> The S, K combinators defined by Moses Schönfinkel on December 7, 1920, are together known to be computation universal. On December 7, 2020, Stephen Wolfram made the suggestion that S alone might also be universal.
I wonder how long in advance Stephen Wolfram first had this thought and waited until the centennial to publicize the suggestion.
That doesn't matter.
You could imagine a system that accumulates two terms: (actual result, junk). Instead of deleting something it just adds it to the junk part of the pair.
Maybe the junk part itself has computation which never ends, but it doesn't matter because you just extract your result from the left part of the pair.
cvoss, ezwoodland, tromp and v64 made a good point. As v64 points I was thinking of a Combinatory Completeness not Turing Completeness. Dropping that requirement as cvoss, ezwoodland, tromp point you can simulate deletion (that's what quantum computing does btw see No-deleting theorem).
I don't follow your argument. Why must we have the ability to "delete" sub-expressions?
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.
as far as I understand it, turing completeness is a weaker property than combinatory completeness, which Craig's theorem is addressing. The nonexistence of a singleton combinatory basis doesn't necessarily imply the nonexistence of a turing complete combinator.
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.
I was thinking in specifically the positive, would compression or encoding potentially allow for more compact representation of programs. Like Kolmogorovs
No not really. It's a thing you can do just because you can, like writing a book without the letter "e". Closer to the context of computation, it's like building a computer using only NAND gates. Or, given how restrictive the S combinator supposedly is, using only AND and OR gates. It won't make an efficient computer because your design is convoluted to all hell to make up for your choice to never use a NOT gate. Even the one made from NAND isn't efficient, because even though NAND can make any circuit without being too convoluted, it's not the most efficient way to make all circuits.
I wonder how long in advance Stephen Wolfram first had this thought and waited until the centennial to publicize the suggestion.
This can be proved by induction. Or you can cite Craig's theorem (the less known one) for that. See [1]
Honestly, I don't see the endgame here.
[1] https://math.stackexchange.com/questions/839926/is-there-a-p...
I see the endgame now, thanks guys.
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.
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.