> https://en.m.wikipedia.org/wiki/Reversi
> Dark must place a piece (dark-side-up) on the board and so that there exists at least one straight (horizontal, vertical, or diagonal) occupied line between the new piece and another dark piece, with one or more contiguous light pieces between them
https://content.wolfram.com/uploads/sites/43/2023/02/sw02142... from https://writings.stephenwolfram.com/2023/02/what-is-chatgpt-...
I imagine it's technically possible to do this in a piecewise manner that doesn't "understand" the larger board. This could theoretically be done with number lines, and not a geometry (i.e. the 8x8 grid and current state of each square mentioned in the comment you replied to). It could also be done in a piecewise manner with three ternary numbers (e.g. 1,0,-1) for each 3 square sets.
I guess this is a kind of geometric representation on the order of Shannon's Theseus.
The material difference is one of scale, not complexity.
Your rules have lookback = 1, while the Othello rules have lookback <= 63 and if you, say, are trying to play A1, you need to determine the current color of all squares on A1-A8, A1-H1, and A1-H8 (which is lookback <= 62) and then determine if one of 21 specific patterns exists.
Both can be technically be modeled with a lookup table, but for Othello that table would be size 3^63.
Could you just generate the subset you need denovo each time? Or the far smaller number of 1-dimensional lines?
There is no way to represent the state of the game without some kind of board model.
So any coherent representation of a sequence of valid game states can be used to infer the game board structure.
GPT is not constructing the board representation: it is looking at an example game and telling us what pattern it sees. GPT cannot fail to model the game board, because that is all it has to look at in the first place.
I agree with the conclusion but not the premise.
The question under debate is about not just a stateful ternary board X but a board endowed with a metric (X, d) that enables geometry.
There are alternative ways you can represent the state without the geometry: such as, an ordered list of strings S = ["A1", "B2", ...] and a function Is-Valid(S) that returns whether S is in the language of valid games.
Related advice: don't get a math degree unless you enjoyed the above pedantry.
But that data is more specific than the set of all possible ordered lists of strings: it's a specific representation of an example game written as a chronology of piece positions.
GPT models every pattern it can find in the ordered list of tokens. GPT's model doesn't only infer the original data structure (the list of tokens). That structure isn't the only pattern present in the original data. There are also repeated tokens, and their relative positions in the list: GPT models them all.
When the story was written in the first place, the game rules were followed. In doing so, the authors of the story laid out an implicit boundary. That boundary is what GPT models, and it is implicitly a close match for the game rules.
When we look objectively at what GPT modeled, we can see that part of that model is the same shape and structure as an Othello game board. We call it a valid instance of an Othello game board. We. Not GPT. We. People who know the symbolic meaning of "Othello game board" make that assertion. GPT does not do that. As far as GPT is concerned, it's only a model.
And that model can be found in any valid example of an Othello game played. Even if it is implicit, it is there.
