Consider a local match that can go in two ways....
Consider a local match that can go in two ways....
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GridGraph[{10,10}]
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GeneralizedGridGraph[{4{"Directed","Circular"},4{"Directed","Circular"}}]
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WolframModel[{{1,2}}{{1,3},{3,2},{3,3}},List@@@EdgeList@GeneralizedGridGraph[{4{"Directed","Circular"},4{"Directed","Circular"}}],3,"StatesPlotsList"]
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WolframModel[{{1,2},{2,4}}{{1,3},{3,2},{3,3},{2,5},{5,5},{5,4}},List@@@EdgeList@GeneralizedGridGraph[{4{"Directed","Circular"},4{"Directed","Circular"}}],2,"StatesPlotsList"]
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WolframModel[{{1,2},{2,4}}{{1,3},{3,2},{3,3},{2,5},{5,5},{5,4}},List@@@EdgeList@GeneralizedGridGraph[{4{"Directed"},4{"Directed"}}],2,"StatesPlotsList"]
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WolframModel[{{1,2},{2,4}}{{1,3},{3,2},{3,3},{2,5},{5,5},{5,4}},List@@@EdgeList@GeneralizedGridGraph[{6{"Directed"},6{"Directed"}}],2,"StatesPlotsList"]
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WolframModel[{{1,2},{2,4}}{{1,2,4}},List@@@EdgeList@GeneralizedGridGraph[{10{"Directed"},10{"Directed"}}],2,"StatesPlotsList"]
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WolframModel[{{1,2},{2,4}}{{1,2,2},{2,4,4}},List@@@EdgeList@GeneralizedGridGraph[{10{"Directed"},10{"Directed"}}],2,"StatesPlotsList"]
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WolframModel[{{1,2},{2,4}}{{1,2,2},{2,4,4}},List@@@EdgeList@GeneralizedGridGraph[{10{"Directed"},10{"Directed"}}],3,"StatesPlotsList","EventOrderingFunction""Random"]
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WolframModel[{{1,2},{2,4}}{{1,2,2},{2,4,4}},List@@@EdgeList@GeneralizedGridGraph[{4{"Directed"},4{"Directed"}}],2,"EventsStatesPlotsList"]
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Edge destroyer
Edge destroyer
Edges are independent, but you can’t an edge again... (because it’s gone)
A rule with an automorphism
A rule with an automorphism
Multiple matches: with different identifications of elements
Multiple matches: with different identifications of elements
Powers of rules
Powers of rules
Analogous to applying to LHSs for total causal invariance......
Implications for this rule
Implications for this rule
Within each fiber, there is invariance wrt relabeling
Initial conditions with these elements swapped with converge (immediately)
We want the effective theory for all relevant initial conditions.....
We want the effective theory for all relevant initial conditions.....
One-step effective rule
One-step effective rule
For each one, get the permutations, then combine them.....
This implies there are 11 nontrivial permutations that leave the original effective rule the same.....
Strings
Strings
Degeneracies of labeling are not possible in strings....