Title: Curriculum Abductive Learning
Venue: NeurIPS 2025
Accepted: yes
Keywords: abductive learning

neuro-symbolic learning

logical reasoning

knowledge base
Abstract: Abductive Learning (ABL) integrates machine learning with logical reasoning in a loop: a learning model predicts symbolic concept labels from raw inputs, which are revised through abduction using domain knowledge and then fed back for retraining. However, due to the nondeterminism of abduction, the training process often suffers from instability, especially when the knowledge base is large and complex, resulting in a prohibitively large abduction space. While prior works focus on improving candidate selection within this space, they typically treat the knowledge base as a static black box. In this work, we propose Curriculum Abductive Learning (C-ABL), a method that explicitly leverages the internal structure of the knowledge base to address the ABL training challenges. C-ABL partitions the knowledge base into a sequence of sub-bases, progressively introduced during training. This reduces the abduction space throughout training and enables the model to incorporate logic in a stepwise, smooth way. Experiments across multiple tasks show that C-ABL outperforms previous ABL implementations, significantly improves training stability, convergence speed, and final accuracy, especially under complex knowledge setting.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes C-ABL, a curriculum-based extension to abductive learning (ABL), aiming to mitigate the large reasoning space and unstable supervision problems faced by standard ABL frameworks. Specifically, C-ABL partitions the logic knowledge base into structured subsets according to dependency relations among logical rules and progressively introduces these subsets into training, facilitating more stable and efficient learning. Empirical results on arithmetic tasks, chess attack prediction, and judicial sentencing demonstrate improved accuracy and faster convergence compared to baseline ABL methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\t\n1. The proposed idea of introducing a logic-guided curriculum into abductive learning is interesting and well-motivated.\n\t\n2. The paper is clearly written and easy to follow.\n\t\n3. Experimental results demonstrate significant improvements in accuracy and training efficiency compared to prior methods.\n\nWeaknesses\n\t\n 1. I still have serious concerns and uncertainties about the soundness of the proposed algorithm, especially regarding the fairness and potential information leakage introduced by the curriculum design (see detailed questions below).\n\nQUESTIONS:\nI should first acknowledge that I am not an expert in ABL, but I believe I have understood most of the paper and have the following questions. If my understanding is incorrect, please feel free to clarify—I would be happy to raise my score.\n\n---\n\n### 1. Data partitioning relies on true concept labels—is this a form of information leakage?\n\nAs I understand it, once the knowledge base is partitioned into curriculum phases, the training data is also filtered accordingly—only examples whose **concept labels lie entirely within the current Zₚ** are used in phase p. But to do this, one must know the true concept labels of each sample in advance. So even though the model is never directly given these labels during training, the data filtering procedure still **relies on ground-truth concept labels**, which the baselines (ABL, A³BL) do not use. This creates an imbalance.\n\n- Could you explain why this setup does not introduce information leakage or unfair advantage?\n- Have you tried giving the same phase-specific training data streams to the baselines to ensure a fair comparison?\n\n---\n\n### 2. Is curriculum learning really necessary?\n\nEven setting aside the above concern, I still have doubts about whether curriculum learning is actually needed. For example, in the digit addition task, what would happen if we train **only in phase 1**, not moving to later phases when performance exceeds random, but simply continuing phase 1 training until convergence, then evaluating? After all, we’re only training the perception model. If it has learned to predict digits well, then at test time we could just pair it with the **full KB** and run inference—do we really need to go through all phases?\n\n- Have you tried this “phase-1-only” setting and compared its performance or convergence time?\n- I can imagine that training only on phase 1 might limit the number of usable samples, but couldn’t we mitigate that by generating more phase 1 data? Given that the task is relatively simple, I’m not convinced that multi-phase curriculum is essential.\n\nI'd appreciate clarification on these points—they are important to my evaluation of whether the gains come from the curriculum design itself, or from more subtle data handling choices.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-17"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a framework for curriculum learning for abductive learning. \n\nAbductive learning (ABL) is a domain where a neural network is trained in tandem with a knowledge base (KB) to learn to recognize concepts from inputs, where the supervision comes from determining which concepts can be combined with rules from the KB to produce the supervised labels. To train a model, existing ABL training frameworks are applied to each KB in the curriculum until above-chance accuracy is reached for all concepts.\n\nThe paper gives an algorithm for constructing a curriculum of increasingly large KB's up to the original KB based on a dependency graph of rules. It also gives a theoretical analysis of how a curriculum, by reducing the size of the the abduction space at each training phase, can speed up convergence of ABL. \n\nThe paper presents experiments on d-digit addition (given MNIST images, rules of addition, and sum outputs, learn to recognize digits and predict correct sums of them), chess attack (given images of chess boards, rules of piece attacks, learn to recognize which pieces are attacking other pieces), and judicial sentencing (given criminal judgement records, predict the sentencing factors. The new method learns faster and is more accurate.\n\nSTRENGTHS AND WEAKNESSES:\n## Summary of strengths and weaknesses\nThe paper is really well written and has strong results. The results are incremental and predictable (applying curriculum learning to a niche neurosymbolic domain that hasn't seen curriculum learning of this type before) but also very solid. My biggest reservation is that the area of research seems to be a toy neurosymbolic domain that is outdated (feels like a pre-2020 type of AI research paper) and is not very significant of a research direction for AI in 2025.\n\n### Strengths\n- The writing, paper organization, description of methods and experiments, and discussion of results are great. The writing is clear and easy to understand. \n- The approach is simple and easy to understand. It is well-designed and described well with pseudocode and text. Helpful clarifying examples are included. \n- The theoretical analysis is relevant and makes sense at a high level. It is a nice cherry on top to justify the curriculum approach. \n- The paper does a good job briefly reviewing abductive learning for unfamiliar readers, and setting up preliminaries with knowledge base concepts, etc. \n- The approach achieves higher accuracy and faster training time for the evaluated domains. \n- Overall, the paper is polished, dots its i's and crosses its t's, and is good.\n- It's nice that the curriculum can be automatically constructed from a knowledge base. \n- (Originality) from what I can tell, the proposed approach has not been applied to ABL before.\n\n## Weaknesses\n- It is an incremental paper (Add on curriculum learning to existing ABL training schemes to improve performance). It is not very surprising that curriculum learning can improve training speed and final performance of the described system. \n- Moreover, the novel proposals in the paper are specific to abductive learning. I don't think there is any insight for domains besides abductive learning, since curriculum learning, as a broad technique, has been applied to a ton of different domains.\n- Abductive learning is a pretty niche problem field. It seems like a toy neurosymbolic problem that is academically studied by a few people, but doesn't have very much real world potential or relevance for AI.\n- Abductive learning and the accompanying neurosymbolic training setups, with the assumptions about what types of data, supervision, and knowledge bases are available for a problem, seems extremely outdated and irrelevant for modern AI. \n- All of the problems studied are contrived setups that only work for research works studying abductive learning. For example, two of the three domains use MNIST images for the perceptual component. All use a logical knowledge base encoding of the rules for each domain.  This type of data isn't that prevalent and especially this combination logical rule systems, simple perception, and then learning perception from the logical rules is academically interesting, but doesn't seem to have any real world usefulness. \n- LLM's seem like a far better way to approach this problem — first learn perception, then learn to reason given existing perception. I think the premise of abductive learning is fundamentally limited. \n\n##  Further reflection on the state of neurosymbolic research\n\nThe main weakness of this paper is that neurosymbolic abductive learning doesn't seem like a very significant, impactful, or important field of computer science research. Neurosymbolic AI has been a popular field of research for the past decade+, and there have been many subareas of research on how to combine neural and symbolic systems for learning and reasoning. Earlier on, there was more \"things are wide open\" for how future AI systems would solve the tasks we're concerned about. Especially in the late 2010's, there was a commonly held sense of the inability for purely neural systems to deal with generalization, compositionality, reasoning, logical rules, and the like. Abductive learning was one such direction, and primarily concerns itself with how reasoning systems can provide learning signal to a perceptual system. I was once heavily immersed in neurosymbolic research of similar varieties.\n\nHowever, as machine learning and especially NLP and LLM research has developed and progressed, a lot of these neurosymbolic methods have become obsolete. \n\nNeurosymbolic AI is not completely obsolete — it has evolved and there are always new ways to usefully combine neural and symbolic systems. There's really exciting work about generating code to reason (ViperGPT), AlphaGeometry which does theorem proving with a LLM and a symbolic rule system, and many other directions too. However, I think the problem setting that abductive learning is built on is fundamentally outdated and not going to provide useful AI research.\n\nQUESTIONS:\nFor chess attack, how do you know where the pieces are located? It looks like the only concepts that then neural network predicts is which piece is in the image. I don't understand this at all. Does the neural network take in one image, and predict multiple things, or is it applied separately to multiple images? Can you fully describe the input space, the concept space, the rule set, and the supervised labels for chess attack? It would probably be good to do this in the appendix for each of the tasks. In the problem setting, the model f is described as mapping each input element to a concept label, but I don't understand how this works for chess attack or judicial sentencing. \n\nWhat does \"pre-training setting\" mean for judicial sentencing? \n\nMy main reservation with this paper is about whether this type of research is significant. A discussion on the doubts raised above that convinces me that this line of research is relevant — either showing how current ABL research can have applications or is elsehow useful to research, or conveying how ABL research gives insight to related fields, or another way of justifying the significance of the research) would convince me to increase my score.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 1, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors study the neurosymbolic learning problem of learning a concept-based neural network with logical supervision. They extend 'abductive learning', which finds the most likely logically consistent prediction as a weak supervision label. This search process is hard to scale, and can introduce instabilities. Therefore, the authors design a curriculum schedule, making the problem progressively harder. They do this via a sequence of increasingly more granular knowledge bases. This results in a more scalable and better performing method.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper is clearly written\n- The idea introduced is interesting and novel as far as I know. It is also theoretically well motivated, and may be extended to other NeSy methods.\n- Experimental results are quite strong\n\nWeaknesses:\n- I have a few concerns about some of the theory and validity of comparisons in terms of experimental setup. \n- Scalability of the method remains unproven compared to the state of the art in the field.\n\nQUESTIONS:\n**Critical** (if these are properly answered I'd be happy to increase my score)\n- The paper suggest they make use of concept labels during training, which (usually) is not done in the baselines. \n    - Lines 191-195 suggest the authors make use of concept labels to determine when to proceed to the next phase. Most baselines assume no access to concept labels. If indeed the authors use these concept labels, they significantly collapse the number of reasoning shortcuts in the problem [1], making it much easier than the problem the baselines have to solve. If this is indeed what is done, I do not believe the experimental results are a valid comparison. If so: Please rerun the method without concept labels, or rerun the baselines with (this limited bit of) concept supervision. \n- Theorem 4.2 has an unlisted assumption that possibly impacts the design of the method. This should be clarified. \n   - The proof of theorem 4.2 relies on the fact that the consistency on concepts in the previous phase remains the same. However, during training (especially if the dataset changes), there is no guarantee (or proof) that this ordering remains valid. Then the search collapses to the exponential bound given in the lemma. \n    - Line 234 suggests the search is only done over the new concepts. Is this correct? If so, is this indeed still valid, given the concern above?\n- Theoretical aspects and details that should be clarified:\n    - The domain knowledge is said to be in first-order logic. Looking at the examples in the appendix, it looks like the authors use a logic programming formalism, maybe equivalent to DataLog? The semantics should be more clearly specified, and it does not look (at least in terms of how it's used in practice) to be equivalent to a full FOL which allows infinite domains. Furthermore, the definition in 2.1 is insufficient for interpreting the rules. \n    - Section 3.1: The algorithm does not assume acyclic graphs, right? How does it handle cycles? \n    - Algorithm 1: It should be clarified how the topological sorting works. The precedence ordering is only a partial order (I think) so topological sorting is not unique. Also, the precedence ordering is a heuristic, right? \n    - How is $\\tau$ optional? Its role is not very clear.\n    - Line 189: Can you make this scheduling more precise? What about data labels $y$ that do not appear in the current phase? \n\n**Minor points & questions**:\n- I really like the explanation of abductive learning, which is minimal and clean. It also made me realise the connection to the top-k inference method common in eg Scallop [2] and other approximate inference methods [3]. Where abductive learning uses $k=1$ (ie: use the MAP state as supervision). Are there any benefits for doing this instead of using weighted model counting, like in eg DeepProbLog? Maybe scalability, although there are also recent efforts on improving this that go beyond what C-ABL is tested on (see eg [4-10]). In particular, [4, 5, 9, 10] also look into decomposing the knowledge base in various methods. \n- Line 114-115: \"For each input x, the correct concept labels should be unique\". This is an assumption, not something that one can state. See Assumption **A1** in [1]\n- In table 1, please specify this is about concept accuracy, not sum prediction accuracy (right?)\n- Line 336-337: These probabilistic logic programming languages are indeed tied to specific logic paradigms, but so is C-ABL, right? There are also probabilistic methods that do not assume a particular language, see eg [1, 4, 5, 6, 8, 9]\n\n\n1. Marconato, Emanuele, et al. \"Not all neuro-symbolic concepts are created equal: Analysis and mitigation of reasoning shortcuts.\" Advances in Neural Information Processing Systems 36 (2023): 72507-72539.\n2. Li, Ziyang, Jiani Huang, and Mayur Naik. \"Scallop: A language for neurosymbolic programming.\" Proceedings of the ACM on Programming Languages 7.PLDI (2023): 1463-1487.\n3. Manhaeve, Robin, Giuseppe Marra, and Luc De Raedt. \"Approximate inference for neural probabilistic logic programming.\" Proceedings of the 18th International Conference on Principles of Knowledge Representation and Reasoning. IJCAI Organization, 2021.\n4. van Krieken, Emile, et al. \"A-nesi: A scalable approximate method for probabilistic neurosymbolic inference.\" Advances in Neural Information Processing Systems 36 (2023): 24586-24609.\n5. van Krieken, Emile, et al. \"Neurosymbolic Diffusion Models.\" arXiv preprint arXiv:2505.13138 (2025).\n6. De Smet, Lennert, Emanuele Sansone, and Pedro Zuidberg Dos Martires. \"Differentiable sampling of categorical distributions using the catlog-derivative trick.\" Advances in Neural Information Processing Systems 36 (2023): 30416-30428.\n7. De Smet, Lennert, and Pedro Zuidberg Dos Martires. \"A Fast Convoluted Story: Scaling Probabilistic Inference for Integer Arithmetic.\" arXiv preprint arXiv:2410.12389 (2024).\n8. Verreet, Victor, et al. \"EXPLAIN, AGREE, LEARN: Scaling Learning for Neural Probabilistic Logic.\" arXiv preprint arXiv:2408.08133 (2024).\n9. Choi, Seewon, et al. \"CTSketch: Compositional Tensor Sketching for Scalable Neurosymbolic Learning.\" arXiv preprint arXiv:2503.24123 (2025).\n10. Ahmed, Kareem, Kai-Wei Chang, and Guy Van den Broeck. \"Semantic strengthening of neuro-symbolic learning.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2023.\n\nLIMITATIONS:\nLimitations are listed in the appendix.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel method called Curriculum Abductive Learning (C-ABL), designed to address the instability and inefficiency of traditional Abductive Learning (ABL) when training under complex knowledge bases. C-ABL structurally partitions the knowledge base into sub-knowledge bases that are introduced progressively, enabling stage-wise model training. This approach reduces the hypothesis space and improves training efficiency. Theoretical analysis demonstrates that the method effectively lowers the reasoning complexity within each stage while ensuring smooth transitions between stages. Experiments across various tasks validate the superiority of C-ABL, showing significant improvements in accuracy, stability, and convergence speed.\n\nSTRENGTHS AND WEAKNESSES:\n#### **Strengths**\n- It's the first work to explicitly leverage the internal logical structure of the knowledge base (KB) in ABL, rather than treating it as a black box, introducing phased sub-knowledge bases to dynamically manage reasoning complexity.  \n\n- Theorems 4.2 & 4.5 prove that C-ABL reduces hypothesis space size, drastically lowering iteration complexity. The method uses stone space and topological continuity analysis (Theorem 4.7) to ensure that no catastrophic forgetting during phase transitions. Sub-base partitioning adheres to logical completeness (Theorem 3.2), ensuring local reasoning conclusions remain valid under the full KB.  \n \n- Benchmarks against not only ABL variants (e.g., A$^{3}$BL) but also leading neuro-symbolic methods (NeurASP, DeepProbLog), highlighting the superiority of C-ABL in terms of training efficiency and stability.\n\n- This work addressed ABL Core Issues. The curriculum design restricts candidate labels per phase, reducing erroneous supervision and the phased learning avoids model divergence among conflicting solutions.  \n\n\n#### **Weaknesses**:   \n- C-ABL requires the knowledge base to have modular or hierarchical structures. Otherwise, the partitioning algorithm (Algorithm 1) may fail, for example the highly interconnected rules and the random or flat rule sets.\n\n- Current phase transitions rely on predefined accuracy thresholds,which may be unable to dynamically adjust phases based on real-time model performance and threshold settings may require task-specific tuning.  \n\n- While KB partitioning is offline, dependency graph construction complexity may become prohibitive for extremely large KBs.\n\nQUESTIONS:\n- Many real-world knowledge bases may lack clear modularity. How can the robustness of this method be validated in unstructured or noisy knowledge bases?\n- Has the impact of different thresholds on final performance been experimentally quantified? Could certain tasks require more conservative/aggressive transition strategies?\n- If new rules are added during training, does the knowledge base need to be repartitioned? Are there incremental partitioning solutions?\n- Many domains use probabilistic logic. Can C-ABL accommodate uncertain reasoning? If not, which core assumptions need to be relaxed?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-06"}