Exact Expressive Power of Transformers with Padding

Chain of thought is a natural inference-time method for increasing the computational power of transformer-based large language models (LLMs), but comes at the cost of sequential decoding. Are there more efficient alternatives to expand a transformer's expressive power without adding parameters? We consider transformers with padding tokens as a form of parallelizable test-time compute. We show that averaging-hard-attention, masked-pre-norm transformers with polynomial padding recognize precisely the class mathsfFO-uniform mathsfTC0 of extremely parallelizable problems. While the mathsfTC0 upper bound was known, proving a matching lower bound had been elusive. Further, our novel analysis reveals the precise expanded power of padded transformers when coupled with another form of inference-time compute, namely dynamically increasing depth via looping. Our core technical contribution is to show how padding helps bring the notions of complete problems and reductions, which have been a cornerstone of classical complexity theory, to the formal study of transformers. Armed with this new tool, we prove that padded transformers with O(logd n) looping on inputs of length n recognize exactly the class mathsfFO-uniform mathsfTCd of moderately parallelizable problems. Thus, padding and looping together systematically expand transformers' expressive power: with polylogarithmic looping, polynomially padded transformers recognize precisely the class mathsfFO-uniform mathsfNC, the best that could be expected without losing parallelism (unless mathsfNC = mathsfP). Our results thus motivate further exploration of padding and looping as parallelizable alternatives to chain of thought for test-time compute.