Limit Analysis for Symbolic Multi-step Reasoning Tasks with Information Propagation Rules Based on Transformers

Transformers are able to perform reasoning tasks, however the intrinsic mechanism remains widely open. In this paper we propose a set of information propagation rules based on Transformers and utilize symbolic reasoning tasks to theoretically analyze the limit reasoning steps. We show that the limit number of reasoning steps is between $2^{L-1}-1$ and $\frac{3^{L-1}-1}{2}$ for a model with $L$ attention layers in a single-pass. In addition, we find that, for problems with a fixed number of reasoning steps, Transformers exhibit a preference for generating sequences that satisfy certain ordering relations. Ultimately, we construct a Transformer and prove that the limiting reasoning depth is achievable.