# Combinator space

Combinator space are terms of combinators which formed a topology space, it is one kind of program space.

## Plain examples

With examples of arithmetic expression space and Church encoding of rational numbers, we can translate arithmetical expression space into a space of terms very intuitively (technically we need handle constructions like Dedekind cut).

## One step further

In SKI system,

 $\displaystyle{ I^n x = x }$ (1)

So we can easily add a tail on each point of any arithmetical expression space, each tail is a line of discrete points. So the tail bundle of arithmetical expression space is another geometry construction.

Even further, we notice $\displaystyle{ I^n }$ here is a group, we can introduce other simple terms groups, to make more examples.

Here we have a program $\displaystyle{ \mathbb{D} }$ can calculate distance of two terms in corresponding arithmetical expression space, we call it distance-inquiry program $\displaystyle{ \mathbb{D} }$, this program can introduce a proper topology on a bigger domain of terms than only the Church numerals.

Xi Li pointed out that, the above idea of a distance-inquiry program $\displaystyle{ \mathbb{D} }$ can only works syntactically, because Rice theorem is the final obstacle. If any extension of the term domain involved semantic properties, it leads to halting problem.

Distance-inquiry program $\displaystyle{ \mathbb{D} }$ can introduce various fibers on the base of arithmetical expression space.

## Discussions

We need to distinguish the two concepts: geometrical computing system and program space. For example, the billiard system can implement a torfilli gate, so it is a geometrical computing system. Even Babbage's unimplemented design is also a geometrical computing system based on mechanics. But combinator space belongs to the second concept, it is a collection of program which have a non-trivial geometry structure.

If we combine geometrical computing system and program space, can they form a new unified geometry space? are they compatible with each other? what operations are available on the space?