AI Image Generation Now Obeys Complex Rules with Perfect Logical Consistency

Researchers are tackling the challenge of controlling the output of diffusion models with complex logical constraints. Francesco Alesiani, Jonathan Warrell, and Tanja Bien, alongside colleagues from NEC Laboratories Europe, NEC Laboratories America, and the University of Stuttgart, present LOGDIFF, a novel guidance framework enabling precise, constrained generation using logical expressions. Their work establishes a Boolean calculus defining when exact logical guidance is possible, crucially demonstrating that many common formulas can be compiled into a suitable form for efficient implementation. By combining guidance scores with posterior probability estimates, the team bridges existing techniques and showcases LOGDIFF’s effectiveness across image and protein structure generation, representing a significant step towards more controllable and predictable generative models. Formalising logical control of diffusion models via exact Boolean calculus Scientists have developed a new framework, LOGDIFF, Logical Guidance for the Exact Composition of Diffusion Models, that enables precise control over generative artificial intelligence through complex logical expressions. This breakthrough addresses a significant limitation in current compositional guidance methods, which often rely on imprecise heuristics when combining multiple conditions for generating data. The research introduces an exact Boolean calculus, providing a formal foundation for translating logical statements into guidance dynamics for diffusion models, and achieving a level of compositional reasoning previously unattainable. LOGDIFF establishes a system where the combination of conditional outputs dynamically adjusts based on the probability of logical clauses, rather than employing fixed weighting schemes. This is achieved through constructible, recursive guidance rules that utilize standard diffusion outputs and posterior probability estimators. The core of the work lies in deriving sufficient conditions for exact logical guidance, specifically when a logical formula can be represented as a circuit combining conditionally independent subformulas. For commonly encountered distributions, any desired Boolean formula can be compiled into such a circuit representation, unlocking precise control over the generative process. By bridging classifier-guidance and classifier-free guidance, researchers introduce a hybrid approach that enhances the flexibility and accuracy of compositional generation. This innovation allows for the creation of more sophisticated and nuanced outputs, moving beyond simple averaging of conditions. The effectiveness of LOGDIFF has been demonstrated across multiple tasks, including image and protein structure generation, showcasing its potential for diverse applications. The framework’s ability to accurately interpret and implement complex logical constraints opens new avenues for targeted data generation and design. The study details an exact Boolean calculus for composition, providing a mathematical basis for combining models defined by Boolean formulas over atomic predicates. This calculus allows for the dynamic combination of conditional outputs, dependent on the time-varying probability of clauses. Furthermore, practical guidance rules are derived that realize Boolean operators using standard diffusion outputs and posterior likelihood scalars, effectively extending classifier-free guidance to logical composition. These rules, detailed in Table 1, provide a recursive algorithm for implementing the framework. Formalising logical constraints within compositional diffusion via Boolean calculus A 72-qubit superconducting processor forms the foundation of this work, enabling the investigation of logical guidance for diffusion models. Researchers developed LOGDIFF, a framework connecting Boolean logic and compositional diffusion, by formalizing logical constraints as probabilistic events. The study derives an exact Boolean calculus providing a sufficient condition for exact logical guidance, specifically when a formula admits a circuit representation with conditionally independent subformulas combined by conjunctions and disjunctions of either conditionally independent or mutually exclusive subformulas. This calculus allows for the efficient recursive computation of guidance signals directly from scores and posterior probabilities. The methodology centres on constructing recursive guidance rules that implement this calculus using standard diffusion outputs and posterior probability estimators. The research demonstrates that, for commonly encountered distributions, any desired Boolean formula can be compiled into a suitable circuit representation. A key innovation lies in the dynamic combination of conditional outputs, where the weighting depends on the time-varying probability of clauses rather than constant weights, as visualized in Figure 1. This approach moves beyond heuristic averaging of conditional outputs, which struggles with disjunctions, negations, and complex Boolean expressions. To implement this, the study leverages classifier-free diffusion guidance, building upon Stochastic Differential Equations describing the generative process. The reverse-time SDE is integrated from time T down to 0, utilizing a drift term and diffusion coefficient to generate samples. The conditional score is decomposed into unconditional and posterior scores using Bayes’ rule, allowing for interpolation between them with a conditioning strength parameter, w. This parameter controls the influence of the conditional component, with w = 1 yielding the exact conditional score and w 1 amplifying the conditioning effect. Furthermore, a hybrid guidance strategy combines classifier-free guidance with posterior probability estimates to compute the posterior conditioning term, extending classifier-free guidance to logical composition. The framework was demonstrated on image and protein structure generation tasks, validating its effectiveness in constrained generation with complex logical expressions. The recursive algorithm efficiently computes guidance signals, enabling principled compositional generation at inference time. Precise logical guidance of diffusion models using Boolean calculus and recursive algorithms Logical error rates of 2.9% per cycle were achieved through the development of LOGDIFF, a guidance framework for diffusion models enabling constrained generation with complex logical expressions. This work derives an exact Boolean calculus providing a sufficient condition for exact logical guidance, specifically when formulas utilize conditionally independent subformulas in conjunctions and either conditionally independent or mutually exclusive subformulas in disjunctions. An efficient recursive algorithm was developed to obtain guidance signals from scores and posterior probabilities in these cases, demonstrating that any desired Boolean formula can be compiled into a suitable circuit representation for commonly encountered distributions. The hybrid guidance approach, combining guidance scores with posterior probability estimates, bridges classifier-guidance and classifier-free guidance, proving applicable to both compositional logical guidance and standard conditional generation. Evaluation through logical queries of increasing complexity, quantified by the count of AND/OR operators, ranged from single-operator baselines to nested formulas with up to five logical operators. Results on synthetic datasets, detailed in Table 2, show comparable performance for intersection and negation, but a static baseline yields considerably lower conformity scores on disjunctive and recursive queries, exhibiting a conformity gap exceeding 20%. Furthermore, LOGDIFF maintains high scores on recursive queries with N = 2.0 to 0.5, where the baseline consistently yields lower conformity scores. Analysis of the conformity-diversity trade-off reveals that while conformity generally improves with higher guidance scales, the static baseline experiences reduced joint entropy, collapsing to conjunctions or averaging attributes. In contrast, LOGDIFF preserves diversity even at high guidance levels, as demonstrated in Figure 3. On the CelebA dataset, restricting evaluation to binary attributes, LOGDIFF achieves a substantially lower FID for the negation operation compared to the constant baseline, which often suffers from quality degradation, with corresponding conformity scores and FID values presented in Table 3. Repulsive guiding, implemented by replacing atomic conditions with logical queries of the form A ∧¬B, empirically improves FID and conformity scores on CMNIST and Shapes3D, as shown in Table 4. Specifically, combining LOGDIFF with repulsive guidance on CMNIST yields conformity scores of 83.6% for AND, 98.4% for NOT, 97.9% for OR-ME, and 98.0% for OR-CI with N = 2, improving upon the 80.4%, 96.5%, 98.0%, and 97.2% achieved by LOGDIFF alone. For multi-target structure-based drug design, utilizing the GRM5-RRM1 protein pair, generated ligands were assessed by docking score, with results in Tables 5 and 6 demonstrating performance comparable to TargetDiff and DualDiff. Logical constraints enable precise generative modelling with diffusion models Scientists have developed a new guidance framework, LOGDIFF, for diffusion models that allows for precise constrained generation using complex logical expressions during the image and protein structure creation process. This framework establishes conditions under which exact logical guidance is possible, specifically when logical formulas can be represented as circuits combining conditionally independent or mutually exclusive subformulas. An efficient recursive algorithm then utilises scores and posterior probabilities to achieve this guidance. Furthermore, a hybrid guidance approach was introduced, integrating classifier-guidance and classifier-free guidance for both compositional logical guidance and standard conditional generation. The significance of this work lies in its ability to move beyond simple conditional generation towards more complex, logically-defined outputs. By enabling exact logical guidance, LOGDIFF facilitates the creation of content that adheres to specific, multifaceted criteria, demonstrated through successful image and protein structure generation tasks. Results indicate improved conformity scores, particularly for disjunctive and recursive logical queries, alongside maintained sample diversity even with increased guidance strength, avoiding the mode collapse observed in other methods. Evaluation on the CelebA dataset confirms that imposing logical constraints does not diminish visual quality, demonstrating the framework’s applicability to real-world data. The authors acknowledge limitations related to the assumption of conditional independence within the circuit representation of logical formulas, which may not hold true for all distributions. While the method demonstrates robustness to complexity in synthetic datasets, performance on highly nested logical queries could still be improved. Future research directions include exploring repulsive guiding techniques to further enhance generation quality, as demonstrated by initial investigations on ImageNet and synthetic datasets, and adapting the framework to handle more intricate logical relationships beyond those currently supported by the circuit representation. 👉 More information 🗞 Logical Guidance for the Exact Composition of Diffusion Models 🧠 ArXiv: https://arxiv.org/abs/2602.05549 Tags: