Definition of universality

By giving initial conditions, one can set it up so that some history reaches a “success” state.
​
Can other, incorrect histories also reach success states? What about reaching a success state later?
https://www.wolframscience.com/nks/p759--undecidability-and-intractability/ :
{445,461,1512}
In[]:=
RulePlot[TuringMachine[445],{{1,9},IntegerDigits[11,2,10]},7]
Out[]=

Typical version: there are many possible answers; each “success” history is a valid answer....

E.g. factoring:
E.g. success could be recognized by halting....

In general, pick the “success” criterion.... More natural to have halting as a success criterion in an NDTM than a DTM.

Multispace visualization

In[]:=
ResourceFunction["MultispacePlot3D"][ResourceFunction["MultiwayTuringMachine"][{1507,2506,3506},{{1,1,0},{0,1,0,1}},4,##]&,"Graph"]
ConstantArrayLayer
:"ConstantArrayLayer setting has been deprecated. Use NetArrayLayer setting instead.
ConstantArrayLayer
:"ConstantArrayLayer setting has been deprecated. Use NetArrayLayer setting instead.
NetGraph
:Incompatible ranks for output of layer "Qs" (​a matrix​) and input "D" of layer "subNet" (​a rank-3 array​).
NetInitialize
:First argument to NetInitialize should be a net or list of nets.
NetTrain
:First argument to NetTrain should be a fully specified net.
NetExtract
:First argument $Failed should be a net, NetEncoder, or NetDecoder.
NetExtract
:First argument $Failed should be a net, NetEncoder, or NetDecoder.
NetExtract
:First argument Normalize[$Failed] should be a net, NetEncoder, or NetDecoder.
General
:Further output of NetExtract::arg1 will be suppressed during this calculation.
Out[]=
In[]:=
MultispaceTM[rule_,t_]:=ResourceFunction["MultispacePlot3D"][ResourceFunction["MultiwayTuringMachine"][rule,{{1,t+1,0},Table[0,2t+1]},t,##]&,"Graph"]
In[]:=
MultispaceTM[rule_,t_,case_]:=ResourceFunction["MultispacePlot3D"][ResourceFunction["MultiwayTuringMachine"][rule,{{1,t+1,0},Table[0,2t+1]},t,##]&,case]
In[]:=
MultispaceTM[{{{1,0}{1,0,1}},{{1,0}{1,1,-1}},{{1,1}{1,0,-1}}},10]
ConstantArrayLayer
:"ConstantArrayLayer setting has been deprecated. Use NetArrayLayer setting instead.
ConstantArrayLayer
:"ConstantArrayLayer setting has been deprecated. Use NetArrayLayer setting instead.
NetGraph
:Incompatible ranks for output of layer "Qs" (​a matrix​) and input "D" of layer "subNet" (​a rank-3 array​).
NetInitialize
:First argument to NetInitialize should be a net or list of nets.
NetTrain
:First argument to NetTrain should be a fully specified net.
NetExtract
:First argument $Failed should be a net, NetEncoder, or NetDecoder.
NetExtract
:First argument $Failed should be a net, NetEncoder, or NetDecoder.
NetExtract
:First argument Normalize[$Failed] should be a net, NetEncoder, or NetDecoder.
General
:Further output of NetExtract::arg1 will be suppressed during this calculation.
Out[]=
In[]:=
MultispaceTM[{{{1,1}{1,0,-1}},{{1,0}{2,0,1}},{{2,1}{2,1,1}},{{2,0}{1,1,-1}},{{2,1}{2,0,1}}},20]
ConstantArrayLayer
:"ConstantArrayLayer setting has been deprecated. Use NetArrayLayer setting instead.
ConstantArrayLayer
:"ConstantArrayLayer setting has been deprecated. Use NetArrayLayer setting instead.
NetGraph
:Incompatible ranks for output of layer "Qs" (​a matrix​) and input "D" of layer "subNet" (​a rank-3 array​).
NetInitialize
:First argument to NetInitialize should be a net or list of nets.
NetTrain
:First argument to NetTrain should be a fully specified net.
NetExtract
:First argument $Failed should be a net, NetEncoder, or NetDecoder.
NetExtract
:First argument $Failed should be a net, NetEncoder, or NetDecoder.