In[]:=
Groupings[Table[s,6],Construct2]
Out[]=
{s[s][s][s][s][s],s[s[s][s][s][s]],s[s[s][s][s]][s],s[s[s[s][s][s]]],s[s[s][s]][s][s],s[s[s[s][s]][s]],s[s[s[s][s]]][s],s[s[s[s[s][s]]]],s[s[s]][s][s][s],s[s[s[s]][s][s]],s[s[s[s]][s]][s],s[s[s[s[s]][s]]],s[s[s[s]]][s][s],s[s[s[s[s]]][s]],s[s[s[s[s]]]][s],s[s[s[s[s[s]]]]],s[s][s[s]][s][s],s[s[s][s[s]][s]],s[s[s][s[s]]][s],s[s[s[s][s[s]]]],s[s][s][s[s]][s],s[s[s][s][s[s]]],s[s][s[s][s]][s],s[s[s][s[s][s]]],s[s[s]][s[s]][s],s[s[s[s]][s[s]]],s[s][s[s[s]]][s],s[s[s][s[s[s]]]],s[s][s][s][s[s]],s[s][s[s][s][s]],s[s[s][s]][s[s]],s[s][s[s[s][s]]],s[s[s]][s][s[s]],s[s][s[s[s]][s]],s[s[s[s]]][s[s]],s[s][s[s[s[s]]]],s[s][s[s]][s[s]],s[s][s[s][s[s]]],s[s][s][s[s][s]],s[s][s][s[s[s]]],s[s[s]][s[s][s]],s[s[s]][s[s[s]]]}
In[]:=
CombinatorExpressionGraph[#,"MatchHighlighting"True]&/@Groupings[Table[s,5],Construct2]
Out[]=
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In[]:=
Graph[CombinatorToDAG[#],VertexLabelsAutomatic]&/@Groupings[Table[s,5],Construct2]
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In these DAGs, the name of every vertex is determined by the lower-level items in the DAG.
Jonathan’s version of operator isomorphism:
operators [fixed]
constants [fixed] + free variables
bound variables [renameable]
operators [fixed]
constants [fixed] + free variables
bound variables [renameable]
In[]:=
ResourceFunction["MultiwayOperatorSystem"][f[x_,y_]f[y,x],f[f[a,b],f[c,d]],3,"StatesGraph"]
Out[]=
In[]:=
ResourceFunction["MultiwayOperatorSystem"][f[x_,y_]f[y,x],f[f[a,b],f[b,a]],3,"StatesGraph"]
Out[]=
In[]:=
ResourceFunction["MultiwayCombinator"][{s[x_][y_][z_]x[z][y[z]],k[x_][y_]x},s[k[s][i]][s[k][k][i]][x][y],6,"StatesGraph"]
Out[]=
Causal dependence for WMs: “tokens” are hyperedges AKA expressions (which are knitted by shared atoms)
[an event produces new hyperedges, which still mostly contain the same atoms]
< There can be two hyperedges that involve the exact same atoms >
[an event produces new hyperedges, which still mostly contain the same atoms]
< There can be two hyperedges that involve the exact same atoms >
Causal dependence for strings: tokens are characters [a given event consumes characters and emits new ones] <characters have to know their ordering in the string> [like a linked list where they need to know]
Combinators / operator system:
token is a subexpression; an event consumes subexpressions
s[x][y][z] -> x[z][y[z]]
token is a subexpression; an event consumes subexpressions
s[x][y][z] -> x[z][y[z]]
Lemma-ization requires causal dependence
Combinator lemma: CaCb [cache of combinator dependence]
Do the RHS x, y have causal dependence on the LHS x, y
Timelike separation, spacelike separation, treelike separation, branchlike separation
Treelike separation in hyperbolic space: e.g. radial displacement in the Poincare disk
Treelike separation vs. scale relativity
This terminates after 26 steps:
26 step halting:
52 step halting:
Innermost order
Innermost order
How local in space or treespace does the head remain?
Alternative approach: “generational updating” (AKA /. updating)
With ∞ there could be an overlap in what is being rewritten
Expression = redex + residuals
Expression = redex + residuals
Usually failures of hyperbolicity are things like closed timelike curves, i.e. cycles in the causal graph.....
Degenerate case of failure of hyperbolicity:
Normally A -> B -> C
[Forces some edge to be zero time]
Normally A -> B -> C
[Forces some edge to be zero time]