Chess, one of the oldest and most universally played board games, presents an ideal testbed due to the wealth of both policy data and language data. In terms of policy data, it is reported that over ten million games are played daily on Chess.com, the most frequented online chess platform.

We further transfer those PGNs into text which helps us build mixed game-language datasets from these sources.

Due to the recurrent structure of RNN, the long information propagation path poses limitations in capturing long-term dependencies, gradient explosion/vanishing issues, and inefficient sequential execution. Based on this, we propose a novel paradigm called Parallel Gated Network (PGN) as the new successor to RNN. PGN directly captures information from previous time steps through the designed Historical Information Extraction (HIE) layer and leverages gated mechanisms to select and fuse it with the current time step information. This reduces the information propagation path to $\mathcal{O}(1)$, effectively addressing the limitations of RNN. To enhance PGN's performance in long-range time series forecasting tasks, we propose a novel temporal modeling framework called Temporal PGN (TPGN).

Language models are traditionally designed around causal masking. In domains with spatial or relational structure, causal masking is often viewed as inappropriate, and sequential linearizations are instead used. Yet the question of whether it is viable to accept the information loss introduced by causal masking on nonsequential data has received little direct study, in part because few domains offer both spatial and sequential representations of the same dataset. In this work, we investigate this issue in the domain of chess, which naturally supports both representations. We train language models with bidirectional and causal self-attention mechanisms on both spatial (board-based) and sequential (move-based) data. Our results show that models trained on spatial board states - \textit{even with causal masking} - consistently achieve stronger playing strength than models trained on sequential data. While our experiments are conducted on chess, our results are methodological and may have broader implications: applying causal masking to spatial data is a viable procedure for training unimodal LLMs on spatial data, and in some domains is even preferable to sequentialization.

In absence of reliable domain expertise, one might resort to inferring the latent graph structure, which is often difficult due to the vast search space of possible graphs.

Thank you to all reviewers for the very careful feedback. Corresponding to R4's comment, we find that while symmetrising the pointers is not strictly necessary, empirically In this sense, a more "global" view of each node is desirable.