WOLFRAM NOTEBOOK

ExtendedSubstitutionSystem[rule_,init_,t_,UpdatingFunctionXXXX,UpdatingFunctionHistoryLengthXXXX]
MultiwaySystem[rule_,init_,t_,EventSelectionFunctionXXXX,UpdatingFunctionHistoryLengthXXXX]
At every step, the ordering function is given a list of events, and returns which events to actually use at that step.
uf[currentevents,{historicalevents}]

Obvious cases:

Pick the first possible substitution (i.e. the one with the shortest prefix)
Greedily pick all substitutions that are independent, starting on the left
Examine all possible subsets of substitutions, and pick the one that updates the most positions independently
Carry along a marker (“mobile automaton style”) that says what to update

Symbolic Formalism

In[]:=
NestList[Replace[#,{x___,1,0,y___}{x,0,1,y}]&,{0,1,1,0,1,1},30]
Out[]=

What are the eigenstates of this rule?

This applies just the first possible replacement at each step....
In[]:=
Graph[(#->Replace[#,{x___,1,0,y___}{x,0,1,y}])&/@Tuples[{0,1},4],VertexLabelsAutomatic]
Out[]=
Here the superselection rule just counts the number of 1s
Eigenstates are any combination of sorted strings.
In[]:=
MultiwaySystem[{{1,0}{0,1}},Tuples[{0,1},3],3]
Out[]=
{{{0,0,0},{0,0,1},{0,1,0},{0,1,1},{1,0,0},{1,0,1},{1,1,0},{1,1,1}},{{0,0,1},{0,1,0},{0,1,1},{1,0,1}},{{0,0,1},{0,1,1}},{}}
In[]:=
MultiwaySystem[{{1,0}{0,1}},Tuples[{0,1},3],3,"StatesGraph"]
Out[]=
In[]:=
MultiwaySystem[{{1,0}{0,1}},Tuples[{0,1},4],5,"StatesGraph"]
Out[]=
In[]:=
Graph[Flatten[Function[s,(If[#==={},ss,Map[s->#&,#]]&[ReplaceList[s,{x___,1,0,y___}{x,0,1,y}]])]/@Tuples[{0,1},4]],VertexLabelsAutomatic]
Out[]=
In[]:=
ReplaceList[{0,0,0,0},{x___,1,0,y___}{x,0,1,y}]
Out[]=
{}
In[]:=
ReplaceList[{1,0,1,0},{x___,1,0,y___}{x,0,1,y}]
Out[]=
{{0,1,1,0},{1,0,0,1}}
In[]:=
MultiwaySystem[{{1,0}{0,1}},Tuples[{0,1},5],5,"StatesGraph"]
Out[]=
There is no causal graph, because no subsequent ever depends on a previous one; each transposition event is on a fresh piece of string, never on one that has been updated before.

Two-way sorting rule

What happens to the non-evolving AAA case?
Why isn’t this a singleton in the states graph? Can it be a single disembodied vertex, with zero edges?

What are the length-preserving string rewrites?

Take any tuple to any other tuple
Wolfram Cloud

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.


I understand and wish to continue anyway »

You are using a browser not supported by the Wolfram Cloud. Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.