MultiwaySystem[{"A""AB","B""A"},"A",4,"CriticalPairs"]
In[]:=
{{AA,ABB},{AAB,ABA},{AAB,ABBB},{ABA,ABBB},{AAA,AABB},{AAA,ABAB},{AABB,ABAB},{AAA,ABBA},{ABAB,ABBA},{AABB,ABBA},{AABB,ABBBB},{ABAB,ABBBB},{ABBA,ABBBB}}
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Table[MultiwaySystem[{"A""AB","B""A"},"A",t,"CriticalPairs"],{t,5}]
In[]:=
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Graph[UndirectedEdge@@@Catenate[%]]
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Graph[UndirectedEdge@@@Catenate[Table[MultiwaySystem[{"A""AB","B""A"},"A",t,"CriticalPairs"],{t,8}]]]
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Graph[UndirectedEdge@@@#]&/@Table[MultiwaySystem[{"A""AB","B""A"},"A",t,"CriticalPairs"],{t,7}]
In[]:=
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We can also compute the branchial graph of the multiway causal network
MultiwaySystem[{"A""AB","B""A"},"A",5,"EvolutionGraph"]//LayeredGraphPlot
In[]:=
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{"A""AB","B""A"}
In[]:=
{AAB,BA}
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MultiwaySystem[{"A""AB","B""A"},"A",5,"CriticalPairs"]
In[]:=
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MultiwaySystem[{"A""AB","B""A"},"A",4,"UnresolvedCriticalPairs"]
In[]:=
{{AAA,AABB},{AAA,ABAB},{AABB,ABAB},{AAA,ABBA},{ABAB,ABBA},{AABB,ABBA},{AABB,ABBBB},{ABAB,ABBBB},{ABBA,ABBBB}}
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MultiwaySystem[{"A""AB","B""A"},"A",5]
In[]:=
{{A},{AB},{AA,ABB},{AAB,ABA,ABBB},{AAA,AABB,ABAB,ABBA,ABBBB},{AAAB,AABA,AABBB,ABAA,ABABB,ABBAB,ABBBA,ABBBBB}}
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{{"AAA","AABB"},{"AAA","ABAB"},{"AABB","ABAB"},{"AAA","ABBA"},{"ABAB","ABBA"},{"AABB","ABBA"},{"AABB","ABBBB"},{"ABAB","ABBBB"},{"ABBA","ABBBB"}}
In[]:=
{{AAA,AABB},{AAA,ABAB},{AABB,ABAB},{AAA,ABBA},{ABAB,ABBA},{AABB,ABBA},{AABB,ABBBB},{ABAB,ABBBB},{ABBA,ABBBB}}
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getEventRenderingFunction["ListSubstitutionSystem",Automatic]:= Text[Framed[Style[getEventRenderingForm[Row,stripMetadata[#2]],Black],FrameMargins->None, FrameStyle->Directive[Opacity[.2],Blue],Background->Directive[Opacity[.1],Blue]],#1,{0,0}]&
getStateRenderingFunction["StringSubstitutionSystem",Automatic]:=Text[Framed[Style[stripMetadata[#2],Black], FrameMargins->None,FrameStyle->Directive[Opacity[.4],Gray],Background->Directive[Opacity[.2],Gray]], #1,{0,0}]&
StateBranchialGraph[rule_,init_,t_]:=Graph[UndirectedEdge@@@MultiwaySystem[{"A""AB","B""A"},"A",t,"UnresolvedCriticalPairs"],VertexShapeFunctionText[Framed[Style[#2,Black], FrameMargins->None,FrameStyle->Directive[Opacity[.4],Gray],Background->Directive[Opacity[.2],Gray]], #1,{0,0}]&]
In[]:=
Table[StateBranchialGraph[{"A""AB","B""A"},"A",t],{t,2,6}]
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VertexDegree
In[]:=
{6,8,10,8,9,10,8,10,8,10,10,8,5}
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TransitiveReductionGraph
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Out[]=
GraphPlot3D
In[]:=
Out[]=
In[]:=
Histogram[{10,13,17,16,13,15,18,16,14,18,16,15,17,18,21,18,21,15,18,12,16,13,16,21,18,16,16,18,16,14,16,16,12,7}]
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Histogram[%]
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Histogram[VertexDegree[#],FrameTrue,ImageSizeSmall]&/@
,
,
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Out[]=
VertexCount/@
,
,
,
In[]:=
{2,3,5,8}
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MultiwaySystem[{"A""AB","B""A"},"A"
MultiwaySystem[{"A""AB","B""A"},"A",8,"StateBranchialGraph"]
In[]:=
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MultiwaySystem[{"A""AB","B""A"},"A",10,"StateBranchialGraphStructure"]
In[]:=
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ConnectedGraphComponents[%]
In[]:=
Out[]=
Histogram[VertexDegree[#]]&/@%
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Histogram[VertexDegree[#]]&/@ConnectedGraphComponents[MultiwaySystem[{"A""AB","B""A"},"A",15,"StateBranchialGraphStructure"]]
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Histogram[VertexDegree[%630]]
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GraphNeighborhoodVolumes
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MultiwaySystem[{"A""AB","B""A"},"A",5,"EventBranchialGraph"]
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MultiwaySystem[{"A""AB","B""A"},"A",10,"EventBranchialGraphStructure"]
In[]:=
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MultiwaySystem[{"AA""AA"},"A",8,"StateBranchialGraph"]
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Testing on Causal Invariant Rules
Testing on Causal Invariant Rules
allres2=Table[Import["/Users/sw/Dropbox/Physics/Data/MWCausalInvariance2/"<>ToString[n]<>".wxf"],{n,12}];
In[]:=
causals=Cases[Catenate[#],(x__Integer)x]&/@allres2;
In[]:=
causals[[2]]
In[]:=
{{AA},{AB}}
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MultiwaySystem[#,"A",5,"StateBranchialGraphStructure"]&/@causals[[2]]
In[]:=
,
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Labeled[MultiwaySystem[#,"A",5,"StateBranchialGraphStructure"],#]&/@causals[[3]]
In[]:=
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Select[Labeled[MultiwaySystem[#,"A",5,"StateBranchialGraphStructure"],#]&/@causals[[4]],VertexCount[First[#]]>0&]
In[]:=
Out[]=
Select[Labeled[MultiwaySystem[#,"A",5,"StateBranchialGraphStructure"],#]&/@causals[[5]],VertexCount[First[#]]>0&]
In[]:=
Out[]=
Select[Labeled[MultiwaySystem[#,"A",5,"StateBranchialGraphStructure"],#]&/@causals[[6]],VertexCount[First[#]]>0&]
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ConnectedGraphComponents[MultiwaySystem[{"A""AA","A""AB"},"A",8,"StateBranchialGraphStructure"]]
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GraphPlot3D
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MeanAround/@TransposeValuesGraphNeighborhoodVolumes
,All,Automatic
In[]:=
{1,21.6±0.5,97.3±2.2,183.5±2.9,230.6±2.2,248.0±1.4}
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Ratios[%]
In[]:=
{21.6±0.5,4.50±0.14,1.89±0.05,1.257±0.023,1.075±0.012}
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ResourceFunction["LogDifferences"][%%]
In[]:=
{4.435±0.031,3.71±0.08,2.21±0.10,1.02±0.08,0.40±0.06}
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ConnectedGraphComponents[MultiwaySystem[{"A""B","B""AAA"},"A",8,"StateBranchialGraphStructure"]]
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Role of weighting in branchial graphs
Role of weighting in branchial graphs
“Equal-time commutator in state branchial graphs”
“Equal-time commutator in state branchial graphs”
Distance in string space corresponding to branchial state graphs
Distance in string space corresponding to branchial state graphs
vl=VertexList
In[]:=
{BBABB,AAAAABB,ABAAAAB,ABABAAA,ABBBB,BAAAAAB,BAABAAA,BABBB,AAABAAB,BBAAAAA,BBBAB,AAABABA,BAAAABA,BBBBA,AAAAAAAAA,AAAABAB,AABAAAB,AAAABBA,AABAABA,ABAAABA,AABBAAA,ABAABAA,ABBAAAA,AAABBAA,BAAABAA,BABAAAA,AABABAA}
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What is the induced string metric?
DistanceMatrix[vl,DistanceFunctionEditDistance]
In[]:=
Out[]=
gdm=GraphDistanceMatrix
;
In[]:=
gdm
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MatrixPlot/@{%696,%698}
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DamerauLevenshteinDistance
Can one characterize distances between states in the branchial state graph by complex numbers?
Can one characterize distances between states in the branchial state graph by complex numbers?
Events Branchial Graphs
Events Branchial Graphs
Select[Labeled[MultiwaySystem[#,"A",5,"EventBranchialGraphStructure"],#]&/@causals[[2]],VertexCount[First[#]]>0&]
In[]:=
{}
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Select[Labeled[MultiwaySystem[#,"A",5,"EventBranchialGraphStructure"],#]&/@causals[[3]],VertexCount[First[#]]>0&]
In[]:=
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Select[Labeled[MultiwaySystem[#,"A",5,"EventBranchialGraphStructure"],#]&/@causals[[4]],VertexCount[First[#]]>0&]
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Select[Labeled[MultiwaySystem[#,"A",5,"EventBranchialGraphStructure"],#]&/@causals[[5]],VertexCount[First[#]]>0&]
In[]:=
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Select[Labeled[MultiwaySystem[#,"A",5,"EventBranchialGraphStructure"],#]&/@causals[[6]],VertexCount[First[#]]>0&]
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Select[Labeled[MultiwaySystem[#,"A",5,"EventBranchialGraphStructure"],#]&/@noncausals[[3]],VertexCount[First[#]]>0&]
In[]:=
{}
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Select[Labeled[MultiwaySystem[#,"A",5,"EventBranchialGraphStructure"],#]&/@noncausals[[4]],VertexCount[First[#]]>0&]
In[]:=
{}
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Select[Labeled[MultiwaySystem[#,"A",5,"EventBranchialGraphStructure"],#]&/@noncausals[[5]],VertexCount[First[#]]>0&]
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Select[Labeled[MultiwaySystem[#,"A",5,"EventBranchialGraphStructure"],#]&/@noncausals[[6]],VertexCount[First[#]]>0&]
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Out[]=
Select[Labeled[MultiwaySystem[#,"A",5,"EventBranchialGraphStructure"],#]&/@noncausals[[7]],VertexCount[First[#]]>0&]
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TransitiveReductionGraph
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Each complete graph is just the critical pairs with a given predecessor
Size of the event branchial graph is a measure of ambiguity at a particular stage.