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Fishing out of the combinator soup
Fishing out of the combinator soup
x⊕y==y⊕x
a⊕((b⊕c)⊕(d⊕(e⊕f)))(((f⊕e)⊕d)⊕(c⊕b))⊕a
Consider combinator equations rather than combinator reductions....
Consider combinator equations rather than combinator reductions....
In[]:=
ResourceFunction["MultiwayCombinator"][{s[x_][y_][z_]->x[z][y[z]],k[x_][y_]->x},ResourceFunction["EnumerateCombinators"][4,{s,k}],4,"StatesGraph"]
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In[]:=
ResourceFunction["MultiwayCombinator"][{s[x_][y_][z_]->x[z][y[z]],k[x_][y_]->x,x_:>Module[{y},k[x][y]],x_[z_][y_[z_]]->s[x][y][z]},k,2,"StatesGraph"]
Out[]=
In[]:=
ResourceFunction["MultiwayCombinator"][{s[x_][y_][z_]->x[z][y[z]],k[x_][y_]->x,x_:>Module[{y},k[x][y]],x_[z_][y_[z_]]->s[x][y][z]},k,3,"StatesGraph"]
Out[]=
In[]:=
ResourceFunction["MultiwayCombinator"][{s[x_][y_][z_]->x[z][y[z]],k[x_][y_]->x,x_:>Module[{y},k[x][y]],x_[z_][y_[z_]]->s[x][y][z]},k,3,"StatesGraphStructure"]
Out[]=
In[]:=
ResourceFunction["MultiwayCombinator"][{s[x_][y_][z_]->x[z][y[z]],k[x_][y_]->x,x_:>Module[{y},k[x][y]],x_[z_][y_[z_]]->s[x][y][z]},k,4,"StatesGraphStructure"]
Out[]=
We could just enumerate axiom systems, but there’s messiness in saying whether something is unary, binary, etc.
We could just enumerate axiom systems, but there’s messiness in saying whether something is unary, binary, etc.
Even if we curry, we’d still have to name the CirclePlus or whatever....
Eventually we will be able to identify computations that correspond to recognizably similar things ... e.g. Plus[1,2] vs Plus[3,4]
Eventually we will be able to identify computations that correspond to recognizably similar things ... e.g. Plus[1,2] vs Plus[3,4]
Each node is a possible mathematical statement (and there has to be some combinator way to represent )
Each node is a possible mathematical statement (and there has to be some combinator way to represent )
In[]:=
VertexList
“Ocean-based computation”
“Ocean-based computation”
With sampling....
This is directly using with its meaning inserted from the beginning
This is directly using with its meaning inserted from the beginning
Alternative: define everything in terms of combinators, including
Alternative: define everything in terms of combinators, including
This is generating lots of semantic expressions
This is generating lots of semantic expressions
To know whether one is “true for a given axiom system” ... find the axiom system in the collection of combinator expressions ... and then look at its entailment cone
Levels of interpretation:
Levels of interpretation:
“That combinator subtree (which occurs often) can be interpreted as Plus[x,y]”
“That combinator subtree (which occurs often) can be interpreted as Plus[x,y]”
That entailment cone is based on a certain initial axiom (which is just combinator combinator)
That entailment cone is based on a certain initial axiom (which is just combinator combinator)
[[[ E.g. for interpretation we can look at a CA example ]]]
Naming the Ultimate Elements
Naming the Ultimate Elements
primon
utom
origon
uratom
utom
origon
uratom
ur
urom
urom
urment
urem
ureme
urem
ureme
eme
emic
emian
emial
emical
emish
emian
emial
emical
emish
“emes of space”
“emes of metamathematics”
“emian level”
repeated emic clusters that represent nameable constructs