  Transformer language models have made tremendous strides in natural language
understanding tasks. However, the complexity of natural language makes it
challenging to ascertain how accurately these models are tracking the world
state underlying the text. Motivated by this issue, we consider the task of
language modeling for the game of chess. Unlike natural language, chess
notations describe a simple, constrained, and deterministic domain. Moreover,
we observe that the appropriate choice of chess notation allows for directly
probing the world state, without requiring any additional probing-related
machinery. We find that: (a) With enough training data, transformer language
models can learn to track pieces and predict legal moves with high accuracy
when trained solely on move sequences. (b) For small training sets providing
access to board state information during training can yield significant
improvements. (c) The success of transformer language models is dependent on
access to the entire game history i.e. "full attention". Approximating this
full attention results in a significant performance drop. We propose this
testbed as a benchmark for future work on the development and analysis of
transformer language models.
