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WOLFRAM NOTEBOOK
The first few rules in the enumeration are immediately causal invariant, as we would expect:
In[]:=
TotalCausalInvariantQ[WolframModel[{{1,2}}{{1,3},{3,1},{1,2},{3,2}}],1]
Out[]=
True
In[]:=
TotalCausalInvariantQ[WolframModel[{{1,2,3}}{{1,2,2},{4,1,3}}],1]
Out[]=
True
In[]:=
TotalCausalInvariantQ[WolframModel[{{1,2,3}}{{1,1,4},{2,4,3}}],1]
Out[]=
True
A pair of similar rules with different causal invariance properties:
In[]:=
TotalCausalInvariantQ[WolframModel[{{1,2},{1,3}}{{2,3},{3,2},{2,4},{1,4}}],1]
Out[]=
False
In[]:=
TotalCausalInvariantQ[WolframModel[{{1,2},{2,3}}{{1,1},{3,1},{3,4},{2,4}}],1]
Out[]=
True
We see that the former rule does not become invariant, even allowing more time steps:
In[]:=
TotalCausalInvariantQ[WolframModel[{{1,2},{1,3}}{{2,3},{3,2},{2,4},{1,4}}],2]
Out[]=
False
We can see this difference between the two rules empirically from their states graphs; both have bifurcations and convergences, but the former seems to produce much more state bifurcation than the latter:
In[]:=
MultiwaySystem[WolframModel[{{1,2},{1,3}}{{2,3},{3,2},{2,4},{1,4}}],{{0,0},{0,0}},3,"StatesGraphStructure"]
Out[]=
In[]:=
MultiwaySystem[WolframModel[{{1,2},{2,3}}{{1,1},{3,1},{3,4},{2,4}}],{{0,0},{0,0}},3,"StatesGraphStructure"]
Out[]=
In[]:=
MultiwaySystem[WolframModel[{{1,2},{2,3}}{{1,1},{3,1},{3,4},{2,4}}],{{0,0},{0,0}},4,"StatesGraphStructure"]
Out[]=
In[]:=
MultiwaySystem[WolframModel[{{1,2},{1,3}}{{2,3},{3,2},{2,4},{1,4}}],{{0,0},{0,0}},4,"StatesGraphStructure"]
Out[]=
An example of a rule with convergent critical pairs, but where more than one step is required for convergence:
In[]:=
TotalCausalInvariantQ[WolframModel[{{1,2},{2,3}}{{3,2},{2,3},{3,4},{1,4}}],1]
Out[]=
False
In[]:=
TotalCausalInvariantQ[WolframModel[{{1,2},{2,3}}{{3,2},{2,3},{3,4},{1,4}}],2]
Out[]=
True
This can again be seen from its states graph, which starts out relatively “disconnected”, but gradually begins to “fill in” over time:
In[]:=
MultiwaySystem[WolframModel[{{1,2},{2,3}}{{3,2},{2,3},{3,4},{1,4}}],{{0,0},{0,0}},3,"StatesGraphStructure"]
Out[]=
In[]:=
MultiwaySystem[WolframModel[{{1,2},{2,3}}{{3,2},{2,3},{3,4},{1,4}}],{{0,0},{0,0}},4,"StatesGraphStructure"]
Out[]=
In[]:=
MultiwaySystem[WolframModel[{{1,2},{2,3}}{{3,2},{2,3},{3,4},{1,4}}],{{0,0},{0,0}},5,"StatesGraphStructure"]
Out[]=
Some more rules which do not appear to become causal invariant, even allowing for more steps:
In[]:=
TotalCausalInvariantQ[WolframModel[{{1,2,3},{4,2,5}}{{2,6,3},{6,1,2},{1,4,2}}],3]
Out[]=
False
In[]:=
TotalCausalInvariantQ[WolframModel[{{1,2,3},{2,4,5}}{{3,6,3},{2,4,6},{4,1,5}}],2]
Out[]=
False
Once again, their state graphs show the expected signature of very little convergence occurring:
In[]:=
MultiwaySystem[WolframModel[{{1,2,3},{4,2,5}}{{2,6,3},{6,1,2},{1,4,2}}],{{0,0,0},{0,0,0}},5,"StatesGraphStructure"]
Out[]=