The Wolfram S Combinator Challenge

Consider a computational model that, rather than work by successively rewriting an expression over and over in a way that honors some equivalence relation over expressions, it works by explicitly building the sequence of such expressions. In that kind of system, every computational state properly contains the previous state. Things grow and grow and never get "deleted". Yet such a system can clearly be universal.

I see the endgame now, thanks guys.

This is somewhat covered in the first link of the background, a Modern Introduction to Combinators:

> So how might this work for S combinator expressions? Basically any sophisticated computation has to live on top of an infinite combinator growth process. Or, put another way, the computation has to exist as some kind of “transient” of potentially unbounded length, that in effect “modulates” the infinite growth “carrier”.

> One would set up a program by picking an appropriate combinator expression from the infinite collection that lead to infinite growth. Then the evolution of the combinator expression would “run” the program. And one would use some computationally bounded process (perhaps a bounded version of a tree automaton) to identify when the result of the computation is ready—and one would “read it out” by using some computationally bounded “decoder”.

I understand it as "you don't need the S combinator application sequence to halt for it to compute something".

In the negative case, it would say the idea doesn't pan out.

In the positive case, it would mean that you can use just S instead of S and K when doing combinator reduction, but doesn't change that this kind of reduction is not super efficient practically speaking.