Makes you wonder if computers will essentially ruin all the novelty of innovating and discovering new tricks/strategies in age-old games. No more will genius' have their moments of brilliance and improvisation that lead to a stunning victory or upset. No more will commentators debate and regal over phenomenal victories in ages past.
With a computer, it's just the optimal play strategy and you were just playing a god who knew the quickest path to terminating your king. When a human runs with non-standard play strategies in almost any game, it's considered brilliant, innovative, game of the century, etc...
Beating human players in go is like finding a good heuristic for an NP-complete problem. Solving 19*19 go is like proving P!=NP, i.e. we don't even have tools that can approach the problem.
I don't know if there are some theorems, thoughts, philosophies about whether this means it can't be solved, but at least it must be extremely difficult.
[1] https://en.wikipedia.org/wiki/Go_and_mathematics#Legal_posit...
There may be other ways to solve the game, but we don't know what they are. Because we know it is theoretically solvable, we cannot rule out a practical approach to solving it by some mathematical magic even if we have no idea what that would look like.
By analogy, we can prove many things about infinitely many integers by mathematical induction, but if we didn't have that technique, such proofs might seem impossible.
[1] https://en.m.wikipedia.org/wiki/Orders_of_magnitude_(numbers...
Two players write a turing machine with at most n states with two symbols. The player that produced a terminating turing machine that produces the most 1 symbols on the tape before terminating wins.
The optimal strategy for this game is producing a busy beaver, a feat shown not to be computable.
I was thinking that for a fixed n that doesn't really work, because there are only finitely many options, but I guess if n>~2000 , ZFC cannot show the winning strategy to be the winning strategy? Is that what you meant?
Given any two machines which halt, finding the one that ends with more ones is computable. Assuming at least one of the two machines halts, which one wins can be computed in the limit? By which I mean, if the process is allowed to have a "who is currently winning" (the one that already halted if only one has, the one that halted later if both have, or neither if both haven't) thing, and the limit of that is whatever it eventually never switches from. I guess that works even if neither halt.
Uh... I'm just saying stuff you already know to try to think through it myself.
Edit: I guess the question is then, what exactly do we mean by solvable?
Do we mean that there is an algorithm that outputs an optimal move on every turn? For any n, there is such an algorithm. The one that has the correct move hard-coded. Maybe we mean that there is an algorithm that probably always outputs an optimal move? In this case, well, I suppose it depends on the axiom system. Hm.
The trivial case of a Turing machine that can be proven to halt is one with only one state: halted.
Compare this to lines that the humans have memorized after deep analysis, like poisoned pawn Najdorf. The play there certainly doesn't look humanlike either.
What you say sounds plausible, but I do not believe it is backed up by any analysis of the games.
When playing against humans, modern engines can accurately evaluate so much deeper than we can that their play is frequently indistinguishable from the purely tactical play of some such perfect computer (at least, once out of the opening, once the number of "good" moves is constrained a bit allowing computers to evaluate to significantly deeper ply).
You say that computers exhibit a different positional judgment, which is true in the "technically correct" sense (their evaluation function is pretty much the literal definition of their positional judgment) but at depths of 25 ply, 30 ply, or even greater, the simple truth is that they are highly willing to enter lines where their king is exposed, they have pawns doubled, they trade away a good bishop and keep their bad bishop, or all of the above (all what we'd consider anti-positional play) just because they can see a concrete outcome (a won piece, a strong attack, etc.) that we can't.
So yeah, by some definition they play positionally. But it's really by a definition that's only perhaps useful to other computers and not us humans; to us, it's significantly closer in practice to what we consider highly tactical play.
I don't think you can "see 25+ moves ahead" in Go. The branchout factor is just too big.
In a nutshell, positional play in chess is simply heuristics we humans use to be able to evaluate a position in lieu of being able to calculate deep non-forced lines. Computers do use this to an extent (as you point out, we coded their evaluation functions) but positional play matters less when you see all the outcomes of every possible tactic with 100% accuracy. So computers tend to play reasonably human-like in the openings, but by the time you reach the middle game they'll happily enter lines where their pawn structures are shattered, pieces appear superficially to have little coordination, and where their king safety appears compromised (all things humans rarely intentionally do), all because they've seen that it works out 25+ moves in advance.