Logical Guidance for the Exact Composition of Diffusion Models

We propose LOGDIFF (Logical Guidance for the Exact Composition of Diffusion Models), a guidance framework for diffusion models that enables principled constrained generation with complex logical expressions at inference time. We study when exact score-based guidance for complex logical formulas can be obtained from guidance signals associated with atomic attributes and constraints. First, we derive an exact Boolean calculus that provides a sufficient condition for exact logical guidance. Specifically, if a formula admits a circuit representation in which conjunctions combine conditionally independent subformulas and disjunctions combine subformulas that are either conditionally independent or mutually exclusive, exact logical guidance is achievable. In this case, the guidance signal can be computed exactly from atomic scores and posterior probabilities using an efficient recursive algorithm. Moreover, we show that, for commonly encountered classes of distributions, any desired Boolean formula is compilable into such a circuit representation. Second, by combining atomic guidance scores with posterior probability estimates, we introduce a hybrid guidance approach that bridges classifier guidance and classifier-free guidance, applicable to both compositional logical guidance and standard conditional generation. We demonstrate the effectiveness of our framework on multiple image and protein structure generation tasks.

Generative AI models for images and proteins, such as diffusion models, are powerful but difficult to control precisely, for instance, when we want to generate a molecule or image that satisfies several conditions at once. We therefore develop LOGDIFF, a framework that can be used to specify rich Boolean rules to guide what a diffusion model generates, with mathematical guarantees that the output exactly satisfies those rules. We compile complex logical conditions recursively into probabilistic circuits, which combine their guidance signals efficiently at generation time. We validate LOGDIFF on image generation and protein structure design, demonstrating that our framework is both theoretically grounded and practically effective.