# Perfect diffusion is mathsfTC0 -- Bad diffusion is Turing-complete

arXiv: https://arxiv.org/abs/2507.12469 · HTML: https://mlsanity.com/paper/2507.12469/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Discrete and text diffusion, one of 14 directions in Diffusion models.
- Builds on 15 of its 20 references that are in the map.
- First posted on arXiv 20 Apr 2025.

## Placed in

- Diffusion models / [Discrete and text diffusion](https://mlsanity.com/diffusion/discrete-and-text-diffusion/)

## Builds on (most cited first, 15 of 15)

- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [DiffuSeq: Sequence to Sequence Text Generation with Diffusion Models](https://mlsanity.com/paper/2210.08933/) (2022)
- [Block Diffusion: Interpolating Between Autoregressive and Diffusion Language Models](https://mlsanity.com/paper/2503.09573/) (2025)
- [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022)
- [Towards Revealing the Mystery behind Chain of Thought: A Theoretical Perspective](https://mlsanity.com/paper/2305.15408/) (2023)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [Chain of Thought Empowers Transformers to Solve Inherently Serial Problems](https://mlsanity.com/paper/2402.12875/) (2024)
- [The Parallelism Tradeoff: Limitations of Log-Precision Transformers](https://mlsanity.com/paper/2207.00729/) (2022)
- [The Illusion of State in State-Space Models](https://mlsanity.com/paper/2404.08819/) (2024)
- [O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions](https://mlsanity.com/paper/2409.18959/) (2024)
- [Spatial Reasoning with Denoising Models](https://mlsanity.com/paper/2502.21075/) (2025)
- [Scaling Properties of Diffusion Models for Perceptual Tasks](https://mlsanity.com/paper/2411.08034/) (2024)

## Built on by (most cited first, 3 of 3)

- [The Serial Scaling Hypothesis](https://mlsanity.com/paper/2507.12549/) (2025)
- [On the Reasoning Abilities of Masked Diffusion Language Models](https://mlsanity.com/paper/2510.13117/) (2025)
- [Fundamental Limits of Crystalline Equivariant Graph Neural Networks: A Circuit Complexity Perspective](https://mlsanity.com/paper/2510.05494/) (2025)

## Abstract (arXiv)

This paper explores the computational complexity of diffusion-based language modeling. We prove a dichotomy based on the quality of the score-matching network in a diffusion model. In one direction, a network that exactly computes the score function of some initial distribution can only perform language modeling within the mathsfTC0 complexity class, reflecting limitations tied to rapid convergence. In the other direction, we show that if there is no requirement for the network to match any score function, then diffusion modeling can simulate any Turing machine in a certain sense. This dichotomy provides a theoretical lens on the capabilities and limitations of diffusion models, particularly concerning tasks requiring sequential computation. We conjecture extensions of our theoretical results, including for the case where the diffusion model is not perfect, but merely good. We also discuss the wider context and practical implications, and hypothesize that a machine learning architecture that can interpolate between sequential and parallel modes of operation would be superior to both Transformers and diffusion models.