  Which transformer scaling regimes are able to perfectly solve different
classes of algorithmic problems? While tremendous empirical advances have been
attained by transformer-based neural networks, a theoretical understanding of
their algorithmic reasoning capabilities in realistic parameter regimes is
lacking. We investigate this question in terms of the network's depth, width,
and number of extra tokens for algorithm execution. Our novel representational
hierarchy separates 9 algorithmic reasoning problems into classes solvable by
transformers in different realistic parameter scaling regimes. We prove that
logarithmic depth is necessary and sufficient for tasks like graph
connectivity, while single-layer transformers with small embedding dimensions
can solve contextual retrieval tasks. We also support our theoretical analysis
with ample empirical evidence using the GraphQA benchmark. These results show
that transformers excel at many graph reasoning tasks, even outperforming
specialized graph neural networks.
