iota combinator

[calcnerd256]

I haven't posted here in awhile. It has been brought to my attention that there are only 35 Google search results for "iota combinator". This needs to be corrected

Comments

history

[calcnerd256]

Any discussion of the history of the combinator should go into this thread.

Re: history

[calcnerd256]

The history of the iota combinator begins with the history of the lambda calculus.

Re: history

[calcnerd256]

A linguist by the name of Chris Barker found that one can simplify the SKI combinator from two symbols (S and K) to just one: a function that takes a parameter, passes S to it, and then passes K to the result of that.

lemmas

[calcnerd256]

Here is a thread to post any useful combinators built from trees of iota.

functional group

[calcnerd256]

ι ι = I = λ x . x
ι I = Church's 0 = Church's false = λ f x . x
ι 0 = K = Church's true = λ x y . x
ι K = S = λ f g x . f x (g x)
ι S = λ x y . x y x
ι (ι S) = I

Re: functional group

[calcnerd256]

∃ a cyclic group with the function ι that generates the following sequence of elements (and repeats):
I, 0, K, S, ιS

Re: functional group

[calcnerd256]

note that although ι is not an element of the group, ιι does produce the first element

[anonymous]

Should totally turn this into a rosetta code entry.

Scheme:

Haskell:

Ruby:

Python:

:D

[calcnerd256]

//Javascript:
function iota(x){
return x(
function S(f){
return function(g){
return function(y){
return f(y)(g(y))
};
};
}
)(
function K(y){
return function(){return y;};
}
);
}
//could be better: those closures are a nightmare to debug