In[]:=
MultiwaySystem[{"A""AA","B""AB"},{"ABA"},3,"StatesGraph"]
Out[]=
In[]:=
MultiwaySystem[{"A""AA","B""AB"},{"ABA"},5,"StatesGraph"]
Out[]=
In[]:=
FindPath[%,"ABA","AAABA"]
Out[]=
{{ABA,AABA,AAABA}}
In a hypergraph system, the only way to generate distinct objects is to have more elements
JG claim: it’s not enough to just add relations; you have to add elements
Pretty soon you’re only adding multiplicity to relations [and that’s trivial when it comes to applying our rules]
Pretty soon you’re only adding multiplicity to relations [and that’s trivial when it comes to applying our rules]
The combinators case
The combinators case
One intermediate case is labeled trees, rather than graphs [combinators]
Combinators:
In[]:=
s[s[s]][s][s[s][k]][k]
Out[]=
s[s[s]][s][s[s][k]][k]
f[x]{rootf,rootx}
f_[x_]Module[{r},{rf,rx}]
In[]:=
Map[f,s[s[s]][s][s[s][k]][k],-1,HeadsTrue]
Out[]=
f[f[f[f[s][f[f[s][f[s]]]]][f[s]]][f[f[f[s][f[s]]][f[k]]]]][f[k]]
In[]:=
Map[Replace[#,f_[x_]Module[{r},{rf,rx}]]&,s[s[s]][s][s[s][k]][k],-1,HeadsTrue]
Out[]=
{r$715413{r$715410{r$715409s,r$715409{r$715408s,r$715408s}},r$715410s},r$715413{r$715412{r$715411s,r$715411s},r$715412k}}[k]
The analog of the graph is tree rewrites, which normally use labels, but could just use tree structure.......
Lambdas
Lambdas
[ trivially the same as the functional case ]
Functional form
Functional form
f[g[x,y],x]
If you don’t have intermediate variables, you’ll end up with identical nested copies (cf the extra hyperedges)
Claim about mass
Claim about mass
The mass is some function of the number of elements
[But if you break the object in 2, how do you up with half the mass?]
[But if you break the object in 2, how do you up with half the mass?]
Edges slicing a hypersurface
Edges slicing a hypersurface
time vector : lapse function + shift vector (indexes leaves in the foliation)
t
μ
Volume of cone is number of events contained in a cone with a certain length of time vector
(Growth rate [i.e. coefficient of t^2] is being interpreted as curvature)
(Growth rate [i.e. coefficient of t^2] is being interpreted as curvature)
Pick a hyperslice: then there a particular space graph
Now evolve to the next hyperslice. We could count the number of edges in the causal graph getting to the next hyperslice. [Which is comparable to the number of elements changed between hyperslices]
Now evolve to the next hyperslice. We could count the number of edges in the causal graph getting to the next hyperslice. [Which is comparable to the number of elements changed between hyperslices]
C
t
Gets a contribution from whatever put edges in the causal graph at a certain density
C(t)~t^d ( 1 + a t^2 )
Out[]=
How do you add nodes without increasing the dimension?
Action: volume average of subleading terms of
C
t
Consequences of Ct behavior
Consequences of behavior
C
t
If increasing faster than t^d, effectively speed of light increases
Things get more connected; independent experiments get harder
Things get more connected; independent experiments get harder
If approaches t^d .... but there could still be a “dimension wave”
C
t
If there is a lump of increased dimension, what are its equations of motion?
Speed of light change
Speed of light change
Multiplies C(X) by a constant
Momentum conservation
Momentum conservation
Continuity equation on nodes of the causal graph
[aggregate has a certain form; therefore individual piece of C(X) have to compensate each other]
[aggregate has a certain form; therefore individual piece of C(X) have to compensate each other]
Dimension vs. scale
Dimension vs. scale
At a small scale, dimension is not defined
Looking at small scale, there are plenty of dimension fluctuations
Dimensionons
Dimensionons
Can a lump of higher connectivity in the causal graph just dissipate? [Has to follow underlying rule]
Thermal conductivity
Thermal conductivity
How does this vary with dimension?