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
In[]:=
MultiwaySystem[{"A""AB","B""BA"},"A",5]
Out[]=
{{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}}
In[]:=
Length/@MultiwaySystem[{"A""AB","B""BA"},"A",8]
Out[]=
{1,1,2,4,8,16,32,64,128}
In[]:=
MultiwaySystem[{"A""AB","B""BA"},"A",5,"StatesGraphStructure"]
Out[]=
In an ordinary states graph, is the branching factor bounded?
In[]:=
MultiwaySystem[{"A""AB","B""BA"},"A",5,"AllStatesListUnmerged"]
Out[]=
In[]:=
LayeredGraphPlot[MultiwaySystem[{"A""AB","B""BA"},"A",5,"StatesGraphStructure","IncludeStepNumber"True,"IncludeStateID"True],AspectRatio1/2]
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
In[]:=
RSolve[{n[t]n[t-1]t2,n[1]1},n[t],t]
Out[]=
{{n[t]Pochhammer[1,t]}}
-1+t
2
In[]:=
FunctionExpand[%]
Out[]=
{{n[t]Gamma[1+t]}}
-1+t
2
In[]:=
Length/@MultiwaySystem[{"A""AB","B""BA"},"A",8,"AllStatesListUnmerged"]
Out[]=
{1,1,2,6,24,120,720,5040,40320}
In[]:=
FindSequenceFunction[%133,n]
Out[]=
Pochhammer[1,-1+n]
In[]:=
FunctionExpand[%]
Out[]=
Gamma[n]
In[]:=
Length/@MultiwaySystem[{"A""AB","B""AB"},"A",5,"AllStatesListUnmerged"]
Out[]=
{1,1,2,6,24,120}
In[]:=
Length/@MultiwaySystem[{"A""ABA","B""ABB"},"A",8,"AllStatesListUnmerged"]
Out[]=
{1,1,3,15,105,945,10395,135135,2027025}
In[]:=
MultiwaySystem[{"A""AA"},"A",3,"AllStatesListUnmerged"]
Out[]=
{{A},{AA},{AAA,AAA},{AAAA,AAAA,AAAA,AAAA,AAAA,AAAA}}
In[]:=
Length/@MultiwaySystem[{"A""AA"},"A",8,"AllStatesListUnmerged"]
Out[]=
{1,1,2,6,24,120,720,5040,40320}
In[]:=
Length/@MultiwaySystem[{"A""AAA"},"A",8,"AllStatesListUnmerged"]
Out[]=
{1,1,3,15,105,945,10395,135135,2027025}
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
In[]:=
MultiwaySystem[{"""A","""B"},"",4]
Out[]=
{{},{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}}
Fastest growth in strings is simply #rules^t
In[]:=
MultiwaySystem[{"""A","""B"},"",3,"AllStatesListUnmerged"]
Out[]=
{{},{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}}
In[]:=
Length/@MultiwaySystem[{"""A","""B"},"",5,"AllStatesListUnmerged"]
Out[]=
{1,2,8,48,384,3840}
In[]:=
FindSequenceFunction[%,n]
Out[]=
-1+n
2
In[]:=
FunctionExpand[%]
Out[]=
-1+n
2
Can a rewrite get to every hypergraph (with a certain arity) of a certain size? [Analog of Alexander moves?]
In[]:=
MultiwaySystem[WolframModel[{{u,v}}{{x,y},{a,b}}],{{1,1}},3,"StatesGraph",VertexSize1]
Out[]=
In[]:=
MultiwaySystem[WolframModel[{{u,v}}{{x,y},{u,v}}],{{1,1}},3,"StatesGraph",VertexSize1]
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?
In[]:=
MultiwaySystem[{"""A","""B"},"",4]
Out[]=
{{},{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}}
In[]:=
StringReplace["ABAA",{"""A","""B"}]
Out[]=
AAABAAAAA
In[]:=
GenerationMultiwaySystem[rule_,init_,tgen_]:=NestList[StringReplace[#,rule]&,init,tgen]
In[]:=
GenerationMultiwaySystem[{"""A","""B"},"",5]
Out[]=
{,A,AAA,AAAAAAA,AAAAAAAAAAAAAAA,AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA}
In[]:=
StringLength/@%
Out[]=
{0,1,3,7,15,31}
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)? ]
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
This is the generation path:
[ 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
Generations evolution
Generations evolution
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:
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