## 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"]

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,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