Foliation
Foliation
A foliation is an update ordering
A foliation is an update ordering
For non-interfering events (i.e. spacelike events) you can do any set you want at the same time
Generations correspond to “thick” collections of update events
Generations correspond to “thick” collections of update events
MultiwaySystem[{"A""BB","B""A"}"Random","A",12,"CausalGraph"]
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Is Sequential a tachyonic foliation?
Is Sequential a tachyonic foliation?
MultiwaySystem[{"A""BB","B""A"}"Sequential","A",12,"CausalGraph"]
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MultiwaySystem[{"A""AB","B""A"}"Sequential","A",12,"CausalGraph"]
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MultiwaySystem[{"A""AA","AAB""BA"}"Sequential","A",12,"CausalGraph"]
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MultiwaySystem[{"AA""A","AB""BAA"}"Sequential","AABA",4,"CausalGraph"]
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MultiwaySystem[{"A""B"}"Sequential","AAA",2,"CausalGraph"]
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MultiwaySystem[{"C""AAA","A""B"}"Sequential","C",4,"CausalGraph"]
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MultiwaySystem[{{0,1,0}{0,0,1},{0}{0,1,0}}"Sequential",{1,0,1,0},20,"CausalGraph"]
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TotalCausalInvariantQ[{{0,1,0}{0,0,1},{0}{0,1,0}},4]
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True
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Sequential is depth-first search..... (which never recurses back up the tree if the rule doesn’t terminate)
Sequential is depth-first search..... (which never recurses back up the tree if the rule doesn’t terminate)
“Generations” is breadth-first search
“Generations” is breadth-first search
Least-recent first....
The standard StringReplace update ordering
The standard StringReplace update ordering
As many as it can without overlapping.......
ResourceFunction["SequentialSubstitutionSystemPlot"][ResourceFunction["SequentialSubstitutionSystem"][{{0,1,0}{0,0,1},{0}{0,1,0}},{1,0,1,0},50]]
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Foliation specification
Foliation specification
What branch of the multiway system are you taking?
What branch of the multiway system are you taking?
At each step, you have a list of possible replacements ... you need EventSelectionFunction
At each step, you have a list of possible replacements ... you need EventSelectionFunction
Preferring the left; preferring the right
First;Last
Sequential updates: traveling at the speed of light
Sort[RandomInteger[100,10]]
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{38,57,59,61,64,84,92,97,99,100}
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Flatten[Table[{t,RandomInteger[100]},10,{t,5}],1]
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Graphics[Point[Flatten[Table[{t,RandomInteger[30]},15,{t,20}],1]]]
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Consider fixed-length strings
UpdatesPicture[Flatten[Table[Table[i,1+RandomInteger[3]],{i,20}]]&,20,5,.2]
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“strong hyperbolicity”
UpdatesPicture[Flatten[Table[Table[i,Ceiling[.3i]],{i,20}]]&,20,5,.2]
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Flatten[Table[Table[i,Ceiling[.3i]],{i,20}]]
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{1,2,3,4,4,5,5,6,6,7,7,7,8,8,8,9,9,9,10,10,10,11,11,11,11,12,12,12,12,13,13,13,13,14,14,14,14,14,15,15,15,15,15,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20}
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lapse function α (“timelike distance”); shift vector βi
lapse function α (“timelike distance”); shift vector
β
i
For each point, pick a sequence of delays between updatings....
ArrayPlot[(Characters[#]/.{"A"1,"B"0})&/@SubstitutionSystem["BA""AB","ABBBBAAAABBBA",10],MeshAll]
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Clear[StringPlot]
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StringPlot[data_]:=ArrayPlot[PadRight[(Characters[#]/.{"A"Yellow,"B"Orange})&/@data,Automatic,White],MeshIf[Length[data]>20,None,All]]
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CharacterStringPlot[data_]:=MapIndexed[{Style[<|"A"Yellow,"B"Orange,0White|>[#1],EdgeForm[Gray],Rectangle[-#2,-#2+{1,1}]],Text[#1,-#2+{1,1}/2]}&,data,{2}]
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LapsedSubstitutionSystem[rule_,init_,lapses_]:=
Priority list where at every step each position gets a priority
(i.e. this is a list of positions to update in order)
(i.e. this is a list of positions to update in order)
{1,5,2,3,4,...}
To make the picture above, coarse grain this, showing only every ___ steps
PrioritySubstitutionSystem[rule_,init_,pos_]:=
"ABBBBAAABBBAAA"
StringLength["ABBBBAAAABBBA"]
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13
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PrioritizedSubstitutionSystem["BA""AB","ABBBBAAAABBBA",RandomInteger[{1,13},1000],"Compressed"]
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{ABBBBAAAABBBA,ABBBBAAAABBAB,ABBBABAAABBAB,ABBBABAAABABB,ABBBAABAABABB,ABBABABAABABB,ABABBABAABABB,ABABABBAABABB,AABBABBAABABB,AABBABABABABB,AABABBABABABB,AABABABBABABB,AABABABABBABB,AABABABABABBB,AABABABAABBBB,AAABBABAABBBB,AAABBAABABBBB,AAABBAAABBBBB,AAABABAABBBBB,AAABAABABBBBB,AAABAAABBBBBB,AAAABAABBBBBB,AAAAABABBBBBB,AAAAAABBBBBBB}
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StringPlot[%]
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PrioritizedSubstitutionSystem["A""B",StringRepeat["A",20],Table[Mod[i,20,1],{i,100}],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem["A""B",StringRepeat["A",20],{10,5,3},"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem["BA""AB",StringJoin[StringRepeat["B",20],StringRepeat["A",20]],Table[Mod[i,40,1],{i,100}],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem["BA""AB",StringJoin[StringRepeat["B",20],StringRepeat["A",20]],Flatten[Table[Table[i,Ceiling[.2i]],{i,40}]],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem["BA""AB",StringJoin[StringRepeat["B",20],StringRepeat["A",20]],41-Flatten[Table[Table[i,Ceiling[.2i]],{i,40}]],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem["BA""AB",StringJoin[StringRepeat["B",20],StringRepeat["A",20]],Flatten[Table[Table[Table[i,Ceiling[.02i]],{i,40}],10]],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem["BA""AB",StringJoin[StringRepeat["B",20],StringRepeat["A",20]],41-Flatten[Table[Table[Table[i,Ceiling[.02i]],{i,40}],10]],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem["BA""AB",StringJoin[StringRepeat["B",20],StringRepeat["A",20]],RandomInteger[{1,40},1000],"Compressed"]//StringPlot
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Show[PrioritizedSubstitutionSystem["BA""AB",StringJoin[StringRepeat["B",20],StringRepeat["A",20]],RandomInteger[{1,40},10000],"Compressed"]//StringPlot,AspectRatio2]
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Show[PrioritizedSubstitutionSystem["BA""AB",StringJoin[StringRepeat["B",20],StringRepeat["A",20]],Flatten[Table[Table[Table[i,Ceiling[.02i]],{i,40}],1000]],"Compressed"]//StringPlot,AspectRatio2]
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Show[PrioritizedSubstitutionSystem["BA""AB",StringJoin[StringRepeat["B",20],StringRepeat["A",20]],Flatten[Table[Table[Table[i,Mod[3i+2j,40,1]],{i,40}],{j,10}]],"Compressed"]//StringPlot,AspectRatio2]
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Show[PrioritizedSubstitutionSystem["BA""AB",StringJoin[StringRepeat["B",20],StringRepeat["A",20]],Flatten[Table[Mod[3i^2,40,1],{i,2000}]],"Compressed"]//StringPlot,AspectRatio2]
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Need an ergodic sequence to do something interesting....
PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Table[Mod[i,40,1],{i,300}],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Table[Mod[i^2,40,1],{i,300}],"Compressed"]//StringPlot
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Flatten[Table[Table[Table[i,Ceiling[.2i]],{i,40}],100]]
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PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Flatten[Table[Table[Table[i,Ceiling[.2i]],{i,40}],10]],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Flatten[Table[Table[Table[i,Ceiling[.3i]],{i,40}],10]],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Flatten[Table[Table[Table[i,Ceiling[.8i]],{i,40}],10]],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Flatten[Table[Table[Table[i,Ceiling[.1i]],{i,40}],10]],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Flatten[Table[Table[Table[i,Ceiling[.02i]],{i,40}],10]],"Compressed"]//StringPlot
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Left-right scan......
PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Flatten[Table[Table[Table[i,1],{i,40}],10]],"Compressed"]//StringPlot
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SubstitutionSystem[{"A""BB","B""A"},"A",10]
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{A,BB,AA,BBBB,AAAA,BBBBBBBB,AAAAAAAA,BBBBBBBBBBBBBBBB,AAAAAAAAAAAAAAAA,BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB,AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA}
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updatepix[pos_]:=Graphics[MapIndexed[Line[{{#,-First[#2]},{#+1,-First[#2]}}]&,pos]]
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updatepix[Flatten[Table[Table[Table[i,1],{i,40}],5]]]
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updatepix[Take[Flatten[Table[Table[Table[i,Ceiling[.2i]],{i,40}],10]],80]]
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Show[updatepix[Flatten[Table[Table[Table[i,Ceiling[.02i]],{i,40}],10]]],AspectRatio2]
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Show[updatepix[Flatten[Table[Table[Table[i,Ceiling[.2i]],{i,40}],10]]],AspectRatio2]
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PrioritizedSubstitutionSystem[{"B""BB"},"B",Flatten[Table[Table[Table[i,Ceiling[.02i]],{i,10}],10]],"Compressed"]//StringPlot
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Show[updatepix[Mod[Range[1000],40,1]],AspectRatio2]
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Show[updatepix[Mod[2Range[1000],41,1]],AspectRatio2]
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Show[updatepix[Mod[3Range[1000],41,1]],AspectRatio2]
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PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Mod[3Range[1000],41,1],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Mod[2Range[200],41,1],"Compressed"]//StringPlot
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PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Mod[2Range[50],41,1]]//StringPlot
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PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Mod[3Range[300],41,1],"Compressed"]//StringPlot
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Table[PrioritizedSubstitutionSystem[{"A""BB","B""A"},"A",Mod[nRange[300],41,1],"Compressed"]//StringPlot,{n,20}]
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Table[Show[updatepix[Mod[nRange[300],41,1]],AspectRatio2],{n,20}]
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