Softmax Transformers are Turing-Complete

Hard attention Chain-of-Thought (CoT) transformers are known to be Turing-complete. However, it is an open problem whether softmax attention Chain-of-Thought (CoT) transformers are Turing-complete. In this paper, we prove a stronger result that length-generalizable softmax CoT transformers are Turing-complete. More precisely, our Turing-completeness proof goes via the CoT extension of the Counting RASP (C-RASP), which correspond to softmax CoT transformers that admit length generalization. We prove Turing-completeness for CoT C-RASP with causal masking over a unary alphabet (more generally, for the letter-bounded languages). While we show that this is actually not Turing-complete for arbitrary languages, we prove that its extension with relative positional encoding is Turing-complete for arbitrary languages. We empirically validate our theoretical results by training transformers for various languages that require complex (non-linear) arithmetic reasoning.