Michael Cardei

Jacob K. Christopher, Michael Cardei, Jinhao Liang et al.

Despite the remarkable generative capabilities of diffusion models, their integration into safety-critical or scientifically rigorous applications remains hindered by the need to ensure compliance with stringent physical, structural, and operational constraints. To address this challenge, this paper introduces Neuro-Symbolic Diffusion (NSD), a novel framework that interleaves diffusion steps with symbolic optimization, enabling the generation of certifiably consistent samples under user-defined functional and logic constraints. This key feature is provided for both standard and discrete diffusion models, enabling, for the first time, the generation of both continuous (e.g., images and trajectories) and discrete (e.g., molecular structures and natural language) outputs that comply with constraints. This ability is demonstrated on tasks spanning three key challenges: (1) Safety, in the context of non-toxic molecular generation and collision-free trajectory optimization; (2) Data scarcity, in domains such as drug discovery and materials engineering; and (3) Out-of-domain generalization, where enforcing symbolic constraints allows adaptation beyond the training distribution.

Michael Cardei, Jacob K Christopher, Thomas Hartvigsen et al.

Discrete diffusion models are a class of generative models that construct sequences by progressively denoising samples from a categorical noise distribution. Beyond their rapidly growing ability to generate coherent natural language, these models present a new and important opportunity to enforce sequence-level constraints, a capability that current autoregressive models cannot natively provide. This paper capitalizes on this opportunity by introducing Constrained Discrete Diffusion (CDD), a novel integration of differentiable constraint optimization within the diffusion process to ensure adherence to constraints, logic rules, or safety requirements for generated sequences. Unlike conventional text generators that often rely on post-hoc filtering or model retraining for controllable generation, CDD directly imposes constraints into the discrete diffusion sampling process, resulting in a training-free and effective approach. Experiments in toxicity-controlled text generation, property-constrained molecule design, and instruction-constrained text completion demonstrate that CDD achieves zero constraint violations in a diverse array of tasks while preserving fluency, novelty, and coherence while outperforming autoregressive and existing discrete diffusion approaches.

Huu Binh Ta, Michael Cardei, Alvaro Velasquez et al.

Discrete diffusion models generate sequences by iteratively denoising samples corrupted by categorical noise, offering an appealing alternative to autoregressive decoding for structured and symbolic generation. However, standard training targets a likelihood-based objective that primarily matches the data distribution and provides no native mechanism for enforcing hard constraints or optimizing non-differentiable properties at inference time. This work addresses this limitation and introduces Search-Augmented Masked Diffusion (SearchDiff), a training-free neurosymbolic inference framework that integrates informed search directly into the reverse denoising process. At each denoising step, the model predictions define a proposal set that is optimized under a user-specified property satisfaction, yielding a modified reverse transition that steers sampling toward probable and feasible solutions. Experiments in biological design and symbolic reasoning illustrate that SearchDiff substantially improves constraint satisfaction and property adherence, while consistently outperforming discrete diffusion and autoregressive baselines.