Consider a local match that can go in two ways....
Consider a local match that can go in two ways....
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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WolframModel[{{1,2},{2,4}}{{1,2,2},{2,4,4}},List@@@EdgeList@GeneralizedGridGraph[{4{"Directed"},4{"Directed"}}],4,"EventsStatesPlotsList","EventOrderingFunction""Random"]
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MultiwaySystem[WolframModel[{{1,2},{2,4}}{{1,2,2},{2,4,4}}],List@@@EdgeList@GeneralizedGridGraph[{3{"Directed"},3{"Directed"}}],2,"StatesGraph"]
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Graph[%138,AspectRatio1/2,VertexSize2]
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MultiwaySystem[WolframModel[{{1,2},{2,4}}{{1,2,2},{2,4,4}}],List@@@EdgeList@GeneralizedGridGraph[{3{"Directed"},3{"Directed"}}],3,"StatesGraphStructure"]
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MultiwaySystem[WolframModel[{{1,2},{2,4}}{{1,2,2},{2,4,4}}],List@@@EdgeList@GeneralizedGridGraph[{3{"Directed"},3{"Directed"}}],2,"CausalInvariantQ"]
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True
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MultiwaySystem[WolframModel[{{1,2},{2,4}}{{1,2,2},{2,4,4}}],List@@@EdgeList@GeneralizedGridGraph[{2{"Directed"},2{"Directed"}}],4,"StatesGraphStructure"]
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MultiwaySystem[WolframModel[{{1,2},{2,4}}{{1,2,2},{2,4,4}}],List@@@EdgeList@GeneralizedGridGraph[{2{"Directed"},2{"Directed"}}],4,"StatesGraph",VertexSize1]
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MultiwaySystem[WolframModel[{{1,2},{2,4}}{{1,2,2},{2,4,4}}],List@@@EdgeList@GeneralizedGridGraph[{2{"Directed"},3{"Directed"}}],5,"StatesGraph",VertexSize1]
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LayeredGraphPlot[MultiwaySystem[WolframModel[{{1,2},{2,4}}{{1,2,2},{2,4,4}}],List@@@EdgeList@GeneralizedGridGraph[{2{"Directed"},4{"Directed"}}],5,"StatesGraph",VertexSize2],AspectRatio1/2]
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Edge destroyer
Edge destroyer
WolframModel[{{1,2},{2,4}}{},List@@@EdgeList@GeneralizedGridGraph[{4{"Directed"},4{"Directed"}}],3,"EventsStatesPlotsList"]
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Graph[MultiwaySystem[WolframModel[{{1,2},{2,4}}{}],List@@@EdgeList@GeneralizedGridGraph[{3{"Directed"},3{"Directed"}}],2,"StatesGraph"],AspectRatio1/2,VertexSize3]
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WolframModel[{{1,2}}{},List@@@EdgeList@GeneralizedGridGraph[{4{"Directed"},4{"Directed"}}],3,"EventsStatesPlotsList"]
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Graph[MultiwaySystem[WolframModel[{{1,2}}{}],List@@@EdgeList@GeneralizedGridGraph[{3{"Directed"},3{"Directed"}}],2,"StatesGraph"],AspectRatio1/2,VertexSize2]
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Edges are independent, but you can’t an edge again... (because it’s gone)
AbsoluteOptions[GeneralizedGridGraph[{3{"Directed"},3{"Directed"}}],VertexCoordinates]
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{VertexCoordinates{{1.,1.},{1.,2.},{1.,3.},{2.,1.},{2.,2.},{2.,3.},{3.,1.},{3.,2.},{3.,3.}}}
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VertexList[GeneralizedGridGraph[{3{"Directed"},3{"Directed"}}]]
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{1,2,3,4,5,6,7,8,9}
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Thread[{1,2,3,4,5,6,7,8,9}->{{1.`,1.`},{1.`,2.`},{1.`,3.`},{2.`,1.`},{2.`,2.`},{2.`,3.`},{3.`,1.`},{3.`,2.`},{3.`,3.`}}]
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{1{1.,1.},2{1.,2.},3{1.,3.},4{2.,1.},5{2.,2.},6{2.,3.},7{3.,1.},8{3.,2.},9{3.,3.}}
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Graph[MultiwaySystem[WolframModel[{{1,2}}{}],List@@@EdgeList@GeneralizedGridGraph[{3{"Directed"},3{"Directed"}}],2,"StatesGraph","StateRenderingFunction"(Inset[WolframModelPlot[#2,VertexCoordinateRules->{1{1.`,1.`},2{1.`,2.`},3{1.`,3.`},4{2.`,1.`},5{2.`,2.`},6{2.`,3.`},7{3.`,1.`},8{3.`,2.`},9{3.`,3.`}}],#1,Center,#3]&)],AspectRatio1/2,VertexSize2]
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Graph[MultiwaySystem[WolframModel[{{1,2}}{}],List@@@EdgeList@GeneralizedGridGraph[{3{"Directed"},3{"Directed"}}],2,"StatesGraphStructure"],AspectRatio1/2]
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Graph[MultiwaySystem[WolframModel[{{1,2}}{}],List@@@EdgeList@GeneralizedGridGraph[{3{"Directed"},3{"Directed"}}],3,"StatesGraphStructure"],AspectRatio1/2]
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A rule with an automorphism
A rule with an automorphism
WolframModel[{{1,2},{2,4}}{{1,2,2},{2,2,4}},List@@@EdgeList@GeneralizedGridGraph[{4{"Directed"},4{"Directed"}}],4,"EventsStatesPlotsList","EventOrderingFunction""Random"]
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Graph[MultiwaySystem[WolframModel[{{1,2},{2,4}}{{1,2,2},{2,2,4}}],List@@@EdgeList@GeneralizedGridGraph[{3{"Directed"},3{"Directed"}}],2,"StatesGraph"],AspectRatio1/2,VertexSize2]
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{{1,2},{2,4}}{{1,2,2},{2,2,4}}
HypergraphAutomorphismGroup[{{1,2},{2,4}}]
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PermutationGroup[{}]
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HypergraphAutomorphismGroup[{{1,2,2},{2,2,4}}]
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PermutationGroup[{}]
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Graph[MultiwaySystem[WolframModel[{{1,2},{2,4}}{{1,2,2},{2,2,4}}],List@@@EdgeList@GeneralizedGridGraph[{3{"Directed"},3{"Directed"}}],2,"StatesGraph"],AspectRatio1/2,VertexSize2]
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......
{{1,2},{2,4}}{{1,2,2},{2,2,4}}
FindCanonicalWolframModel[{{x,y},{x,z}}{{x,y},{x,w},{y,w},{z,w}}]
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{{1,2},{1,3}}{{1,2},{1,4},{2,4},{3,4}}
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({{1,2},{1,3}}{{1,2},{1,4},{2,4},{3,4}}/.#)&/@(Thread[Range[4]#]&/@Permutations[Range[4]])
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FindCanonicalWolframModel/@%
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({{1,2},{1,3}}{{1,2},{1,4},{2,4},{3,4}}/.#)&/@(Thread[Range[4]#]&/@Cases[Permutations[Range[4]],{1,___}])
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FindCanonicalWolframModel/@%
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({{1,2},{1,3}}{{1,2},{1,4},{2,4},{3,4}}/.#)&/@(Thread[Range[4]#]&/@Cases[Permutations[Range[4]],{___,4}])
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FindCanonicalWolframModel/@%
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(({{1,2},{1,3}}/.#){{1,2},{1,4},{2,4},{3,4}})&/@(Thread[Range[3]#]&/@Permutations[Range[3]])
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FindCanonicalWolframModel/@%
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#===({{1,2},{1,3}}{{1,2},{1,4},{2,4},{3,4}})&/@%206
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{True,True,False,False,False,False}
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Implications for this rule
Implications for this rule
Permutations[Range[3]]
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{{1,2,3},{1,3,2},{2,1,3},{2,3,1},{3,1,2},{3,2,1}}
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Within each fiber, there is invariance wrt relabeling
Initial conditions with these elements swapped with converge (immediately)
MultiwaySystem[WolframModel[{{1,2},{1,3}}{{1,2},{1,4},{2,4},{3,4}}],{{1,2},{1,3}},2,"StatesGraph","IncludeEventInstances"True,VertexSize1]
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MultiwaySystem[WolframModel[{{1,2},{1,3}}{{1,2},{1,4},{2,4},{3,4}}],{{1,2},{1,3},{2,3}},2,"StatesGraph","IncludeEventInstances"True,VertexSize1]
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MultiwaySystem[WolframModel[{{1,2},{1,3}}{{1,2},{1,4},{2,4},{3,4}}],{{1,2},{1,3},{2,3},{3,1}},2,"StatesGraph","IncludeEventInstances"True,VertexSize1]
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We want the effective theory for all relevant initial conditions.....
We want the effective theory for all relevant initial conditions.....
getOverlaps[{{1,2},{1,3}}]
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{}
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HypergraphUnifications[{{1,2},{1,3}},{{1,2},{1,3}}]
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First/@%
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{{{2,1},{2,4},{2,3}},{{2,3},{2,1}},{{2,1},{2,3},{2,4}},{{2,3},{2,1}},{{2,1},{2,4},{2,3}},{{2,1},{2,3},{2,4}}}
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Flatten[Function[i,Map[i#&,Last[MultiwaySystem[WolframModel[{{1,2},{1,3}}{{1,2},{1,4},{2,4},{3,4}}],i,2]],1]]/@%269]
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One-step effective rule
One-step effective rule
Flatten[Function[i,Map[i#&,Last[MultiwaySystem[WolframModel[{{1,2},{1,3}}{{1,2},{1,4},{2,4},{3,4}}],i,1]],1]]/@%269]
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RulePlot[WolframModel[%]]
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Flatten[Function[i,Map[i#&,Last[MultiwaySystem[WolframModel[{{1,2},{1,3}}{{1,2},{1,4},{2,4},{3,4}}],i,1]],1]]/@%269]
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FindCanonicalWolframModel[%]
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{{{1,2},{1,3}}{{2,1},{2,3},{1,3},{4,3}},{{1,2},{1,3},{1,4}}{{2,1},{2,3},{1,3},{2,4},{5,3}}}
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RulePlot[WolframModel[%]]
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For each one, get the permutations, then combine them.....
permuteRule[lhs_rhs_]:=((lhs/.Thread[#])rhs)&/@(Function[u,(u#)&/@Permutations[u]]@Union[Flatten[lhs]])
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permuteRule[{{1,2},{1,3},{1,4}}{{2,1},{2,3},{1,3},{2,4},{5,3}}]
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Counts[FindCanonicalWolframModel/@Tuples[permuteRule/@{{{1,2},{1,3}}{{2,1},{2,3},{1,3},{4,3}},{{1,2},{1,3},{1,4}}{{2,1},{2,3},{1,3},{2,4},{5,3}}}]]
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This implies there are 11 nontrivial permutations that leave the original effective rule the same.....
permuteRuleX[lhs_rhs_]:=(Last[#]->((lhs/.Thread[#])rhs))&/@(Function[u,(u#)&/@Permutations[u]]@Union[Flatten[lhs]])
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permuteRuleX[{{1,2},{1,3},{1,4}}{{2,1},{2,3},{1,3},{2,4},{5,3}}]
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gencases[rule_]:=((First/@#)->FindCanonicalWolframModel[Last/@#])&/@Tuples[permuteRuleX/@rule]
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gencases[{{{1,2},{1,3}}{{2,1},{2,3},{1,3},{4,3}},{{1,2},{1,3},{1,4}}{{2,1},{2,3},{1,3},{2,4},{5,3}}}]
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checkcases[rule_]:=Cases[gencases[rule],(x_rule)x]
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checkcases[{{{1,2},{1,3}}{{2,1},{2,3},{1,3},{4,3}},{{1,2},{1,3},{1,4}}{{2,1},{2,3},{1,3},{2,4},{5,3}}}]
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Last/@%//Counts
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{1,2,3,4}2,{1,2,4,3}2,{1,3,2,4}2,{1,3,4,2}2,{1,4,2,3}2,{1,4,3,2}2
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Keys[%]
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{{1,2,3,4},{1,2,4,3},{1,3,2,4},{1,3,4,2},{1,4,2,3},{1,4,3,2}}
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PermutationGroup[PermutationCycles/@%]
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PermutationGroup[{Cycles[{}],Cycles[{{3,4}}],Cycles[{{2,3}}],Cycles[{{2,3,4}}],Cycles[{{2,4,3}}],Cycles[{{2,4}}]}]
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GroupElements[%]
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{Cycles[{}],Cycles[{{3,4}}],Cycles[{{2,3}}],Cycles[{{2,3,4}}],Cycles[{{2,4,3}}],Cycles[{{2,4}}]}
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GroupMultiplicationTable[%%]
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{{1,2,3,4,5,6},{2,1,4,3,6,5},{3,5,1,6,2,4},{4,6,2,5,1,3},{5,3,6,1,4,2},{6,4,5,2,3,1}}
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ArrayPlot[%]
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CayleyGraph[%325]
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3!
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6
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FiniteGroupData["Icosahedral"]
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Icosahedral
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ArrayPlot[FiniteGroupData["Icosahedral","MultiplicationTable"]]
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FiniteGroupData["Icosahedral","CayleyGraph"]
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Strings
Strings
MultiwaySystem["AA""AAA","AA",2,"StatesGraph","IncludeEventInstances"True,VertexSize1]
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Degeneracies of labeling are not possible in strings....