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WOLFRAM NOTEBOOK
Types of Separation
Types of Separation
Timelike
Timelike
Successive function application < Apply an operation at all possible times >
In[]:=
ResourceFunction["TraceGraph"][1+(1+(1+1))]
Out[]=
In[]:=
ResourceFunction["TraceGraph"][1+(1+(1+1)),_Plus]
Out[]=
f[g[x]]
In[]:=
ResourceFunction["TraceGraph"][f[g[x]],TraceOriginal->True]
Out[]=
{a->b,b->c}
ResourceFunction["TraceGraph"][(#+1)&,TraceOriginal->True]
In[]:=
f[1]=1;f[x_]:=f[x-1]
In[]:=
ResourceFunction["TraceGraph"][f[4]]
Out[]=
In[]:=
ResourceFunction["TraceGraph"][f[4],f[_]]
Out[]=
In[]:=
Clear[f]
Timelike can terminate [cf black holes]
[ Can measure timelike distance by counting events ]
Spacelike
Spacelike
[ Operations applied at different positions in the expression ] < Apply an operation in all possible places >
In[]:=
ResourceFunction["TraceGraph"][{1+1,1+1,1+1,1+1}]
Out[]=
In[]:=
Clear[f]
In[]:=
f[n_]:=f[n-1]+f[n-2]
In[]:=
f[1]=f[2]=1;
In[]:=
ResourceFunction["TraceGraph"][f[5],_f]
Out[]=
Different ways to evaluate this ... with depth-first, breadth-first, etc.
[ SMP attributes determine equivalencing ]
[ The “states” are f[x1]+f[x2]+f[x3]+ ....; the individual f[x]’s are like tokens ]
[Can measure spacelike distance by looking for common ancestors AKA shared light cones]
(Only works when space exists.. Spacelike distances are consistent when there is some uniformity in the system...)
(Only works when space exists.. Spacelike distances are consistent when there is some uniformity in the system...)
[ Standard WL evaluation is normally “generational evolution” ] < Unless you have Return[ ] in the middle >
If some positions terminate and others do not, there is a BH at the terminating positions....
Branchlike
Branchlike
[ Operations applied in multiple different ways ] < Apply an operation in all possible ways >
Pattern overlaps ; multiple outputs ; ( multiple rules )
(cf automated theorem proving) (quintessential NP case) [cf multiway Turing machines]
[ “Factoring graph” shows different trial divisions ... ] [? reverse of graph above]
[ If you are maintaining the Multi, there’s no reason to have an event selection order ]
Rulelike
Rulelike
[ Different operations applied ] < Apply all possible operations > [[ Given a universe of all possible rules ]]
Multiple rules
? different sorting networks
[Treelike]
[Treelike]
< Breaks the independence of possible events. > < Does not factor DoFs >
Can determine treelike vs. spacelike separation by looking at part numbers
Structural dependence
The Concept of Multi
The Concept of Multi
A Multi could store a superposition of paths (AKA a slice across this graph)
In this case there is no entanglement; each element of the Multi can be handled separately... [like spacelike separation] < There are 1-token states here ... >
should give entangled Multi’s, where you have to keep all the elements in the Multi together to be able to compute the next step....
[ For more optimization, the Multi should store the “next events”, so it doesn’t have to recompute from the states what events are possible, and potentially redo them ... ] < Effectively needs to store matches >
Everything as a Hypergraph
Everything as a Hypergraph
Can get rid of treelike separation. “Linearize” everything.
Cost is that you have to introduce e.g. IDs to prevent “wrong” matches [ The IDs are just pointers in internal implementations ... ]
Representing the Process of Evaluation
Representing the Process of Evaluation
[ Write the computation graph, not a loop ... ] [ cf CTCs ]
[ Works best when it is pure feed-forward ... ]
[ Works best when it is pure feed-forward ... ]
Computation graph = causal graph
Infinite Computations
Infinite Computations
Symbolic representation of an infinite computation graph / AKA causal graph
As a practical matter, needs to be terminated; what does the termination front look like?
Does this terminate?
Cf. some combinator evaluations will terminate; others will not....
Like there is a branchial black hole + some infinite branchial space [?]
Branchial black hole = “inevitable measurement” (AKA objective collapse)
Black holes are always about a failure of choice:
timelike BH : no operation at all to apply [no “time” motion]
spacelike BH : no choice of where to apply the rule (cycles forever) [no spatial motion]
branchlike BH : no choice of which branch [no branchial motion]
rulelike BH : no choice of which rule [ possibly: one reached the null element ?? ]
timelike BH : no operation at all to apply [no “time” motion]
spacelike BH : no choice of where to apply the rule (cycles forever) [no spatial motion]
branchlike BH : no choice of which branch [no branchial motion]
rulelike BH : no choice of which rule [ possibly: one reached the null element ?? ]