# Continuous Diffusion Models Can Obey Formal Syntax

arXiv: https://arxiv.org/abs/2602.12468 · HTML: https://mlsanity.com/paper/2602.12468/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Discrete and text diffusion, one of 14 directions in Diffusion models.
- Builds on 18 of its 27 references that are in the map.
- First posted on arXiv 12 Feb 2026.

## Placed in

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

## Builds on (most cited first, 18 of 18)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [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)
- [Diffusion-LM Improves Controllable Text Generation](https://mlsanity.com/paper/2205.14217/) (2022)
- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [Dream 7B: Diffusion Large Language Models](https://mlsanity.com/paper/2508.15487/) (2025)
- [Efficient Guided Generation for Large Language Models](https://mlsanity.com/paper/2307.09702/) (2023)
- [Grammar-Constrained Decoding for Structured NLP Tasks without Finetuning](https://mlsanity.com/paper/2305.13971/) (2023)
- [Likelihood-Based Diffusion Language Models](https://mlsanity.com/paper/2305.18619/) (2023)
- [JSONSchemaBench: A Rigorous Benchmark of Structured Outputs for Language Models](https://mlsanity.com/paper/2501.10868/) (2025)
- [Grammar-Aligned Decoding](https://mlsanity.com/paper/2405.21047/) (2024)
- [Flexible and Efficient Grammar-Constrained Decoding](https://mlsanity.com/paper/2502.05111/) (2025)
- [Continuous Diffusion Model for Language Modeling](https://mlsanity.com/paper/2502.11564/) (2025)
- [Constrained Sampling for Language Models Should Be Easy: An MCMC Perspective](https://mlsanity.com/paper/2506.05754/) (2025)
- [Constrained Decoding of Diffusion LLMs with Context-Free Grammars](https://mlsanity.com/paper/2508.10111/) (2025)

## Abstract (arXiv)

Diffusion language models offer a promising alternative to autoregressive models due to their global, non-causal generation process, but their continuous latent dynamics make discrete constraints -- e.g., the output should be a JSON file that matches a given schema -- difficult to impose. We introduce a training-free guidance method for steering continuous diffusion language models to satisfy formal syntactic constraints expressed using regular expressions. Our approach constructs an analytic score estimating the probability that a latent state decodes to a valid string accepted by a given regular expression, and uses its gradient to guide sampling, without training auxiliary classifiers. The denoising process targets the base model conditioned on syntactic validity. We implement our method in Diffinity on top of the PLAID diffusion model and evaluate it on 180 regular-expression constraints over JSON and natural-language benchmarks. Diffinity achieves 68-96% constraint satisfaction while incurring only a small perplexity cost relative to unconstrained sampling, outperforming autoregressive constrained decoding in both constraint satisfaction and output quality. Diffinity is open-sourced at github.com/large-loris-models/Diffinity.