The Expressive Power of Low Precision Softmax Transformers with (Summarized) Chain-of-Thought

Existing expressivity results for transformers typically rely on hardmax attention, high precision, and other architectural modifications that disconnect them from the models used in practice. We bridge this gap by analyzing standard transformer decoders with softmax attention and rounding of activations and attention weights, while allowing depth and width to grow logarithmically with the context length. As an intermediate step, we construct hardmax transformers with ternary activations and well-separated attention scores that simulate Turing machines using Chain-of-Thought (CoT). This lets us convert the constructions to equivalent softmax transformers without the unrealistic parameter magnitudes or activation precision that prior approaches would require. Using the same technique, we analyze a recently proposed summarized CoT paradigm (InftyThink) and show that it simulates Turing machines more efficiently, with model size scaling logarithmically in a space bound rather than a time bound. We empirically demonstrate this advantage by training small transformers to solve Sudokus almost perfectly with summarized CoT, even on instances requiring more than a million reasoning tokens.