#### Claim: Mt(S) never increases faster than exponentially with the generation time t

Claim: (S) never increases faster than exponentially with the generation time t

M

t

MultiwaySystem[{"A""AB","B""BA"},"A",5]

In[]:=

{{A},{AB},{ABA,ABB},{ABAA,ABAB,ABBA,ABBB},{ABAAA,ABAAB,ABABA,ABABB,ABBAA,ABBAB,ABBBA,ABBBB},{ABAAAA,ABAAAB,ABAABA,ABAABB,ABABAA,ABABAB,ABABBA,ABABBB,ABBAAA,ABBAAB,ABBABA,ABBABB,ABBBAA,ABBBAB,ABBBBA,ABBBBB}}

Out[]=

Length/@MultiwaySystem[{"A""AB","B""BA"},"A",8]

In[]:=

{1,1,2,4,8,16,32,64,128}

Out[]=

MultiwaySystem[{"A""AB","B""BA"},"A",5,"StatesGraphStructure"]

In[]:=

Out[]=

In an ordinary states graph, is the branching factor bounded?

MultiwaySystem[{"A""AB","B""BA"},"A",5,"AllStatesListUnmerged"]

In[]:=

Out[]=

LayeredGraphPlot[MultiwaySystem[{"A""AB","B""BA"},"A",5,"StatesGraphStructure","IncludeStepNumber"True,"IncludeStateID"True],AspectRatio1/2]

In[]:=

Out[]=

After t events, not only do we have 2^t strings, we also have distinctness based on the lineage of each string (i.e. what sequence of rules was applied)

From each string at each step, there are (length of string)*(number of rules) possible outcomes

n[t] = n[t-1] len[t] #rules

https://www.wolframscience.com/nks/p937--multiway-systems/

RSolve[{n[t]n[t-1]t2,n[1]1},n[t],t]

In[]:=

{{n[t]Pochhammer[1,t]}}

-1+t

2

Out[]=

FunctionExpand[%]

In[]:=

{{n[t]Gamma[1+t]}}

-1+t

2

Out[]=

Length/@MultiwaySystem[{"A""AB","B""BA"},"A",8,"AllStatesListUnmerged"]

In[]:=

{1,1,2,6,24,120,720,5040,40320}

Out[]=

FindSequenceFunction[%133,n]

In[]:=

Pochhammer[1,-1+n]

Out[]=

FunctionExpand[%]

In[]:=

Gamma[n]

Out[]=

Length/@MultiwaySystem[{"A""AB","B""AB"},"A",5,"AllStatesListUnmerged"]

In[]:=

{1,1,2,6,24,120}

Out[]=

Length/@MultiwaySystem[{"A""ABA","B""ABB"},"A",8,"AllStatesListUnmerged"]

In[]:=

{1,1,3,15,105,945,10395,135135,2027025}

Out[]=

MultiwaySystem[{"A""AA"},"A",3,"AllStatesListUnmerged"]

In[]:=

{{A},{AA},{AAA,AAA},{AAAA,AAAA,AAAA,AAAA,AAAA,AAAA}}

Out[]=

Length/@MultiwaySystem[{"A""AA"},"A",8,"AllStatesListUnmerged"]

In[]:=

{1,1,2,6,24,120,720,5040,40320}

Out[]=

Length/@MultiwaySystem[{"A""AAA"},"A",8,"AllStatesListUnmerged"]

In[]:=

{1,1,3,15,105,945,10395,135135,2027025}

Out[]=

#### Without merging, what is the maximum number of states after t events? (i.e. Mt )

Without merging, what is the maximum number of states after t events? (i.e. )

M

t

#### With merging, what is the maximum number of states after t events? (i.e. Mt )

With merging, what is the maximum number of states after t events? (i.e. )

M

t

Presumably it is exponential in t [e.g. because it simply reaches each state of size c t ]

Note number of hypergraphs of size n grows much faster than the k^n for strings

Note number of hypergraphs of size n grows much faster than the k^n for strings

MultiwaySystem[{"""A","""B"},"",4]

In[]:=

{{},{A,B},{AA,AB,BA,BB},{AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB},{AAAA,AAAB,AABA,AABB,ABAA,ABAB,ABBA,ABBB,BAAA,BAAB,BABA,BABB,BBAA,BBAB,BBBA,BBBB}}

Out[]=

Fastest growth in strings is simply #rules^t

MultiwaySystem[{"""A","""B"},"",3,"AllStatesListUnmerged"]

In[]:=

{{},{A,B},{AA,BA,AA,AB,AB,BB,BA,BB},{AAA,BAA,AAA,ABA,AAA,AAB,ABA,BBA,BAA,BBA,BAA,BAB,AAA,BAA,AAA,ABA,AAA,AAB,AAB,BAB,AAB,ABB,ABA,ABB,AAB,BAB,AAB,ABB,ABA,ABB,ABB,BBB,BAB,BBB,BBA,BBB,ABA,BBA,BAA,BBA,BAA,BAB,ABB,BBB,BAB,BBB,BBA,BBB}}

Out[]=

Length/@MultiwaySystem[{"""A","""B"},"",5,"AllStatesListUnmerged"]

In[]:=

{1,2,8,48,384,3840}

Out[]=

FindSequenceFunction[%,n]

In[]:=

-1+n

2

Out[]=

FunctionExpand[%]

In[]:=

-1+n

2

Out[]=

Can a rewrite get to every hypergraph (with a certain arity) of a certain size? [Analog of Alexander moves?]

MultiwaySystem[WolframModel[{{u,v}}{{x,y},{a,b}}],{{1,1}},3,"StatesGraph",VertexSize1]

In[]:=

Out[]=

MultiwaySystem[WolframModel[{{u,v}}{{x,y},{u,v}}],{{1,1}},3,"StatesGraph",VertexSize1]

In[]:=

Out[]=

Conjecture: with merging, there is no way to grow hypergraphs faster than exponential....

#### With merging, what is the maximum number of states after T generations?

With merging, what is the maximum number of states after T generations?

In a generation, events visit everything....

Two effects: 1. visit each element in each state, 2. visit each state

For a string system, how do we compute the states per generation rather than per event?

MultiwaySystem[{"""A","""B"},"",4]

In[]:=

{{},{A,B},{AA,AB,BA,BB},{AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB},{AAAA,AAAB,AABA,AABB,ABAA,ABAB,ABBA,ABBB,BAAA,BAAB,BABA,BABB,BBAA,BBAB,BBBA,BBBB}}

Out[]=

StringReplace["ABAA",{"""A","""B"}]

In[]:=

AAABAAAAA

Out[]=

GenerationMultiwaySystem[rule_,init_,tgen_]:=NestList[StringReplace[#,rule]&,init,tgen]

In[]:=

GenerationMultiwaySystem[{"""A","""B"},"",5]

In[]:=

{,A,AAA,AAAAAAA,AAAAAAAAAAAAAAA,AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA}

Out[]=

StringLength/@%

In[]:=

{0,1,3,7,15,31}

Out[]=

## Observer’s version

Observer’s version

In an elementary time (as measured by the observer), every update can happen in parallel

In terms of states in the multiway system, there is a certain rate of branching/expansion in total number of states per generation

[in a generation, everything got updated]

[in a generation, everything got updated]

Our impression of time is based on the progression of branchlike surfaces

How many κ-ings have there been in total universe states? [ How many separate end-to-end events have occurred that affect any given spatial node (or full state node)? ]

How many κ-ings have there been in total universe states? [ How many separate end-to-end events have occurred that affect any given spatial node (or full state node)? ]

In[]:=

8.1×

60

10

Out[]=

%/Sqrt[10.^116]

In[]:=

806.468

Out[]=

#### tgen is the generation counter

tgen is the generation counter

Perceived generation time interval : time per generation as perceived by an observer [numerically tbar in cosmological frame] < could sculpt their frame to give a different perceived time > < like with time dilation, can force events to be sequential by you moving through branchial space, as opposed to just letting things happen around you >

Generation time interval : all events happen in parallel (tbar)

Generation time interval : all events happen in parallel (tbar)

Generation counter = tH / tbar

MultiwaySystem[{"A""AB","B""BA"},"A",5,"StatesGraph"]

In[]:=

Out[]=

LayeredGraphPlot[MultiwaySystem[{"A""AB","B""BA"},"A",5,"EvolutionEventsGraph"],AspectRatio1/2]

In[]:=

Out[]=

This is the generation path:

NestList[StringReplace[#,{"A""AB","B""BA"}]&,"A",5]

In[]:=

{A,AB,ABBA,ABBABAAB,ABBABAABBAABABBA,ABBABAABBAABABBABAABABBAABBABAAB}

Out[]=

MultiwaySystem[{"A""AB","B""BA"},"A",5]

In[]:=

{{A},{AB},{ABA,ABB},{ABAA,ABAB,ABBA,ABBB},{ABAAA,ABAAB,ABABA,ABABB,ABBAA,ABBAB,ABBBA,ABBBB},{ABAAAA,ABAAAB,ABAABA,ABAABB,ABABAA,ABABAB,ABABBA,ABABBB,ABBAAA,ABBAAB,ABBABA,ABBABB,ABBBAA,ABBBAB,ABBBBA,ABBBBB}}

Out[]=

StringJoin/@%

In[]:=

{A,AB,ABAABB,ABAAABABABBAABBB,ABAAAABAABABABAABABBABBAAABBABABBBAABBBB,ABAAAAABAAABABAABAABAABBABABAAABABABABABBAABABBBABBAAAABBAABABBABAABBABBABBBAAABBBABABBBBAABBBBB}

Out[]=

[ Should have been joined with an overlap of 1 ]

We could get this by on each layer in the evolution events graph by sorting the events in which they sample the “generational string”

#### What are the generations

What are the generations

MultiwaySystem[{"A""AB","B""BA"},"A",5]

In[]:=

{{A},{AB},{ABA,ABB},{ABAA,ABAB,ABBA,ABBB},{ABAAA,ABAAB,ABABA,ABABB,ABBAA,ABBAB,ABBBA,ABBBB},{ABAAAA,ABAAAB,ABAABA,ABAABB,ABABAA,ABABAB,ABABBA,ABABBB,ABBAAA,ABBAAB,ABBABA,ABBABB,ABBBAA,ABBBAB,ABBBBA,ABBBBB}}

Out[]=

#### Generations evolution

Generations evolution

GenerationMultiwaySystem[rule_,init_,tgen_]:=NestList[StringReplace[#,rule]&,init,tgen]

In[]:=

GenerationMultiwaySystem[{"""A","""B"},"",5]

In[]:=

{,A,AAA,AAAAAAA,AAAAAAAAAAAAAAA,AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA}

Out[]=

StringPosition["ABBBAAABABAAAB","AA",OverlapsAll]

In[]:=

{{5,6},{6,7},{11,12},{12,13}}

Out[]=

NonOverlapTest[list_]:=Catch[(Scan[If[#[[2,1]]≤#[[1,2]],Throw[False]]&,Partition[list,2,1]];True)]

In[]:=

FindGenerationUpdates[string_,lhs_]:=Module[{u=StringPosition[string,lhs,OverlapsAll]},u=Select[Subsets[u],NonOverlapTest];With[{m=Max[Length/@u]},Select[u,Length[#]m&]]]

In[]:=

FindGenerationUpdates["ABBBAAABABAAAB","AA"]

In[]:=

{{{5,6},{11,12}},{{5,6},{12,13}},{{6,7},{11,12}},{{6,7},{12,13}}}

Out[]=

FindGenerationUpdates["AAAAAAAAAAA","AA"]

In[]:=

Out[]=

NumberLinePlot/@%278

In[]:=

Out[]=

#### Claim: we can make branchlike surfaces that capture all states which, by simple gluing, will give the states at that generation

Claim: we can make branchlike surfaces that capture all states which, by simple gluing, will give the states at that generation

Every different update event leads to a different branch.

Some different update events are spacelike separated as well [lack of causal dependence] <spacelike separation of events implies branchlike separation>

Some different update events are spacelike separated as well [lack of causal dependence] <spacelike separation of events implies branchlike separation>

Given a branchial graph:

MultiwaySystem[{"A""AB","B""BA"},"A",4,"BranchialGraph"]

In[]:=

Out[]=

MultiwaySystem[{"A""AB","B""BA"},"A",3,"BranchialGraph"]

In[]:=

Out[]=

## How close can time dilation get to infinite?

How close can time dilation get to infinite?

Size of spacelike hypersurface * elementary time i.e. the maximum gamma is #nodes in spacelike hypersurface