I’m currently studying data structure, regular grids from strings (m¹) to (mⁿ), and topologies of non-regular grid n-dimensional database topologies (tensors, etc.) to probabilistic database and potential quantum databases.

So of course I had to start with Gödel databases (prime factorization) which have the useful quality of reversibility, which can be a function of quantum databases. (QTMs are interesting because they have infinite dimensional state spaces.)

It occurs to me that there are a set of games where the tree maps to the boardstate, nimbers as an example. Each nimber is an address, a location in a linear array. But it’s not natively reversible. Unless, perhaps, we use prime factorization to extend the values?

I must consider this more because, if such a set of games exists, I’d like to see how they extend into n-dimensions. The thrust of my paper is that, now we’ve extended data structure into so many robust new models, it might be good to examine how we ultimately represent, and order them as a set of physical states, from the perspective of a cost-function.²

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PS: I’ve already come to see sudoku as a Shannon database via the entropy of any given board state, and plan to see what Von Neumann might have had to say about it. I seem to recall that the math for some of the work in MOLS was similar in spirit to QT, and I’ve been finding that myself. Let me know if you have any thoughts on that or suggestions for further math reading.

The schedule will appear on the faculty scheduled list of lectures. I don’t have the information yet.

]]>– Aeron ]]>

Very interesting!

]]>It seems to me that this version of recursive chess should result in some meaningful game. Furthermore, it should have the property that the main game alone should look, for observers not aware of the recursive sub-games, like a valid classical game of chess which is played reasonably by the two players. It also seems that it might be possible to prove something along the line that the corresponding classical game can only be won (given a state) if the recursive one can be won, or vice versa. However, there is some non-trivial sequence dependency (there might be two valid next moves for which each a piece would be captured next, and for each of which the to be captured piece might be able to prove that the other one should be taken). ]]>

A little change in the game play which prevents infinite recursion: So, the main game is played by the kings, which is the reason they must not die. Assume a capture is attended, and for simplicity assume white tries to capture a piece of black. Then, a sub-game played by the two involved pieces is started given that the black piece (the one which should be captured and thus would “die”) is yet not already playing a game further up the recursion (see below why this makes sense). This sub-game starts one turn before the current state of its super-game, i.e. the last move of black is reset) and has the same rules as the main game, except that the sub-game is lost if the respective playing piece is lost OR if any piece playing a super-game is lost. In other words, the playing pieces become _additional_ “proto-kings”. Obviously, the recursion is finite since, at one point, every piece is a proto-king and thus no sub-game can be started anymore.

It makes sense that a sub-game is only started if the piece which should be captured is not yet a proto-king because, in the current super-game, the player already tries to protect this piece at all cost.

If the white capturing piece wins the sub-game, the capture is successfull and the super-game continues normally. If the black piece which should be captured wins the sub-game, or if there is a draw, the last turn of black in the super-game is reset (i.e. the super-game is set to the start-position of the sub-game). However, the black player is not allowed to make the same move in this turn again. As always, if a player cannot make a valid move anymore, he looses the game.

The underlying idea is very simple: The piece which should be captured forsees that it would be captured and “vetoes” the last move of the super-game given that it can prove (by playing the sub-game) that there is the possibility to at least achieve a draw in the super-game without itself being captured. Conversely, a piece only accepts a move after which it is captured if it is anyway captured under all possible moves or if the super-game would be lost otherwise.

Ah, I get it now – I guess I was biased after looking at that N and forced my own intuition! Classic

]]>Actually, it is not true that every finite order can be generated. Indeed, there is an order with four elements that cannot. The earlier commentator has got it, if you can take the hint there.

]]>Yes, thank you! I have now edited.

]]>Another commenter suggested $\mathbb{N}$ as the smallest order, and you answered this is correct, but isn’t the countable antichain (i.e. identity, or $1 \sqcup 1 \sqcup 1 \cdots$) arguably “smaller” than $\mathbb{N}$ (or any other infinite order, at least in my meaning above, as it will trivially be a suborder of any of those)?

Really enjoying this series of posts on order theory!

]]>Nitpick: is there a typo in this sentence?

“One generally conceives of the finite sets as clustering near the Earthly bottom of the lattice, finitely close to $\emptyset$, whereas the coinfinite sets soar above, finitely close to the heavenly top”

I believe “coinfinite” should be “cofinite” here.

]]>Thank you for your answer, which is correct. Can you answer my other questions?

]]>What an extremely interesting idea! It would lead to a radical change in game play, since we’d have temporarily to think of a particular pawn or bishop as proto king. But I think this would actually be quite interesting to do.

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