,0,1,0,0,1,0,0}}
In[]:=
Length[%]
Out[]=
42
In[]:=
Length/@%
Out[]=
{0,1,2,1,2,3,1,2,3,4,5,6,4,5,6,4,5,6,4,5,6,7,8,9,7,8,9,7,8,9,7,8,9,7,8,9,7,8,9,7,8,9,7,8,9}
In[]:=
AllInits[3]
Out[]=
{{0,{0,0,0}},{0,{0,0,1}},{0,{0,1,0}},{0,{0,1,1}},{0,{1,0,0}},{0,{1,0,1}},{0,{1,1,0}},{0,{1,1,1}},{1,{0,0,0}},{1,{0,0,1}},{1,{0,1,0}},{1,{0,1,1}},{1,{1,0,0}},{1,{1,0,1}},{1,{1,1,0}},{1,{1,1,1}},{2,{0,0,0}},{2,{0,0,1}},{2,{0,1,0}},{2,{0,1,1}},{2,{1,0,0}},{2,{1,0,1}},{2,{1,1,0}},{2,{1,1,1}}}
In[]:=
Length[AllInits[3]]
Out[]=
24
In[]:=
TSAllInits[3]
Out[]=
{{1,0,0,1,0,0,1},{1,0,0,1,0,0,1,0},{1,0,0,1,0,0,1,0,0},{1,0,0,1,0,0,0},{1,0,0,1,0,0,0,0},{1,0,0,1,0,0,0,0,0},{1,0,0,0,0,0,1},{1,0,0,0,0,0,1,0},{1,0,0,0,0,0,1,0,0},{1,0,0,0,0,0,0},{1,0,0,0,0,0,0,0},{1,0,0,0,0,0,0,0,0},{0,0,0,1,0,0,1},{0,0,0,1,0,0,1,0},{0,0,0,1,0,0,1,0,0},{0,0,0,1,0,0,0},{0,0,0,1,0,0,0,0},{0,0,0,1,0,0,0,0,0},{0,0,0,0,0,0,1},{0,0,0,0,0,0,1,0},{0,0,0,0,0,0,1,0,0},{0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0}}
In[]:=
Length[%]
Out[]=
24

Investigating Freeways

In[]:=
With[{g=NestGraph[TSStep,TSAllInits[UpTo[3]],400]},HighlightGraph[Graph[g,VertexSize(#.1Length[#]&/@VertexList[g]),VertexLabels(#->Length[#]&/@VertexList[g])],{Style[Subgraph[g,FindCycle[g,{1,Infinity},All]],Thick,Red],Pick[VertexList[g],VertexOutDegree[g],0]}]]
Out[]=
In[]:=
stg5=NestGraph[TSStep,TSAllInits[UpTo[5]],20000];
In[]:=
stg5
Out[]=
In[]:=
ReverseSortBy[WeaklyConnectedGraphComponents[stg5],VertexCount]
Out[]=

,
,
,
,
,
,
,
,

In[]:=
VertexList

Out[]=
{1}
[[ Probably meaningless... ]]
Along the freeway, the length differences have to be ±1 ....

Is there only one long eel/freeway?

What is the inverse image of the various attractors?

Of inits evolving to a given attractor, how long did it take them?

Ultimate goal: find which waypoints to store....

Basin of attraction is large + evolution time is large
The base of the tail is worth storing....
Because the main freeway is mostly big values, but its tail (as well as its head) is smaller values...

Final vs. Initial Length

Cycle Finding

At step t, store and see if you ever return to the step t value; then store at step 2t; .... and continue exponentially....
Note: the 1 attractor is not a cycle...... [it’s termination]
Claim: there is a way to combine cycles.... by concatenation....
With different cyclic tapes, they can be combined to make another cycle......

[ Maybe there is only one base cycle, being “genetically combined” ]

Now need to clean these cycles.....

Specific Evolutions

Random Tests

Cycles we know:

1000-gated....

Parallelism (failed)