  Transformers have revolutionized machine learning across diverse domains, yet
understanding their behavior remains crucial, particularly in high-stakes
applications. This paper introduces the contextual counting task, a novel toy
problem aimed at enhancing our understanding of Transformers in quantitative
and scientific contexts. This task requires precise localization and
computation within datasets, akin to object detection or region-based
scientific analysis. We present theoretical and empirical analysis using both
causal and non-causal Transformer architectures, investigating the influence of
various positional encodings on performance and interpretability. In
particular, we find that causal attention is much better suited for the task,
and that no positional embeddings lead to the best accuracy, though rotary
embeddings are competitive and easier to train. We also show that out of
distribution performance is tightly linked to which tokens it uses as a bias
term.
