We thank the reviewer for their valuable comments, and are glad they found our approach novel.

The reviewer wonders about the benefit of Neurosymbolic Diffusion Models (NeSyDMs) compared to increasing the number of parameters in weighted model counting (WMC). We highlight how NeSyDMs use neural networks that predict the same number of parameters as baseline SotA NeSy methods that independence over concepts. Instead, increasing the number of WMC parameters also requires larger neural networks. This allows it to scale to more challenging reasoning problems. The reviewer also asked for a motivation for the ECE measure in studying Reasoning Shortcut Awareness. We motivate this measure in this context and provide an example. We will add both clarifications to our manuscript, and provide detailed answers below.

Question: Given that WMC and Problog are already fully expressive, with better bounds on the number of required parameters than what is proposed in the paper, then why should one use diffusion?

The number of parameters is precisely our motivation. We assume a fixed number of concepts, say $n$ binary variables. Over these, we assume conditional independence, predict parameters with a neural diffusion model from an input $\mathbf{x}$, and estimate $n$ WMC parameters, as is standard in our baselines. One could alternatively increase the expressivity by introducing additional variables and changing the program. Then one can indeed achieve full expressivity at $2^{n}-1$ parameters. However, in NeSy one needs to predict these additional parameters with the neural network, significantly increasing the neural network size and training requirements. For full expressivity on our largest path planning task, our diffusion method requires a neural network that predicts only $5*900$ rather than $5^{900}-1$ parameters. We explain how to think of these numbers below.

Question: Can the authors show that these two notions, RS-Awareness while maximizing data likelihood and constraint satisfaction, and ECE are mutually compatible or even consistent with each other?

Certainly! We understand RS-awareness as in BEARS [1]: we aim for high accuracy on concepts unaffected by reasoning shortcuts, while being calibrated on concepts that are affected; therefore, perfect concept accuracy is unattainable. A model which predicts concepts like this while mixing over reasoning shortcuts also maximises data likelihood. To measure calibration over ambiguous concepts, we use ECE. We provide a detailed example below.

Detailed response

Independence assumption

Thank you for the useful comments. We will clarify our understanding of this problem and highlight the benefit of using diffusion for NeSy.

Weighted Model Counting is fully expressive wrt any discrete distribution, given that one adds additional parameters

The reviewer is correct that WMC can be more expressive only by adding variables apart from those used to characterise concepts and changing the constraint.

• We understand independence in our paper as follows: given a fixed program and a fixed number of concepts, we define a factorised distribution over them (conditioned on the input $\mathbf{x}$). In the graphical modelling literature, one would therefore say these concepts are conditionally independent.
• By increasing the number of variables beyond those characterising concepts, we can indeed represent more flexible distributions over concepts than conditionally independent ones. This is an alternative to diffusion for increasing the expressivity beyond independence, which is done in eg SPL [1] (mentioned in paper). SPL cannot be directly applied to our setting where concepts are latent, and all NeSy baselines we compare against do not introduce these additional variables to make WMC more expressive. As a matter of fact, all baselines we consider (except BEARS) assume concepts to be independent and hence are limited in expressiveness and uncertainty quantification.
• We agree that this possibility is not clearly highlighted in the paper; we will clarify this point in the camera-ready version.

Secondly, the reviewer shows a parameterisation that attains the uniform distribution. Note that this parameterisation requires access to the label $y$ to compute the partition function. But in our NeSy setting, the model does not know what the label $y$ is at test time, as that is what the model needs to predict. Therefore, this particular parameterisation is not applicable here.

It is possible to have an unconditioned uniform WMC distribution when using three parameters; however, this would go beyond what we understand in our paper as the independence assumption.

Now, since theoretically WMC can be fully expressive, why use diffusion? As the reviewer mentions, to attain increased expressivity via WMC, we need to increase the number of WMC parameters. However, consider our NeSy setting: these WMC parameters are predicted from the high-dimensional input $\mathbf{x}$ using our neural network. So we would need to train a (potentially very large) neural network that predicts $2^n-1$ WMC parameters, which is certainly infeasible for the problems we study. For the largest path planning task in our experiments, this requires predicting $5^{900}-1$ WMC parameters for full expressivity.

Instead, by having our neural network predict an independent distribution / multiplicative parameterisation, the network only needs to predict $5*900$ WMC parameters, creating a practically viable expressive parameterisation.

As Theorem C.1 only guarentees full expressivity as $T\rightarrow \infty$ which in my understanding requires infinitely many parameters to be modeled

Increasing $T$ does not increase any requirements of parameter counts. $T\rightarrow \infty$, mentioned in Theorem C.1 refers to the number of diffusion timesteps, not number of parameters. We assume $T\rightarrow \infty$ (continuous-time diffusion) throughout our paper.

ECE validity

Question: Can the authors show that these two notions, RS-Awareness while maximizing data likelihood and constraint satisfaction, and ECE are mutually compatible or even consistent with each other?

We give an example of consistency. Consider again the XOR problem, which has 1 reasoning shortcut (RS) that maps MNIST digits of 1s to 0s and MNIST digits of 0s to 1s. This RS cannot be distinguished from the ground-truth mapping. The ideal model under this ambiguity would assign 50% confidence to the ground-truth mapping and 50% to the RS.

How would one achieve this? Using a neural network that, 1) given two distinct MNIST digits outputs a uniform distribution over (0, 1) and (1, 0), and 2) given two equivalent MNIST digits, outputs a uniform distribution over (0, 0) and (1, 1). Then for case 1), running the XOR function returns 1 with probability 1, and for case 0), it will return 0 with probability 1.

This attains maximum data likelihood: The model will always return the correct output label. Furthermore, it always assigns 0.5 probability to 1 and 0. Its concept accuracy will also be ~0.5 in our synthetic setup. Therefore, the ECE will also be ~0, highlighting its calibration. Instead, a non RS-aware method finding the RS will attain an ECE of ~1: it always predicts exactly the opposite of the correct concept.

For concepts that are not affected by RSs, the accuracy will be 1 and maximum likelihood will also attain high confidence, resulting also in a low ECE value.

Now, ECE is a proxy for measuring RS-awareness and also not the only / perfect metric for uncertainty quantification. A future study in the limitations of ECE for measuring RS-awareness could be of significant value, even if it is useful for the problems studied in this paper. We would be happy to add alternative metrics the reviewer suggests.

We will add a paragraph motivating the ECE measure in the appendix with this example included.

1. Ahmed, K., Teso, S., Chang, K. W., Van den Broeck, G., & Vergari, A. (2022). Semantic probabilistic layers for neuro-symbolic learning. Advances in Neural Information Processing Systems, 35, 29944-29959.

We thank the reviewer for their helpful comments and feedback, and are glad they found our paper clear and well-organized. They believe NeSyDM is a theoretically grounded solution with a solid learning objective, while providing significant experimental improvements.

The reviewer had several important questions on the fit of NeSyDMs to other tasks and applications, which we discuss in our response below. They also gave the valuable suggestion to include an ablation study for the weighting of our loss components, which we ran and also provide below. We will add these clarifications and new results to our manuscript.

How would NESYDMs behave on tasks whose outputs are not easily factorised and whose programs lack a compilable circuit? A deeper discussion on this matter would clarify better the scope and the limitations of the approach.

Yes, this also is a challenge for all our NeSy competitors. However, the strength of NeSyDM lies in its compatibility with approximate inference methods for WMC [1, 2]. So, if one has a problem for which a good WMC approximation is available, this can be directly applied in NeSyDM by updating the gradient estimation procedure.

In particular, we believe a significant scalability boost can be obtained by using pretrained models. For instance, Reinforcement Learning with Verifiable Rewards (RLVR) methods use pretrained LLMs that already have a large probability of obtaining a positive reward. Similarly, we can use pretrained diffusion models, such as LLADA or MMADA [3, 4], to bootstrap the learning process.

We will include this discussion in the camera-ready version.

How do the authors envision applying NESYDMs to reasoning-heavy tasks like visual question answering, where the symbolic structure is more complex and less explicit?

Indeed, when the symbolic structure is not explicit, we have the additional challenge of predicting a suitable symbolic structure. Luckily, this can be neatly integrated with NeSyDMs. For instance, we can use a pretrained model (again, like LLADA or MMADA [3, 4]) that can already generate programs ($\approx$symbolic structures). Then, we can evaluate the resulting program to predict the answer. If this aligns with the ground-truth answer, we can reinforce the probability of this program. This is similar to popular approaches like ViperGPT, VISPROG, Faithful CoT and Logic-LM [5-8], which can improve the quality of the generated program. We will include a discussion of this extension in our camera-ready.

Since L_y , L_c , and L_H are critical to performance (L234), what is the effect of varying these weights? Can the authors provide ablation studies or sensitivity analysis?

Yes we can provide those, see below! We ran the conditioned model NeSyDM_Cond with 3 repeated runs on MNISTHalf with various values of the entropy weight and concept unmasking weight to understand their effect on the various metrics. We will include these results and our analysis in our appendix.

L234 is somewhat imprecise, we only experimented with the concept unmasking weight and the variational entropy weight, keeping the output unmasking weight to 1 (affected only by learning rate). We will update L234 and refer to the new appendix.

For the output unmasking weight $\gamma_{\mathbf{c}}$, we find that all test values achieve high label accuracy. However, its value significantly influences the concept accuracy both in-distribution and OOD, and the OOD label accuracy. We observe lower values (<1.0) are effective, which is in line with our observation on L947. We suspect the output unmasking loss can provide the model information that it is useful, but it should not dominate the loss, as this results in significantly lower concept accuracy and OOD performance, and very poor calibration. This suggests the model converged onto a single reasoning shortcut.

We also observe a significant influence of the entropy weight $\gamma_{\text{H}}$. All values <1.3 seem to converge on a single reasoning shortcut, exhibiting very poor calibration and worse concept accuracy. These runs do not balance the maximisation of entropy as the weight is too low, resulting in models with low entropy. Increasing the entropy weight beyond the value we used (1.6), we find that this can also impact the performance (>2.0), resulting in reduced label and concept accuracy and calibration. As the optimal range of values is quite tight, we recommend tuning $\gamma_{\text{H}}$ (as mentioned in L949).

Citations

1. van Krieken, E., Thanapalasingam, T., Tomczak, J., Van Harmelen, F., & Ten Teije, A. (2023). A-nesi: A scalable approximate method for probabilistic neurosymbolic inference. Advances in Neural Information Processing Systems, 36, 24586-24609.
2. De Smet, L., Sansone, E., & Zuidberg Dos Martires, P. (2023). Differentiable sampling of categorical distributions using the catlog-derivative trick. Advances in Neural Information Processing Systems, 36, 30416-30428.
3. Nie, S., Zhu, F., You, Z., Zhang, X., Ou, J., Hu, J., ... & Li, C. (2025). Large language diffusion models. arXiv preprint arXiv:2502.09992.
4. Yang, L., Tian, Y., Li, B., Zhang, X., Shen, K., Tong, Y., & Wang, M. (2025). Mmada: Multimodal large diffusion language models. arXiv preprint arXiv:2505.15809.
5. Surís, D., Menon, S., & Vondrick, C. (2023). Vipergpt: Visual inference via python execution for reasoning. In Proceedings of the IEEE/CVF international conference on computer vision (pp. 11888-11898).
6. Gupta, T., & Kembhavi, A. (2023). Visual programming: Compositional visual reasoning without training. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition (pp. 14953-14962).
7. Lyu, Q., Havaldar, S., Stein, A., Zhang, L., Rao, D., Wong, E., ... & Callison-Burch, C. (2023, November). Faithful chain-of-thought reasoning. In The 13th International Joint Conference on Natural Language Processing and the 3rd Conference of the Asia-Pacific Chapter of the Association for Computational Linguistics (IJCNLP-AACL 2023).
8. Pan, L., Albalak, A., Wang, X., & Wang, W. Y. (2023). Logic-lm: Empowering large language models with symbolic solvers for faithful logical reasoning. arXiv preprint arXiv:2305.12295.

We thank the reviewer for their encouraging comments, and would like to confirm their summary is pretty spot-on! Furthermore, we are happy they thought our paper is well-motivated, discussing the related work properly while providing extensive significant technical novelty, and appreciate they believe the exposition is thorough.

We will extend our paper to give a linear overview of the loss computation and sampling, and provide extra discussion on our RLOO gradient estimator, as suggested by the reviewer.

I believe it would facilitate understanding if the authors included a “linear” summary of what training and sampling look like at the end of the day when all components are put together.

That is a great suggestion, thank you. We will add this to our paper.

Is it somehow unavoidable to denoise outputs in addition to concepts? If not unavoidable, did you try at any point in your exploration process to only use diffusion for the concepts? What were the limitations?

It is not unavoidable, but it makes the resulting algorithms more tricky. If outputs are not denoised, they’d from a generative perspective only come in for completely clean data. This requires full simulation of the diffusion model during training, which is more akin to standard RL training in LLMs and diffusion. While this is theoretically possible, it is highly compute-intensive and has much a more challenging credit assignment problem: Ie, which of the many denoising steps were relevant to producing a good result?

How did you arrive at the RLOO estimator as a solution to computing ? As in, what was the reasoning process for trying this approach specifically?

RLOO is the simplest RL-based estimator to implement, really. It scales well with number of samples, and is unbiased [1]. Recent work in RLHF also shows it is a great, simple alternative to, e.g., PPO [2] and it is closely related to (but simpler than) the currently very popular GRPO algorithm [3]. For more expensive symbolic programs, one could also consider LOOP [4], a combination of PPO and RLOO. We will add this discussion to our paper.

1. Kool, Wouter, Herke van Hoof, and Max Welling. "Buy 4 reinforce samples, get a baseline for free!." (2019).
2. Schulman, J., Wolski, F., Dhariwal, P., Radford, A., & Klimov, O. (2017). Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347.
3. Shao, Z., Wang, P., Zhu, Q., Xu, R., Song, J., Bi, X., ... & Guo, D. (2024). Deepseekmath: Pushing the limits of mathematical reasoning in open language models. arXiv preprint arXiv:2402.03300.
4. Gupta, S., Ahuja, C., Lin, T. Y., Roy, S. D., Oosterhuis, H., de Rijke, M., & Shukla, S. N. (2025). A simple and effective reinforcement learning method for text-to-image diffusion fine-tuning. arXiv preprint arXiv:2503.00897.

We thank the reviewer for their comments and feedback, and appreciate they believe our work is well-motivated and clear, and technically sound.

The largest problem studied is MNIST classification. To me, these experiments do not show the "scalability" of the approach, as minst classification is a rather small-scale problem.

The scalability described here is regarding reasoning. Our method also significantly surpasses baseline methods on large reasoning problems like visual path planning that are challenging for current NeSy methods [1-3]. Furthermore, we also study real-world images with the BDD-OIA dataset [4].

The experimental validation is limited to few tasks

We are very happy to add additional tasks if the reviewer has a concrete list in mind.

1. van Krieken, E., Thanapalasingam, T., Tomczak, J., Van Harmelen, F., & Ten Teije, A. (2023). A-nesi: A scalable approximate method for probabilistic neurosymbolic inference. Advances in Neural Information Processing Systems, 36, 24586-24609.
2. Verreet, V., De Smet, L., De Raedt, L., & Sansone, E. (2024). EXPLAIN, AGREE, LEARN: Scaling Learning for Neural Probabilistic Logic. AAAI 2025.
3. Maene, J., Derkinderen, V., & Martires, P. Z. D. (2024). Klay: Accelerating arithmetic circuits for neurosymbolic ai. ICLR 2025.
4. Xu, Y., Yang, X., Gong, L., Lin, H. C., Wu, T. Y., Li, Y., & Vasconcelos, N. (2020). Explainable object-induced action decision for autonomous vehicles. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 9523-9532).