Curriculum Abductive Learning

Thank you for your supportive and insightful comment. We will address your concerns as follows:

---

Q1 Why this setup does not introduce information leakage or unfair advantage? Can you give the same phase-specific training data streams to the baselines to ensure a fair comparison?

A1 First, we would like to emphasize that the machine learning model $f$ in C-ABL does not access ground-truth concept labels during training. On one hand, as you noted, the supervision signal for $f$ still comes solely from the target label $y$; on the other hand, each sub-base in C-ABL often contains multiple logically related concepts rather than focusing on a single one, so the process will not reduce to supervised learning of individual concepts and will not diminish the effect of reasoning shortcuts. The partitioning is performed before training as part of the curriculum design, and the concept labels are never exposed to $f$ or used for its supervision—exactly as in the baselines—ensuring that no information leakage occurs.

To ensure fairness, we also trained the baselines using the same phase-specific data streams on the MNIST addition (hexadecimal) task, and the final accuracies are reported below. As seen, simply applying the same schedule to the data without partitioning the knowledge base leads to very poor performance. This is primarily because unrelated or noisy rules can produce incorrect pseudo-labels, which severely undermines both learning stability and final accuracy.

The above findings demonstrate that the main performance gain of C-ABL stems from the principled partitioning of the knowledge base, which forms the foundation of our curriculum approach. Without this structured logic-level partitioning, no scheduling strategy alone would deliver the observed improvements.

---

Q2 Is curriculum learning really necessary? What would happen if we train only in phase 1？

A2 Thank you for the thoughtful question. Following your suggestion, we conducted an additional ablation experiment on the “phase-1-only” setting (with more data generated but only partial knowledge base $\mathcal{KB}_1$). As expected, the model trained solely on phase 1 converges more quickly in the early stages but exhibits extremely poor final performance. We will include this ablation study, along with full results and convergence curves, in the appendix of the final version. Briefly: on the MNIST Addition task (hexadecimal, $d=3$), test accuracy drops to 21.26% (C-ABL: 66.67%); on Chess Attack, to 12.59% (C-ABL: 86.79%); and on Legal Judgement Prediction, the F1 score falls to 0.325 (C-ABL: 0.904).

The underlying reason is that the knowledge base in phase 1 is structurally incomplete—it omits essential rules required to handle the full task (e.g., necessary legal rules for handling more comprehensive sentencing scenarios). Simply generating more phase-1 data cannot compensate for this, as the model is never exposed to the full set of reasoning patterns. These findings further confirm that a multi-phase curriculum is necessary for progressively introducing the complete rule structure and enabling generalizable learning.

---

Thank you again for recognizing the clarity, motivation, and empirical strength of our paper. We will reflect these revisions in the final version of our paper.

Thank you for your supportive and insightful comment. We will address your concerns as follows:

---

Q1 What is the significance of your paper?

A1 While curriculum learning has indeed been widely applied, our work is the first to design a curriculum from a logic-level perspective, by explicitly partitioning the knowledge base according to logical dependencies and complexity. This is not a trivial extension of standard curriculum strategies that merely reorder data samples or tasks. Our partitioning algorithm (Algorithm 1) formalizes how to decompose a rule set into sub-bases that enable smoother abductive reasoning and provable reductions in the search space (Theorem 4.2), which is a novel insight applicable to any neuro-symbolic framework requiring structured reasoning. In this sense, our approach provides a general methodology for curriculum design in knowledge-driven systems, potentially extending beyond abductive learning.

---

Q2 Why is ABL/NeSy research significant and relevant?

A2 We appreciate the reviewer’s perspective on the evolution of neuro-symbolic AI. It is true that recent advances—particularly in LLMs and code-generating systems like ViperGPT and AlphaGeometry—have opened exciting new paths for integrating symbolic reasoning. However, we believe that abductive learning (ABL) retains distinct value, especially in domains where explicit, verifiable, and logic-consistent reasoning remains critical.

ABL differs from many other paradigms in that it maintains the full formal rigor of symbolic reasoning, rather than relying on implicit heuristics or approximations. This makes it especially well-suited for real-world domains that possess both data and well-curated expert knowledge—such as judicial sentencing, historical document understanding, medical diagnosis, etc.—where knowledge bases are often readily accessible and incorporates a wealth of valuable information, yet the challenge lies in leveraging this rigorous and insightful domain knowledge effectively. ABL methods, explicitly driven by both data and knowledge without sacrificing integrity of each side, have demonstrated strong results in such domains [1,2,3,4]—an example is also given in our legal sentencing task (Sec. 5.3). LLM-based methods, while powerful, typically rely on implicit mechanisms (e.g., retrieval or prompting), which fall short in fully exploiting structured knowledge and lack transparency.

Moreover, in safety-critical domains like industrial control or defense, ensuring strict adherence to rules is not optional. ABL and related logic-based frameworks enforce such constraints by design [5,6], making them suitable for trustworthy, interpretable, and legally or ethically constrained AI—an area where black-box models remain insufficient.

Our contribution in this paper—C-ABL—advances this line of research by introducing the first curriculum framework for ABL that strategically organizes symbolic knowledge over learning phases. This allows reasoning complexity to grow in a controlled, data-efficient, and semantically stable way, further broadening the practical applicability of ABL systems, and potentially applicable to a wide range of neuro-symbolic paradigms.

---

Q3 Further clarification on experiment of chess attack and judicial sentencing.

A3 Thank you for the question and the opportunity to clarify. In the chess attack task, the positions of the pieces are given and fixed as part of the input layout, and the learning model $f$ is applied to each individual board cell image to identify what piece is present at that location. The concept label $z_i$ corresponds to the identity of the piece in cell $i$, not its position. Each input $x_i$ is a small image of one board cell, and $f$ maps it to a concept label $z_i$. These predicted concepts are assembled into a board-level representation, and logical rules (e.g., for attack range) are then applied to determine if the king is under threat. We will make this clearer in the final version of the paper.

In the judicial sentencing task, as described in Lines 323–325 and Appendix F.3, the model $f$ classifies whether the criminal jugement record expresses certain sentencing factor (e.g., “voluntary surrender”), and then reasoning is performed over these predicted factors using the legal knowledge base.

For the "pre-training setting", thank you for pointing this out. The term may have been used imprecisely in the original draft. Consistent with the protocol in the original ICDM 2020 paper [1], we assume that a small portion of the dataset (e.g., 10% or 50%) includes ground-truth concept labels. In addition to the abductive learning process, the model also learns to predict concepts from this limited supervision. We will revise the text to clarify this terminology and avoid potential confusion.

---

Thank you again for your high recognition of the quality, clarity and originality of our paper.

---

[1] Huang et al. "Semi-Supervised Abductive Learning and Its Application to Theft Judicial Sentencing." ICDM 2020.

[2] Gao et al. "Knowledge-Enhanced Historical Document Segmentation and Recognition." AAAI 2024.

[3] Wu et al. "Abductive Multi-instance Multi-label Learning for Periodontal Disease Classification with Prior Domain Knowledge." Medical Image Analysis 2025.

[4] Hu et al. "Efficient Rectification of Neuro-Symbolic Reasoning Inconsistencies by Abductive Reflection." AAAI 2025 (AAAI outstanding paper).

[5] Hoernle et al. "MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks." AAAI 2022.

[6] Belle. "On the Relevance of Logic for AI, and the Promise of Neuro-Symbolic Learning" Neurosymbolic Artificial Intelligence 2025.

Thank you for your thoughtful follow-up. While we agree that some ABL tasks, including some used in our work, may appear simplified, we believe the core idea—designing a curriculum through knowledge base partitioning to reduce reasoning complexity—has broader implications and can benefit a wide range of knowledge-driven AI systems and tasks.

We truly appreciate your high recognition of our paper’s quality, clarity, and originality. Please feel free to reach out if further discussion would be helpful.

Thank you for your supportive and insightful comment. We will address your concerns as follows:

---

Q1 Does the method use concept labels during training?

A1 To enable consistent transitioning across experiments, the phase transition check described in Lines 191–192 indeed involves access to concept label information. Yet it is worth noting that during experiments, this check is performed exclusively on a validation set, and the concept labels are not used during training or backpropagation.

Thank you for pointing this out, and, as a further exploration, we re-ran the MNIST Addition (hexadecimal) experiment using a fixed-interval schedule that do not require access to concept labels—specifically, the transition to the next phase occurs every 200 iterations. As shown below (each cell reports test accuracy and training time), C-ABL still maintains a clear advantage over the baselines, demonstrating that the improvements arise from the structured curriculum itself rather than from any concept label usage.

Finally, we would like to emphasize that the learning model $f$ in C-ABL never accesses ground-truth concept labels during training: the only supervision signal comes from the task label $y$, just as in the baselines. The validation-based threshold is used solely for determining when to trigger phase transitions, and does not introduce information leakage into the learning process.

---

Q2 What assumptions underlie Theorem 4.2? How is the stability of previous phase predictions ensured in Theorem 4.2?

A2 We agree with you that Theorem 4.2 relies on the stability of concepts learned in previous phases. In our method, this stability is ensured by the curriculum schedule (Lines 191–195), where a new phase only begins once $z \in \mathcal{Z}_p$ achieve accuracy above the random baseline, providing a sufficiently reliable foundation. Empirical evidence in Figure 3 (and Appendix F.1) shows that earlier concepts retain their accuracy during later training, validating this assumption in practice.

Moreover, as is common in neuro-symbolic learning, we assume that the semantics of concepts remain constant across the dataset [1,2], characterized as $p(z\mid x)$ remain unchanged as the dataset or knowledge base evolves. This is a natural and implicit assumption in many settings—for example, an image of the digit three consistently corresponds to the concept $z = `three`$ regardless of training phase. It enables C-ABL to build on previously learned reasoning without semantic drift. We appreciate your comment on this point and will revise Theorem 4.2 to explicitly state this assumption and clarify why the abduction space reduction remains valid and does not revert to the exponential bound in Lemma 4.1.

---

Q3 What is the logical formalism used to represent domain knowledge?

A3 The domain knowledge, as in ABL or DeepProbLog, is indeed expressed in a logic programming formalism, implemented using Prolog, which is a well-known subset of first-order logic focusing on Horn clauses. While we use the term “first-order logic” broadly to emphasize its logical expressiveness, we agree that specifying the logic programming would improve clarity, and we will revise Section 2.1 accordingly in the final version of the paper.

---

Q4 Details about the knowledge base partitioning algorithm.

A4 We address below several questions regarding the partitioning algorithm (Algorithm 1). Thank you for raising these points—clarifications will be added in the final version.

• (Cycles) You are correct that our algorithm does not assume acyclic dependency graphs. In cases where the concept dependency graph contains cycles, Line 4 of Algorithm 1 typically groups the involved rules into the same sub-base, which is then introduced as a whole in the curriculum. However, in practice, the knowledge bases rarely exhibit cycles during the initial graph construction—a cycle would imply that the conclusion of one rule is also used to justify its own premise, which is uncommon.

• (Topological sort) Regarding the topological sort in Algorithm 1 Line 8, we acknowledge that the original explanation may have been too brief due to space limitations; below is a more detailed clarification. The ordering is derived as follows: if there exists an edge $(r_i, r_j) \in E$ with $r_i \in C_a$ and $r_j \in C_b$, then we set $C_a \prec C_b$. This reflects that sub-base $C_b$ depends on $C_a$, and should appear later in the curriculum. Since this forms only a partial order, multiple valid topological sorts are possible; for unordered pairs, their relative order has no effect on the algorithm’s behavior—just like in standard topological sorting procedures, where such pairs are typically ordered arbitrarily. In our implementation, we heuristically place smaller clusters earlier, but we have observed almost identical experimental results when using a random ordering. We will add this in the final version of the paper.

• (Hyperparameter $\tau$) As for the use of $\tau$, as stated in Line 167, it serves to avoid overly fine-grained partitions by allowing small clusters to be optionally merged with their immediate successors until a minimum number of concept predicates per phase (namely, $\tau$) is reached. This parameter is optional—if one does not mind having a larger number of phases, it may be omitted. We provide a sensitivity analysis of $\tau$ in Table 4 of Appendix F for reference.

---

Q5 Connection between ABL and top-k inference.

A5 Regarding the choice of using the MAP state (i.e., $k=1$) as supervision, this design is indeed intentional. First, it provides computational advantages over approaches like weighted model counting, as it avoids the combinatorial cost of summing over all possible explanations—particularly valuable in early training when many candidates are noisy or semantically implausible. Second, using a single best explanation greatly simplifies the abduction process, making it more tractable to identify, validate, and refine hypotheses throughout training.

We appreciate your references to recent work on probabilistic inference methods, and will update the discussion in the final version to better contextualize our approach within this broader landscape.

---

Q6 Other minor points.

A6 Thank you for the thoughtful comments and references—we appreciate the reviewer’s broad perspective and insightful connections across related work.

• (Unique assumption) We will revise Line 114–115 to clarify that the uniqueness of concept labels per input is an assumed property of the datasets, and we will reference Marconato, Emanuele, et al., 2023 to contextualize this more carefully.

• (Accuracy) Table 1 indeed reports concept prediction accuracy, as in ABL and A³BL—we will make this explicit in the caption to avoid ambiguity.

• (Related work) Regarding Lines 336–337, we will clarify that C-ABL primarily uses first-order logic, which, while specific, remains expressive enough to capture a wide range of domain knowledge. Additionally, according to the official code of ABL [3], ABL framework can also be implemented with other logical formalisms, e.g., propositional logic and satisfiability modulo theories; similarly, C-ABL can be adapted to these settings with minor modifications.

---

Thank you again for your recognition of our paper. We also sincerely appreciate your review comments, which have provided us with valuable new insights. We will incorporate the corresponding revisions in the final version.

---

[1] Marconato et al. "Neuro-Symbolic Continual Learning: Knowledge, Reasoning Shortcuts and Concept Rehearsal." ICML 2023.

[2] Marconato et al. "BEARS: Make Neuro-Symbolic Models Aware of their Reasoning Shortcuts." UAI 2024.

[3] Huang et al. "ABLkit: A Python Toolkit for Abductive Learning." Frontiers of Computer Science, 2024.

Thank you for your supportive and insightful comment. We will address your concerns:

---

Q1 How can the robustness be validated in unstructured or noisy knowledge bases?

A1 While C-ABL benefits from some degree of modularity, it does not assume perfectly clean or pre-modularized knowledge bases: It does not rely on manually specified modules or expert-defined clusters, instead, constructs a dependency graph based on actual logical use relations between predicates, and induces a curriculum in a structure-agnostic manner. It also allows fine-grained control via the threshold $\tau$ to avoid over-fragmentation in poorly structured KBs, therefore even when modularity is weak or noisy, C-ABL still yields meaningful phases by capturing local dependency clusters and avoids overfitting to artificial structure.

In real-world tasks such as legal judgment prediction and chess attack, the knowledge bases we use are indeed not explicitly modular—many rules are loosely connected, overlapping, or even noisy—yet C-ABL still shows consistent improvements over ABL and A³BL, demonstrating its robustness in practice.

---

Q2 Has the impact of different thresholds on final performance been experimentally quantified?

A2 Our current transition strategy—advancing to the next phase once performance exceeds random—is already among the most aggressive. Even under this strict setting, C-ABL achieves strong results. We have also experimented with higher thresholds and observed that final test accuracy can improve slightly further. Thank you for the suggestion—designing more adaptive or fine-grained transition strategies is a promising direction we plan to explore.

---

Q3 If new rules are added during training, does the knowledge base need to be repartitioned?

A3 Thanks to the design of Algorithm 1 and the theoretical guarantee provided by Theorem 4.6, repartitioning is not necessary when new rules are added during training.

---

Q4 Can C-ABL accommodate uncertain reasoning?

A4 While the current version of C-ABL is built on deterministic abductive logic, its design does not preclude extension to probabilistic reasoning. Specifically, the core assumption that would need to be relaxed is the binary truth valuation used in standard logic programming (i.e., each concept is either true or false). To support probabilistic logic, one could extend the abductive engine to reason over distributions—e.g., by using weighted abduction or probabilistic logic programming frameworks, e.g., ProbLog or (perhaps) distributional extensions of ABL.

The curriculum structure itself—i.e., the phase-wise introduction of logically related sub-bases—can remain intact. In fact, introducing probabilistic knowledge in a staged manner may help mitigate the complexity of inference. We consider such probabilistic C-ABL variants to be an exciting direction for future work.

---

Thank you again for recognizing the overall ideas and significance of our paper.