Curriculum Abductive Learning
Authors: Wen-Chao Hu
Qi-Jie Li
Lin-Han Jia
Cunjing Ge
Yu-Feng Li
Yuan Jiang
Zhi-Hua Zhou
Venue: NeurIPS 2025 poster ()
Keywords: abductive learning
neuro-symbolic learning
logical reasoning
knowledge base
Primary area: general_machine_learning
TL;DR: We propose Curriculum Abductive Learning (C-ABL), a method that explicitly leverages structural properties of domain knowledge bases to improve training efficiency, stability and reasoning accuracy in ABL.

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.

Curriculum Abductive Learning
Wen-Chao Hu 1.2 Qi-Jie Li Lin-Han Jia'
Cunjing Ge
Yu-Feng Li Yuan Jiang' Zhi-Hua Zhou 1.2
National Key Laboratory for Novel Software Technology, Nanjing University, China
School of Artificial Intelligence, Nanjing University, China
(huwc,liqj jialh, jangy, zhouzh)@1 @ladda...aaamdmmmmmmaaa .nju. aaj..duu..u. edu..n, cn, cec.genninngnn, edu cn
Abstract
Abductive Learning (ABL) integrates machine learning with logical reasoning loop: learning model predicts symbolic concept labels from raw inputs, which are revised through abduction using domain knowledge and then fed back for retraining. Due 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 knowl- edge base as a static black box. In this work, we propose Curriculum Abductive Learning (C-ABL), method that explicitly leverages the internal structure of the knowledge base address the ABI. training challenges. C-ABI. 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 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.
Introduction
Recent advances in data-driven machine learning methods have achieved remarkable success across a wide range of tasks [Zhou, 2021]. However, when it comes to leverage structured, formal knowl- edge-such as logic rules or domain-specific constraints- these methods often fall short, therefore mitigating their interpretability and reliability. This has motivated the emergence of data- and knowledge-driven artificial intelligence, which aims to tightly integrate learning from data with logical reasoning [Hitzler, 2022, Yang et al., 2025].
Abductive Learning (ABL) [Zhou, 2019, Zhou and Huang. 2022] is a representative framework that integrates machine learning with logical reasoning balanced loop. In ABL, machine learning model first predicts symbolic concept labels from input data, which may initially violate domain knowledge when the model is under-trained; abduction then generates a of concept label candidates compatible with domain knowledge, and from which the best candidate is selected via consistency optimization. The selected candidate is subsequently used to supervise the model's 'ss next update. This loop iterates, enabling the system to gradually align predictions with logical constraints.
While elegant in principle, this loop setup often introduces training inefficiency and instability [Yang et al., 2024, He and Li, 2025]. Due to the nondeterminism of abduction [Magnani et al., 2009]. often multiple candidates compatible with the domain knowledge exist. When the knowledge base is complex, the resulting abduction space can become prohibitively large, causing the model to oscillate among many plausible yet incorrect concept labels. This undermines the reliability of the supervision signal and hinders the training process.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Prior work [Huang et al., 2021, Tao et al., 2024, He et al.. 2024] has attempted to improve ABL stability by refining consistency optimization, i.e., the selection process within the abduction space. While these methods offer incremental gains, they remain fundamentally limited when the space itself too large. key limitation that they treat the knowledge base as a static black box, ignoring its internal structure [Hu et al., 2025]. We argue that a more principled solution requires leveraging the structure of knowledge base and actively manage its complexity throughout training.
Many knowledge bases exhibit inherent staged or hierarchical structural properties [Glanois et al., 2022]. Take the legal domain as an example [Huang et al., 2020], some laws and regulations are relatively simple and foundational, while others involve more complex conditions or exceptions. Such rules can be organized into phases of increasing complexity. Motivated by this commonly observed structure, we propose logic-level Curriculum Abductive Learning (C-ABL) method. Instead of exposing the model to the full knowledge base from the beginning. C-ABL incrementally introduces logic rules over several phases during training. In this way, abduction space is substantially reduced throughout training, enabling faster and more stable optimization. As training progresses, increasingly complex logic is introduced, allowing for a gradual alignment with full-domain constraints.
To realize this idea, we first propose a principled algorithm that leverages the structural properties of knowledge base and partitions it into a sequence of sub-bases with strong local dependencies, increas- ing complexity, and self-contained reasoning. These are then integrated in ABL via curriculum training process. Our theoretical analysis shows that this curriculum design achieves efficient and stable training within phases by leveraging prior knowledge learned in earlier phases, and ensures smooth transitions across phases without catastrophic forgetting. Experiments on digit addition, chess attack, and world legal judgment tasks demonstrate that C-ABL. significantly improves training stability, convergence speed, and final accuracy over strong ABL and neuro- seuro-symbolic baselines.
In summary, our contributions are as follows:
We identify the uncontrolled abduction space as the core bottleneck in ABL training and highlight the limitations of prior implementations.
We propose the C-ABL method that explicitly structures and stages the logical knowledge base, transforming abduction from a static process into a dynamic and adaptive one.
We theoretically and empirically demonstrate that our curriculum design leads to a more efficient and stable training across various domains.
To our knowledge, this is the first work to explicitly leverage the internal structure of the knowledge base in ABI-an an essential yet previously overlooked somponent-shitinng from black-box invocation to structure-aware, logic-enhanced training. This perspective opens new directions for improving ABL and extends its potential for broader and more reliable applications.
2 Problem Setting and Preliminary
2.1 Knowledge Base Concepts
We begin with a brief introduction to the basic concept of a knowledge base. This paper focuses on domain knowledge expressed in logic programming [Huth and Ryan, 2004, Ben-Ari, 2012], a formalism rooted in first-order logic that offers expressiveness and human interpretability. while re- maining computationally tractable for reasoning. Specifically, the knowledge base KBB is represented as set of logic programming rules. Each rule is written in the Horn clause form [Hom, 1951]:
A + B1 A ... Bn:
where "t" denotes logical implication, meaning that A (left-hand side, the head of the rule) holds when the conjunction of B; (right-hand side, the body of the rule) holds. Each A and B, is a logical statement expressed with logical predicates.
Given such a knowledge base, both deduction and abduction [Josephson and Josephson, 1996, Magnani et al., 2009] can be performed. Deduction allows deriving A (head) from the premises A," B, (bodies), while abduction seeks possible explanations A, B, (bodies) that would account for the observation of A (head).

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2.2 Problem Setting
The task of this paper is as follows: Given an input data sequence a = (x())... where each element z(()) is sampled from an imput space X' and corresponds concept label 2(( drawn from a symbolic concept label set Z = {z1...., } the entire sequence æ is associated with a target label y that reflects a target reasoning outcome. In addition, we are provided with a logical knowledge base K,B, expressed in first-order logic, that defines how concept labels can entail specific target labels. Specifically, each concept label E appears as predicate in the bodies of rules in K.B, and target labels appear in the heads.
(m)) of length m.
The goal is to learn a model f that maps each x+0) to a concept label e0)) such that the sequence (2 ,.... mm)) together with K.B, can be used to deduce the target label y, in other words, satisfies logical condition = A KB 1 y. Example 2.1 (Decimal d-digit Addition). Consider a task of predicting the sum of two decimal d-digit numbers, represented by sequence of images [Manhaeve et al., 2018]. The input a = 2d (x(2), x (2)) consists 2d images from a visual input space X. such as MNIST [LeCun et al.,
The concept labels are drawn from the symbol set Z {zero,.... nine), and the target label y is their sum. The knowledge base KB encodes the rules of multi-digit addition. The goal is to map each x8 to its corresponding concept label 2 such that the resulted sequence (e.g.. lone, three, seven, threel) and KBB allows deduction of the correct sum y (e.g.. 86).
1998]ar or CIFARIO (Krichevsky et al., 2009], (e.g., E
2.3 Brief Introduction of Abductive Learning
Abductive Learning (ABL) [Zhou, 2019, Zhou and Huang, 2022] consists of two components: a machine learning model f + Z for concept perception, and logical reasoning module that utilizes the first-order rules in K.B. During the training process, when the input x is received, ABL uses f to map to perceptive concept labels ε, then the logical reasoning module checks whether & A KBB F y holds, i.c., & A KB can correctly deduce y. If the condition is not satisfied, ABL utilize the reasoning module to get revised concept labels ž, and then, Σ is used as the ground-truth concept label for æ to train f. This process is repeated iteratively until the training of f converge.
Specifically, the process by which the reasoning module obtains the revised concept labels ă involves the following two sequential steps [Dai et al., 2019, Huang et al., 2024]: (1) Abduction: Return the abduction space, i.e., the set of all concept label candidates that are compatible with KBB, denoted as S = { z A KB F y): and (2) Selection: Performing consistency optimization to select the candidate S that is most consistent with the model's current prediction p = f(z):
112
II p (2(0),
arg min EE
i=1
where (z)0) denotes the probability assigned by the model to label z(()
Challenges. Due to the nondeterminism of abduction, for a given y. the size of abduction space [S] is often large, i.e., there may be multiple z satisfying z A KBB + y. However, for each input æ, the correct concept labels should be unique [He et al., 2024]. Therefore, selecting from introduces erroneous supervision to the perception model f, which could then be reinforced through subsequent iterations in the learning loop. This creates vicious cycle that slows convergence, destabilizes training, and ultimately degrades final model accuracy.
incorrect
Example 2.2. In Example 2.1, given y === 86 and KBB, the resulting set 3 would be of size L = 87 (S = ([zero, zero, eight, s siz). [zero, one, eight, five). ...). However, the concept labels that consistent with x A 7 3 should be unique:
[one, three, seven, three). In this case, all other elements in S may introduce false labels that mislead the training of f-
Several solutions have been proposed to address this training challenge, including methods by Huang al. [2021], Tao al. [2024], He et al. [2024], which primarily focus on consistency optimization, i... how to select the best candidate within S. However, such methods remain limited when $ is large-no selection strategy can fully compensate for the challenges brought by an overwhelming hypothesis space YYang et al., 2024, Jia et al., 2025]. Addressing the root of training instability requires going beyond improving candidate selection and actively managing the size and structure of the abduction space itself.
3

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3
Our Method
This section proposes Curriculum Abductive Learning (C-ABL), method that addresses the limitations in previous ABL methods. C-ABL actively managing the complexity of the knowledge base during training: Instead of exposing the model to the full knowledge base all once, divides ABL training into multiple curriculum phases, with each phase guided by sub-base K.pp progressively expanding towards K.B.
3.1 Curriculum Design over Knowledge Base
To enable curriculum-style progression, first introduce a principled algorithm that partitions the full knowledge base K.B into a sequence of structured sub-bases K.B,,... K.Bp used in each curriculum phase. Our partitioning strategy follows three intuitive principles that together ensure each sub-base builds upon the last in stable and constructive manner:
P1 Dependeney Cohesion (What should be grouped together?): Rules depending on each other, such as those frequently co-occur or form tight reasoning chains, appear in the same phase.
P2 Stepwise Complexity (What should be introduced first?): Simpler rules are placed in earlier phases, while more complex rules built on prior predicates are introduced later.
P3 Self-contained Reasoning (Can reason independently?): Each phase includes all rules necessary for reasoning over the involved concepts, without relying on unseen rules.
To implement these principles, we introduce the following structure:
Definition 3.1 (Dependency Graph). Given a knowledge base KBB consisting of first-order logic rules, its dependency graph G = (V, E)) is directed graph where each node E V corresponds to rule, and there is an edge (ri, ry) E E if the head predicate of r; appears in the body of rj.
This graph captures the structural reasoning flow: r relies on conclusions from vú, there is an edge from ry to r.. Based on this graph, design partitioning algorithm shown in Algorithm 1.
Partitioning Algorithm. The algorithm begins by scanning each concept label x (as introduced in Section 2.2, each is predicate that appears in the body of some rule) and forms initial cluster C₂ containing all rules that directly reference z (Line 3). This cluster then recursively expanded by traversing the dependency graph, incorporating additional rules that are required to derive the final target label y. as well as rules that define intermediate predicates used this process (Line 4). In cases where the dependency graph contains cycles (e.g., formed by recursive rules). the above process will cluster all rules involved in the cycle together.
Algorithm 1: KBB partitioning
Input: Knowledge base KB (a set of first-order logic rules), minimum sub-base size 7 (optional)
Output: A sequence of sub-bases KB K.B..... KBp
Construct the dependency graph G (V, F.) of KB for each concept label E 2 do
>> from Definition 3.1;
Initialize cluster C₂ with all rules referencing z;
Recursively expand Cs2 via edges in G;
for each pair (Ca: C&) do
if C. Cb then Merge clusters;
else if (ri, E with rs E Cas ry E C2 then
Set C. x Os;
10 Topologically sort clusters under <;
Set phase counter p 1;;
12 for each sorted cluster C; do
13
| < T, merge C, with the next cluster Assign sub-base: KB, + U3=11, Cj;
> Z,; the set of concept labels C, involves;
15 p p 1;

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Once the initial clusters are formed, we refine them in two ways: (1) Merge any duplicate clusters with identical rule sets (Line 7); (2) Define a precedence ordering <, and perform topological sort (Line 9-10). Specifically, this ordering is based on the interdependencies of clusters: If there is an edge from a rule C2 to a rule Co. i.e., reasoning C₁ relies on conclusions from Ca. we assign C. < C (when bidirectional dependencies exist, we let the cluster with fewer rules appear earlier).
Sub-bases are then formed incrementally by aggregating clusters up to given index (Line 14). This process satisfies PI by grouping logether rules in the same reasoning chain, keeping each phase logically localized around certain concept predicates. also satisfies P2, the precedence ordering introduces foundational rules and simpler concepts earlier, and allows subsequent phases to build upon prior ones, forming a progression from easy to complex reasoning throughout training.
To avoid overly fine-grained partitions, small clusters can be optionally merged with their immediate successors (Line 13). This step is governed by optional threshold which specifies the minimum number concept label predicates per phase; if not set, no merging is performed.
Illustrative examples of the above process are shown in Appendix B.I. We also provide the com- putational cost analysis in Appendix B.2, showing that compared to the ABL training pipeline, the partitioning is a one-time, offline preprocessing step with negligible computational overhead.
Guarantee of Self-Contained Reasoning. To ensure P3, we define the concept label domain of each phase p. denoted ž,. the set of concept labels K.B,p involves. We require that K.B,, is logically sound and complete for reasoning over Zp. without relying on future rules. This is formalized below.
Theorem 3.2. For any phase p. let be any logical statement over concepts in Zp. Then:
1. Soundness: KBp } ip - KB - y;; that is, KB,, does not derive any conclusions that are invalid under the full knowledge base.
2. Completeness: KBp p KB F si that is, K.Bpp is sufficient to derive all valid conclusions over its concept label domain.
We then have the following corollary, indicating that the final sub-base KBP is logically equivalent to the full knowledge base K.B in terms of reasoning over all concept labels.
Corollary 3.3. For the final phase P, for any formula xp over 2, we have: KBp F = KB = +
All proofs are provided in Appendix C.
3.2 Curriculum-Guided Training
Building on the sub-bases from Algorithm 1, we now present C-ABL, which conducts ABL training across curriculum phases. The pseudocode is provided in Algorithm 2 in Appendix D.
In phase p. training is guided by sub-base KCpp. To align the model's prediction space with the reasoning scope of CBB, dynamically schedule training data whose concept labels fall within the domain 2, [Bengio et al., 2009, Marconato et al., 2023a]. Then, each phase follows the standard ABL procedure, stated in Section 2.3.
Training in phase p continues until the prediction accuracy for every label a E Z, exceeds the uniform guessing bascline 1/ and then C-ABL proceeds to the next phase + 1. This is mild criterion requiring only a weak signal for each concept to facilitate progression. Nonetheless, as shown in Section 4, reaching this level is sufficient for one phase to benefit learning in the next, enabling the model to build upon previously acquired knowledge a curriculum-style progression. Note that concept labels are evaluated a held-out validation set and not used for training backpropagation, and the only supervision signal during training comes from the target label
This process repeats until reaching the final phase P, at which point the full knowledge base K.B is employed and training continues until a termination condition met, such as reaching a maximum number of training iterations.
5

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Theoretical Analysis: Efficient and Smooth Optimization
In this section, provide formal analysis of the Curriculum Abductive Learning (C-ABL) method. We demonstrate how C-ABL improves the training process of ABL by addressing two core aspects: (I) Improving the efficiency and stability of training within each phase, and (2) Ensuring smooth transitions across phases. All proofs are provided in Appendix C.
4.1 Efficient Training within Phase
We begin by analyzing each phase individually. In particular, analyze how our method reduce the size the abduction space by benefiting from curriculum training paradigm, and therefore contributes more efficient reasoning and stable optimization.
The key bottleneck in ABL training arises from the size of the abduction space S { = A KB - y). This space 3 includes all possible concept labels compatible with the knowledge base, constituting the set of candidates ABL search and select from.
When using the full knowledge base, the size of S can grow exponentially:
Lemma 4.1. If the knowledge base KBB involves N concept labels and the imput has m positions, then the size of the abduction space with full knowledge base is bounded by [S] < N
To mitigate this, C-ABL partitions the knowledge base to reduce complexity. At phase p. the subset 2p S Z is active, and the abduction space is: Sp { 2 | = A K.B,p } ) The theorem below shows that, under the curriculum strategy proposed in Section 3.2, the size of S, largely reduced. Theorem 4.2. Assuming that all previously introduced concepts E Zp-1 are predicted with accuracy exceeding random chance (i.e., Acc(z) > [2]), and pseulo-labels are selected via Eq. (1). Then the size of the abduction space in phase p is bounded by |s,] S Z,, 1p-11"".
Remark 4.3. The above theorem relies on natural assumption that the semantics of concepts remain constant when the knowledge base evolves (c.g., the image 3 consistently corresponds to the concept three regardless of training phase) [Marconato et al., 2023b, 2024]. This theorem suggests that the abduction space in each phase effectively confined to newly introduced concepts Zp Zp-1. benefiting from the accumulated knowledge of earlier phases: The previously learned concepts provide a foundation with lower risk of noisy supervision [Van Krieken al., 2024, Maene al., 2024], enabling the model to focus resolving the new concepts introduced in the current phase.
Combining Lemma 4. and Theorem 4.2, the relative upper bound of the phase-level abduction space satisfies
Abduction Space dog scale)
10
Phase
S S
22p
2p-11
10a
Full
Z
10?
which shrinks exponentially in m. The following example illustrates this reduction:
10
Example 4.4. In the d-digit addition task, as shown in Fig- ure 1, the size of increases rapidly with d when applying the full knowledge base. In contrast, with phase partitioning (using a phase threshold T = in Algorithm 1), the abduction space remains consistently small across all values of d.
100
d
d
Figure 1: Comparision of abduction space w/ and w/o phase partitioning across different digit length.
This reduction yields two advantages:
1. Improved computational efficiency: A smaller abduction space significantly lowers the reason- ing cost per training iteration, especially when the knowledge base is large, where the number of logically compatible candidates explodes exponentially, making abduction prohibitively expen- sive [Yu et al., 2016, Hu et al., 2025];
2. Faster and stabilized convergence: A large abduction space introduces many plausible yet in- correct candidates, especially early training when the model's predictions are still noisy. These incorrect candidates can mislead training and cause the model to oscillate or reinforce suboptimal hypotheses. Limiting the abduction space will promote faster, more stable convergence, which captured in the following theorem:

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Theorem 4.5 (Iteration Complexity of ABL). Let f()) be the model at iteration t in ABL. Assuming that: (1) At each iteration, the model receives pseudo-label z E S that satisfies z0) A KB F y: (2) The model updates are driven by minimizing a loss function {(f(z), that is convex and p-Lipschitz: (3) The learning rate 1j > 0 is fixed. Then, the number of iterations T required to reach
an expected consistency error less than satisfies: T - [3]
This theorem formally shows that the number of required iterations grows quadratically with the size of the alduction space. Therefore, reducing 3 in each phase not only improves the speed of logical reasoning but also accelerates overall convergence, particularly in large knowledge base scenarios.
4.2 Smooth Phase Transition
Besides improvements within each phase, we now examine transitions across phases. Two properties are key to maintaining stability: logical consistency, ensuring conclusions made in earlier phases remain valid, and topological continuity, ensuring the optimization landscape evolves with no abrupt changes. Together, these properties help prevent catastrophic forgetting [Bengio et al., 2009], common challenge in curriculum learning where new updates may destabilize previously acquired knowledge, by preserving previously acquired knowledge both semantically and parametrically as the curriculum progresses.
To ensure logical consistency, we require that conclusions derived in one phase remain valid in all subsequent phases. The following theorem shows sub-bases from Algorithm satisfy this property:
Theorem 4.6. For any formula yp over 2p. we have KBpp } yp if and only if KBpp1 F sp
To analyze topological continuity, we adopt the notion of Stone spaces [Johnstone, 1982], which offer a topological view of logical model sets (a brief introduction is provided in Appendix C.7). Assuming each concept label 2 is unary predicate with Boolean assignment, the set of formulas over Zp then forms a Boolean algebra Bp. Its corresponding Stone space (BB))~ the set ultrafilters over captures the space of logically consistent models. For these model spaces, we have:
Theorem 4.7. For each p, ((pp) S ((Bp+1)
As a result, the model spaces form a nested sequence of compact topological spaces, ensuring that admissible models evolve smoothly, without causing abrupt shifts in the optimization landscape.
Experiments
In this section, evaluate method across three tasks. On digit addition and its variants, the most widely used benchmarks in the neuro-symbolic field, we thoroughly validate the effectiveness of our method. On chess attack task, test robustness when only Boolean target label is provided, leading to a significantly larger abduction space. Finally, we apply our method to real-world legal judgment task demonstrate its practical applicability in complex domains.
5.1 d-digit Addition
This task, as outlined in Example 2. 1, the input consists of image sequences representing two d-digit numbers, and the goal to predict their sum. We aim to assign concept label to each image, so that the final result is obtained through reasoning. The standard decimal setting includes 10 concept labels, and we use images from CIFAR [Krizhevsky et al., 2009] to represent digits. To extend the challenge, we introduce hexadecimal variant with 16 concept labels, constructed using the first 16 classes CIFARI00 [Krizhevsky et al., 2009]. More details and additional experimental results are provided in Appendix F.1.
We compare our method against two ABL baselines: (1) the original ABL [Huang et al., 2024], and (2) the improved variant A BL [He et al., 2024]. We also include several representative neuro-symbolic methods: (1) Neur [Yang et al., 2020], (2) LTN [Serafini and Garcez, 2016), (3) DeepProbLog [Manhaeve et al., 2018, 2021a], and (4) DeepStochlogg [Winters et al., 2022]. Introduction of compared methods are provided in Appendix
All methods use ResNet18 [He et al., 2016] as the perception module, and are trained for total of 5,000 iterations.
7

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Table Comparison of prediction accuracy (top) and total training time in minutes (bottom) on digit addition tasks. "N/A" indicates runtime exceeding 12 hours. C-ABL consistently achieves higher accuracy and faster training, with its benefits more pronounced as reasoning complexity increases.
Decimal
Hexadecimal
Method
=
d 2
d d=3
d 4
=
=
d ==3
NeurASP
10.00:0000
9.73-20.46
9.732046
N/A
6.27=
N/A
N/A
DeepProbLog
7.55
N/A
N/A
6.76
N/A
N/A
DeepStuchLog
70.56+00.9
73.25+140
72.88+1.00
71800+150
49.25+234
42.42+1172
35.10#2mz
LTN
68.25
71.10+152
71.77+11.8
70.95+1..6
12.85
53.59+2.
51.14
ABL
10.05
14.49
74.50.1.06
73.05
60.87:1122
61.75
64. 1..32
A
1.10
75.62
14.65
1..28
22.45
6..75
65.81 11.12
C-ABL
7..07
1..74
1..65
76.30
63.02 1.03
64.25 1112
66.67 1.11
NeurASP
113.2478
214.7+136
376.5
108.5+103
N/A
N/A
DeepProbLog
269.8477
N/A
N/A
214.3+14.2
N/A
N/A
DeepStochLog
32.812.1
59.2238
1014419
492.51224
34.5-23
97.3=59
182 189.119.1
LTN
30.2.22.
55.613.5
96,72668
473.2125.1
33.1=20
93.5
176.0188
ABL
12.61111
27.4
47.1
253.61148
20.8-122
49.1=
99.7
BBL
20.6
29.1
60.1
291.2+182
31.2-26
68.2=43
109.8
C-ABI.
16.2#14
21.5219
39.8=28
132.6275
16.4=1.9
32.0=2m
55.5=42
Accuraci C
D6
Accuracy
0.5
ABL
..5
Accuracy
0.4
ABI
0.4
ABL
0.3
C-A
0.2
C-ABL
C-ABL
1000
2000
3000
4000
3000
1000
2000
3000
4000
5000
1000
2000
3000
4000
5000
Iteration
Iteration
Iteration
Figure 2: Comparison of training curves the hexadecimal addition task with varying digit length d.
Experiments were conducted decimal addition (d 1 4) and hexadecimal addition (d 1 to 3). As shown in Table 1, compared to previous ABL and other neuro-symbolic methods, C-ABL consistently achieves higher accuracy and requires significantly less total training time, indicating more efficient reasoning process within each iteration. This improvement becomes especially evident in more complex settings (e.g.. higher-digit or hexadecimal tasks), where the knowledge base is larger and reasoning becomes more expensive, highlighting the benefits of curriculum-style learning.
Analysis of the Training Process. We examine how curriculum learning affects ABL training
process. Figure shows the training curves on the hexadecimal setting with varied d. We may see that C-ABL achieves faster and more stable convergence than ABL and A ABBL across all settings: While the baselines often exhibit noisy or stagnant training curves, C-ABL rapidly improves within the first few hundred iterations and reaches near-maximun performance by iteration 1,000. This aligns with our theoretical findings: C-ABL mitigates training instability and achieves faster convergence.
0.6
Figure 3 further shows the training curves of three randomly selected concept labels in C-ABL (shown in green curves), each introduced in a different phase, with three vertical dashed lines indicating the start of each phase. (Here we present the first 2,000 iterations; the full training curves are provided Figure in Appendix F.1.) As shown, each concept improves imme- diately and stably once its corresponding phase begins, while learned concepts from earlier phases remain unaffected. In con- trast, ABL introduces all labels at once (shown in red curves), leading early-stage oscillations and slower convergence for all labels. This support our claim: C-ABL enables stable, stepwise learning with smooth transitions across curriculum phases.
Accuracy
0.8
0.3
ABL
C-ABL
0.1
0
500
1000
1500
2000
Iteration
Figure 3: Training curve of three concept labels (hex., d = 1), each introduced in different phase.

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Table 2: Results on chess attack task.
attack
Iterations to Converge +
Method
Accuracy 1
attack
ABL
73. 13.75
4560
BL
74.964007
3389+ +
C-ABL
86.79.9966
2553
Figure 4: Examples of chess attack task.
Table 3: Results judicial sentencing task. C-ABL. achieves improved F1, MAE and MSE scores, and takes fewer tokens to converge (defined as the point where Fl reaches 99% of its final value).
Method Labeled Data
F1
MAE +
MSE 4
Tokens to Converge 1
ABI 10%
0.895 100024
0.762
000
4. 4.000M 3.199M
C-ABL
2.904
0.762
0.865
ABL C-ABL
0.910+n.00
0.773 +-0.000
0.874+0022
3.101M
50%
0.920+..006
0.722++0.014
0.743
2.106M
5.2
Chess Attack
We consider chess attack task [Dai et al., 2019, Huang et al., 2023]. The input is randomly generated chessboard populated with chess pieces, each represented by an image, and the goal is to determine whether any pair of pieces are in attacking relationship, aim to assign a concept label to each piece from Z (rook, pawn, bishop, king, knight, queen), } such that the assignments can logically determine the boolean target label (attack). The knowledge base defines the attack behavior each piece. In our setup, chess pieces are represented using randomly sampled MNIST digits, and we LeNet-5 [LeCun al., 1998] as the perception model. More details are provided in Appendix F.2.
examples in Figure 4. We
We compare C-ABL with ABL. and BI, with results shown in Table 2. C-ABL achieves higher accuracy with faster convergence. Notably, since the target label in this task binary, the abduction space is particularly large, highlighting the advantage of C-ABL in complex reasoning scenarios.
5.3 Judicial Sentencing
In this section, we apply C-ABL. to real-world task: judicial sentencing. This task aims to predict the final sentence length y e -+ based on the input criminal judgment records. The model first predicts concept labels representing sentencing factors (c.g., "voluntary surrender", "repeat offense"), and then reasons with legal knowledge base KB to deduce We use the dataset from Huang et al. [2020], which includes 687 records. We use the pretrained al., 2019] as the learning model. More details are provided in Appendix F.3.
chinese [Devlin
We compare C-ABL with prior ABL implementations under two sumi-super training settings, where 10% and 50% of the data include ground-truth concept labels available for training supervision. As shown in Table C-ABL achieves higher Fl scores, lower MAE and MSE, and requires fewer tokens to converge, showing superior accuracy and training efficiency in real-world setting.
6 Related Work
Recently, there has been notable progress in combining data-driven machine learning with knowledge- driven logical reasoning. Some methods embed logical structure into neural networks [Serafini and Garcez, 2016, Xu et al., 2018, Badreddine et al., 2022, Yang et al., 2022, Ahmed et al., 2022], often by relaxing logical constraints, which can undermine reliability. Probabilistic neuro-symbolic systems preserve classical reasoning by treating neural outputs as distributions over symbols [Manhaeve al., 2018, Yang et al., 2020, Kikaj al., 2025]. Their inference typically involves weighted model counting [Chavira and Darwiche, 2008], whose exact computation can be intractable for large or complex knowledge bases. Recent efforts have introduced approximate inference methods to alleviate this issue [Manhaeve et al., 202 lb, Winters et al., 2022, van Krieken et al., 2023, Li et al., 2023]. Abductive Learning (ABL) [Zhou, 2019, Zhou and Huang, 2022] integrates learning and reasoning in balanced loop, where abduction and consistency optimization provide a bridge between the two

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components. Supported by an souree toolkit [Huang et al., 2024], ABL has shown success in diverse applications [Huang et al., 2020, Cai et al., 2021, Wang al., 2021a, Gao al., 2024, Hu et al., 2025, Wang et al., 2025]. However, current implementations treat the knowledge base as black box and perform reasoning over the full logical space, often resulting in unstable supervision and inefficient convergence when the logical knowledge base is large complex.
Curriculum learning improves generalization by organizing the learning process from easier to harder [Bengio et al., 2009, Wang et al., 2021b]. In the neuro-symbolic field, prior work has incorporated curriculum strategies by gradually increasing the complexity of input data or task structure [Mao et al., 2019, Chen et al., 2020, Le- ee-Phooc et al., 2021, Marconato et al., 2023a, Abbe al.. 2023, Lorello et al., 2024, Wei et al.. 2025]. While they mainly focus on the data side, C- ABL introduces a curriculum over the knowledge base itself lever-levnggging its symbolic structure to guide reasoning progression. This allows the model to engage with increasingly complex logic in controlled, interpretable way, and better align with the goal of exploiting structured knowledge.
Another line of research investigates ambiguity and shorteut behavior in neuro-symbolic systems and ABL framework [Marconato et al., 2023a, 2024, Li et al., 2024, Yang et al., 2024, He et al., 2024], which shares an underlying connection with our analysis: Large abduction spaces in ABL introduce ambiguity in concept label selection, which can in turn result in unstable or shortcut-driven supervision [He al., 2024]. C-ABL tackles this structually by actively managing the reasoning space of knowledge base, thereby enhancing the reliability of reasoning supervision.
Partitioning knowledge bases into structured components has long been studied in the logic and knowledge representation communities [Amir and Mcllraith, 2000, 2005, Gocht and Balyo, 2017, Simiński, 2017]. While originally developed for formal reasoning, these ideas motivate our logic- guided curriculum for improving learning dynamics in ABL.
7
Conclusion
This paper presents Curriculum Abductive Learning (C-ABL), a method designed to improve the training stability and efficiency of Abductive Learning (ABL). Unlike prior methods that treat the knowledge base as a fixed black box, C-ABL leverages structual properties of knowledge base, partitions the logic into sub-bases and introduces them progressively. This curriculum design reduces the abduction space, stabilizes supervision, and enables the model to incorporate domain knowledge from simple to complex. We provide theoretical guarantees for phase-level improvements and smooth transitions, and demonstrate strong empirical results on synthetic and real-world tasks. By structurally organizing domain knowledge, C-ABL offers principled path toward scalable and efficient ABL..
Acknowledgements
This research was supported by the Jiangsu Science Foundation Leading-edge Technology (BK20232003) and the National Natural Science Foundation China (62176117). Cunjing Ge was supported by the National Natural Science Foundation of China (62202218). The authors would like to thank Reviewer #dk43 of OpenReview for the thorough review and insightful suggestions.
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A Limitation
While Curriculum Abductive Learning (C-ABL) demonstrates substantial improvements in training stability and reasoning performance, it also has several limitations. First, our partitioning strategy assumes that the knowledge base exhibits certain structural regularity- pendency chains-tha that supports meaningful curriculum design. This assumption holds for many real-world domains, including law, mathematics, medicine, structured game, etc. However, in some cases where the rule is densely entangled lacks clear structure, such partitioning may be less effective. Second, the current design of phase transitions is based on fixed structural principles and a simple accuracy threshold; more adaptive or learnable curriculum schedules could further improve performance but remain unexplored. Third, like standard ABL,, C-ABL assumes access to pre-defined, human-curated knowledge base. While our focus is on leveraging such structured knowl- edge, future work may explore methods for acquiring or refining the knowledge base, potentially by leveraging information from data.
as modularity or de-
B Additional Analysis of Algorithm
B.1 Examples of Knowledge Base Partitioning
This section illustrates examples of partitioning the knowledge base into sub-bases based on Algo- rithm I in Section 2.1.
Chess Attack. We use the chess attack task as the first example. As stated in Section 5.2, in this task, the input is a chessboard configuration with several pieces, and the concept label set is Z (rook, pawn, bishop, king, knight, queen). KB contains rules of the attack behavior for each piece, with the target label y = attack, indicating whether any pair of pieces in the chesshoard are an attacking relation.
Figure 5 illustrates a portion the aforementioned KBB, and provides visual example of how the dependency graph is constructed well how sub-base clusters are formed based on Algorithm 1.
Each node in the dependency graph corresponds to rule in the knowledge base, and a directed edge drawn from rule r; ry if the head predicate of r, appears in the body of ry- For example, in the top portion of the figure, the rule for Ishape depends on both left and fwd, and in turn supports the rule for attack when the piece is a knight. This forms a dependency path: left, fwd Ishape - attack.
Our algorithm begins by constructing an initial cluster C² for each concept label = E E by collecting all rules that directly reference z in their bodies. In the figure, these clusters are color-coded by background:
The green-shaded region corresponds Cinight which includes the attack rule with concept knight, the Ishape rule, and the supporting geometric rules left and fud.
The red-shaded region represents Crook. which includes the attack rule with concept rook, and the supporting 1ine predicate rules that it depends on.
The yellow-shaded region represents Chishop. which includes the attack rule with concept bishop, and the supporting diag predicate rules that depends on.
The blue-shaded region shows Cqueen. which includes the attack rule involving queen and the line or_ predicate, along with its dependencies on both 1ine and diag.
Edges the figure indicate predicate-level dependencies between rules, which guide the recursive expansion of clusters and the establishment of inter-cluster precedence. For example, Cqueen depends on both Crook and Cosshop due to its reliance on line and diag via line_orr _r__iagg so both Crook and Coiehop must precede Cqueea in the curriculum.
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(eft((Xt,Y1) (1), (2))
lahapa Y1), (X2, (X2,Y2))
1eft((X1, Y1), (X3,13, Y3)), fwd X3). (4)).
I1-X241, Y1-Y2.
attack Y1), ,,, (X2, Y2)>
fvd((X1,Y1), (2) 6- Y1-Y2+1, X1-X2.
fud fvd (4). (X2,Y2)).
(P)
lahape ((11, ((X1,1)), (X2, Y2)).
1shape ( (X1, Y1), (X2,2)) 6-...
line ((11, (X2, Y2)) X1=X2, Y1<Y2
(xtack((X1,11)). ,P, (x2, V2)) +
line (X), (x2,
pook(P),lined (X1,, (X2, Y2)).
line.or (xr.diaa((1111)))))) Y11, (X2,Y2)) 1ine ((I1((X))))) (22,
attack (X1,Y1), v1), (X2,, Y2)) <- queen (P).
line.or.diag ((11 Y1), (X2,Y2)) -
Line or www.oorrdrgg............... (X1, Y1), (X2, Y2)).
diag (X1, (X), Y1), (X2, Y2)),
attack (11,,1), (x2,22)
(X1,y1), (x2,1))) Y2))
bishop (P), ding (X),Y1), (x2,
Figure 5: A portion of knowledge base in chess attack task.
The resulting sub-bases, for example, follow an incremental structure such as:
KB₁ Cknight
KB2 Cmigght Czook
(2)
KB3 Cknight U Crook U Chiahop
KBA Cenight Crook U Chiehop Cqueen
d-digit Addition. In the digit addition task (outlined in Example 2.1 and Section 5.1), the knowledge base XB encodes multi-digit addition through symbolic predicates for digits and arithmetic operations. It includes logic for digit-wise computation and mappings from concept labels (c.g., zero, one) to numeric values, as illustrated below:
addition(Numl, Num2, Y) < number(Num1, Res1),
number(Num2, Res2),
Y is Res1 + Res2.
number (0, Res, Res)
number((H[]), Acc, Res) t digit(H, D),
Acc1 + 10 Acc, number(T, Acc1, Res).
number(X,N] < number(X.0,1)
digit(Pos, 0) < zero(Pos).
digit(Pos, 1) < one(Pos).
digit(Pos, 2) <- two(Pos).
digit(Pos, <- three(Pos).
digit(Pos, 4) <- four(Pos).
digit(Pos, 5) five(Pos).
digit(Pos, 6) <- six(Pos).
digit(Pos,, 7) seven(Pos).
digit(Pos, 8) <- eight(Pos).
digit(Pos, 9) < nine(Pos).
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When applying Algorithm 1, rules are partitioned based on concept label dependencies. For example, KB, may include rules for 2 {zero, one), then expand KB,, with 2n (zero, one, two, three), and so on. Fach K.B,p includes (1) digit-mapping rules relevant to 2p. and (2) shared arithmetic rules for addition (rules with predicate addition as head) and number construction (rules with predicate number head). While the core arithmetic logic remains fixed across phases, digit-related rules are introduced gradually.
The above two visualizations on chess attack and digit addition make clear how our algorithm traces dependencies to group semantically related rules into coherent sub-bases, and defines curriculum order based on the reasoning hierarchy encoded in the knowledge base structure.
B.2 Computational Cust Analysis
We now analyze the computational overhead introduced by the knowledge base partitioning algorithm, i.e., Algorithm 1 proposed in Section 3.1.
Time Complexity. Let n and be the number of nodes (also, number of rules in KBB) and edges in the dependency graph of KBB. The partitioning process involves the following major steps:
Dependency Graph Construction: For each rule, we inspect its head and compare it against all other rule bodies. This results in O(n²) comparisons in the worst case. Also, since predicates tend to be sparse in practice, hash-based indexing scheme can reduce this to O(n + +e.
Cluster Initialization and Expansion: For each concept label z E Z (with Z N), initiate a cluster and expand it by traversing the dependency graph, the overall time complexity of this process O(N (n e)).
3. Precedence Ordering and Topological Sort: After forming clusters, we identify precedence relations based on cross-cluster dependencies. This forms a directed acyclic graph over clusters, with topological sorting in O(N ez) time, where e, is the number of inter-cluster edges (typically e < e).
Putting all steps together, the overall complexity is approximately O(N(n e)) under practical sparsity assumptions.
Empirical Partition Time. In practice, evaluated the partitioning time on the two examples described in Appendix B.1. For the digit addition task, the knowledge base contains 14 rules, and partitioning takes 0.05 seconds. For the chess attack task, which includes 58 rules, partitioning completes in 0.9 seconds. All measurements were conducted on standard CPU (Intel Xeon Gold 6226R, single thread), with no GPU required.
To put this context, full ABL training run (e.g., under 1,000 iterations) typically takes several minutes to hours, depending on the task and model. Therefore, the knowledge base partitioning accounts for minimal portion the total runtime. More importantly, is one-time, offline preprocessing step, independent of data size iteration count, and its cost amortizes over training. In summary, Algorithm I incurs negligible computational overhead while providing benefits in reasoning efficiency, supervision quality, and convergence speed.
C Theorem Proof
C.1 Proof of Theorem 3.2
Proof. We prove the two directions separately.
This follows from the fact that KBB K.B. Any logical derivation that holds under KBB, , also holds under the full knowledge base KB, since KB contains all rules in KCB, Therefore, have:
Soundness.
KB, F p = KB - s.
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Completeness. We assume that KB F i.e., there exists proof (derivation) of yp using rules from K.B. Let R S KB denote the minimal of rules used in this derivation. We now argue that all rules in R are also contained in K.Bpp
As stated in Algorithm 1, for any predicate 2p. sub-base involving it is constructed by: (1) Create a cluster C2 by collecting all rules whose bodies mention z. (2) Recursively expand C2 by traversing the dependency graph G: If a rule depends on predicate a, and concluded by another rule, then that rule is also included. (3) Continue until all rules needed to derive any conclusions involving 2 are included. Therefore. for each on are included in K.B.p by construction. Since by assumption, yp only involves predicates in 2p, and the derivation of yp in KBB uses only predicates and rules these chains, we conclude that R C K.B.
Zp. all reasoning chains involving and the predicates it depends
Hence, the same inference of yp can be carried out within KBpp- Therefore, we have:
KB F 4 == KB = 4.
(4)
C.2 Proof of Corollary 3.3
Proof. Since the final phase P includes all clusters and thus covers all predicates in Z, we have
Zp = Z. Therefore, for any formula yp Z, we trivially have:
KBp Fx - KB
(5)
C.3 Proof of Lemma 4.1
Proof. Each of the m positions independently selects label from Z, giving N possible assignments in total. Logical constraints may eliminate some assignments, but the maximum remains N
C.4 Proof of Theorem 4.2
Proof. We first denote Zp Zp-1 J Ap. where AZ, Zp \ 2p-1 the new concept labels introduced in phase and for any z E Sp, we write 2 (Zuld, )new) accordingly.
Recall that the consistency score used for pseudo-label selection is:
(zn(2,f) f) II 1 (2(0).
(6)
where (2) denotes the model's predicted probability for concept ( at position i.
Now fix two candidate concept sequences (a) and (b) in Sp. such that: z(a) and 2b)) agree all positions i where 200) E Z, Zp-1, 1, but disagree some positions where 2 2p-1. Then their consistency scores differ only in terms involving 2,-1. By the semantic stability assumption, previously learned concepts retain their meaning across phases. Combined with the assumption that for any previously introduced concept 2 Zp-1, the model achieves accuracy exceeding random chance, so that for positions i with 200) E Zp-1, the predicted probabilities (2(() tend to assign higher mass to correct concepts learned in prior phases. Therefore we have Con(2(") f'
Con( z 2(h), f), meaning the selection strategy will typically prefer zaa) over (B)
This shows that, under consistency-baset selection, the model tends preserve earlier learned concepts (those in 1,, and variations in Sp primarily arise from the newly introduced concepts Z, 2p-1- Therefore, the number of valid configurations in Sp. is bounded by the number of possible combinations of knew:
Spl S S|AZ,''
2p Zp-1 m
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C.5 Proof of Theorem 4.5
Proof. Let the loss function iteration be defined as:
L = (z),
(8)
where z(() E S is selected via abduction given y and K.B, we aim bound the average suboptimality of the iterates:
1 EL se,
(9)
T
where I, miny =*) is the best possible loss under the true concept label z".
By chapter 14 in Shalev-Shwartz and Ben-David [2014], under the assumptions that: (1) loss function (1) convex, (2) the gradient norm bounded by the Lipschitz constant p. i.e., [|V S p. and
(3) the model lies in bounded domain of diameter D, then the regret over I steps is bounded by:
D)2
TT
Σ
100
L'
(10)
29
Choosing the optimal learning rate = MVT.
D
we obtain:
E L (1) 1-
Dp /T
(11)
T
To ensure this average suboptimality is below e, it suffices that:
Dp VT
Dp
2
72
(12)
Now, suppose that due to the abductive process, at each iteration f, the pseudo-label (+) is selected uniformly from a candidate set of size [S], then the effective variance or inconsistency in supervision may grow linearly with [S). To conservatively account for this, assume the domain diameter D = ((S)). which reflects the number of pseudo-label variants over which the model needs to generalize. Therefore, we have
T 0
(13)
which completes the proof.
C.6 Proof of Theorem 4.6
Proof. Recall that for each phase we have KBB, C KB and that Z, S Zp+1, therefore, by Theorem 3.2, both ICB,, and K.B++1 are sound and complete with respect to all formulas over Zp. Then, for any formula over Zp, we have:
KB F p + KBF
KBB KBp+1 F
(14)
where the first equivalence comes from applying Theorem 3.2 to KBB, and the second from applying it to KB p+1, since both sub-bases fully capture all reasoning over the same concept domain Zp.
Hence, KBB F to if and only if KBB 1 }= 4p completing the proof.
C.7 Background on Stone Spaces and Proof of Theorem 4.7
We first provide backgrounds on stone spaces [Johnstone, 1982]: Stone spaces provide topological perspective on logic by characterizing the space of all models (truth assignments) that satisfy a given set of logical formulas. Formally, given a Boolean algebra B, its Stone space S(B) is defined as the
25

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set of all ultrafilters over B, where an ultrafilter is maximal consistent set of formulas formulas-iituutivvlly corresponds complete and consistent truth assignment over the logic.
Stone's representation theorem [Stone, 1936] states that every Boolean algebra is isomorphic to the algebra of clopen (simultaneously closed and open) subsets of its Stone space. This construction enables logical entailment to be interpreted as topological containment: if E B holds in all ultrafilters SBB), then yp logical consequence of B.
our setting, for each phase p. we define a Boolean algebra Bp as the closure of all logical formulas over the predicate set 2p. The Stone space S(Bp) thus encodes all logically consistent concept-level assignments admissible under p.
Below is the proof of Theorem 4.7.
Proof. Recall that for each phase p. we construct sub-base K.Bpp such that KBB,, S KBB++ and that Zp _ 2p+1- Let Bp denote the Boolean algebra of formulas over 2p, and Bp+1 denote that over 2p+1. Since 2p 2p+1. every formula over 2p is also a formula over 2p+1, that is, the Boolean algebras satisfies B C Bp+1.
We now argue that any ultrafilter over B, be extended to ultrafilter over B pp+ Following Zorn's is Lemma [Conrad, 2016], any consistent set of formulas over Boolean algebra can be extended ultrafilter over larger Boolean algebra containing therefore, any model over 2p can be extended to a model over Zp+1 that preserves the truth values of formulas in By. Then, for every ultrafilter Up S(Bp). there exists least one ultrafilter Up+1* E S(Bp+1) such that Up Up. This directly implies:
S(B,) S S(B,+1),
(15)
since every model in phase p remains valid (or extendable) in phase p + 1.
D Pseudocode for Curriculum Abductive Learning
This section shows the pseudocode (see Algorithm 2) for the Curriculum Abductive Learning (C-ABL) training process, as stated in Section 3.2.
Algorithm 2: Curriculum Abductive Learning (C-ABL) Training
Input: Sub-bases KCBI.... K.B.p training dataset Output: Trained model f Initialize randomly;
2 for p = 1 to P do
Schedule training stream D, S D with concept labels in Zp:
repeat
for each x E D, do
2 < [(x);;
2 KB, | then
else
10
S - {z z A AAB, F y):
11
Z ト min zc3
(=))
from Eq. (1);
12
Update f using z;
13
until Phase transition condition met (p < P) or termination condition met (p = P);
E
Comparison Methods
In this section, we provide brief supplementary introduction to the compared baseline methods used in experiments.
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ABL and its Variants.
(1) ABL [Dai et al., 2019], the original implementation of Abductive Learning, supported by an efficient open-source toolkit [Huang et al., 2024]. For consistency evaluation, we adopt the confidence-based selection strategy defined in Section 2.3 (see Eq. (1));
(2) A 'BL [He al.. 2024], one of the most effective ABL variants, which improves concept label selection by evaluating all candidates in the abduction space.
Other variants such as Huang et al. [2021], Tao et al. [2024] also focus on refining label selection but perform less robustly than BL. We do not compare with methods like ABL-Reff [Hu et al., 2025] or ABL-PSP [Jia et al., 2025], they process the entire input sequence jointly and hence require different perception modules, making direct comparison less meaningful.
Neuro-Symbolic Methods.
(1) NeurASP [Yang et al., 2020], an extension of answer set programs [Brewka et al., 201 1] by treating the neural network output as the probability distribution over atomic facts;
(2) LTN [Serafini and Garcez, 2016], a neural-symbolic framework that uses differentiable first- order logic language. This is representative work of relaxing logical rules as soft constraints in neural networks;
(3) DeepProbLog [Manhaeve et al., 2018], an extension of ProbLog [De Raedt et al., 2007] by introducing neural predicates;
(4) DeepStochLog [Winters et al., 2022], a related strategy to DeepProbLog that improves inference efficiency through stochastic logic program.
F Experiment Details and Additional Results
All experiments are performed on a server with Intel Xeon Gold 6226R CPU and Tesla A100 GPU, and each experiment is repeated 5 times.
F1 d-digit Addition Tasks
Curriculum Design. The curriculum partitioning of KBB for this task is detailed in Appendix B. We set the minimum phase size to r = 2.
Additionally, Table 4 presents sensitivity analysis different values of conducted under the hexadecimal addition setting with d = 3. In this configuration, there are a total of 16 concept labels, and we experimented with values ranging from to 4. Results show that method consistently outperforms both ABL and A 'L across different т settings. In general, a smaller leads to finer-grained sub-bases and smaller abduction space, which often improve reasoning accuracy and training efficiency. Indeed, when the phases are too fine-grained, the model will go through more curriculum, which slightly brings implementation complexity.
Cross-Datase Analysis. We further validate our method on alternative datasets. In the decimal setting, we now use digit images from MNIST dataset [Krizhevsky et al., 2009], and in the hex- adecimal setting, we use digits 0-9 and letters A-F from EMNIST dataset [Cohen et al., 2017]. All methods use LeNet-5 [LeCun et al., 1998] as the perception model, and are trained for total of 1,000 iterations. Results in Table 5 show that C-ABL continues to yield superior reasoning accuracy and training efficiency compared to basclines.
Full Training Curve (Complementing Figure 3). Previously in Figure 3, we illustrated segment of the training curves for three randomly selected concept labels, each introduced in a different phase (with three vertical dashed lines indicating the start of their respective phases). Here in Figure 6, we present the full training curve of these concept labels (left). We also provide a zoomed-in view (right) focusing on the early training period of C-ABL, which highlights that C-ABL, each concept label improves immediately and stably upon the start of its corresponding curriculum phase, while the performance of previously learned concepts remains unaffected demonstrating the smoothness and stability of phase transitions.
27

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Table Sensitivity analysis on different values across different tasks. C-ABL outperforms ABL and BL across different 7 settings.
Digit Addition
Chess Attack
Method
Test
Training Time (min)
Test Accuracy
Training Time (min)
Accuracy
ABL BL
64. 14
99.7=sa
73.75
36.7228
65.81
109.8
74.96
45.6
66. 16.78
53.7=27
85.86
20.9+16
66.67+111
55.5=42
86.79
23.1
C-ABL
65. 15.76+108
64.8=49
+0.7
26.3+21
66.01
70.1
Table 5: Comparison of prediction accuracy (top) and total training time in minutes (bottom) on digit addition tasks with MNIST EMNIST dataset. "N/A" indicates runtime exceeding hours. C-ABL consistently achieves higher accuracy and faster training.
Decimal
Hexadecimal
Method
d
d=
d=3
d 4
d =
d d=3 2
d d=3
NeurASP
97.01
10.21 1.0.9
N/A
N/A
17.65
N/A
N/A
DeepProbLog
97.42
N/A
N/A
N/A
96.5940.48
N/A
N/A
DeepStochLog
99.00+0.3
98.72+0.14
98,59#0112
98.46+a.11
97.56+0.12
97.96+0.1
97.13
LTN
18.64.0018
98.53
98.48
98.344..19
97.1810.14
96.5319.1
97.1 17.11
ABL
98.7910.10
18.9910.08
98.59
98.2910.11
97.1710.10
97.491000
97.50
'LLL
98.944t012
99.03
98.68-t0.1
98.42+t..1
96.76
97.6 07.62
97.65
C-ABL
99.04
99.17 10,5
99.09-0004
99.17
98.48
98.41
98.83
NeurASP
33.1
87.8
N/A
N/A
48.3444
N/A
N/A
DeepProhL.og
44.8+115
N/A
N/A
N/A
69.8+77
N/A
N/A
SeepStochLog
3,9+0.
7.6+:09
38.5++1.
166.8
6.8405
24.0+20
139.449.1
LTN
4.2+004
8,0=110
29.1+
140.2+59
7.1
24.5+22
155.0+133
ABL
1.9402
2.5
11.9#112
80.3+78
1.4002
4.8
30.8=24
A A'BL
2.9403
4.6.20.
15.5
105,8165
3.210.
6,5
54.2=37
C-ABL
0.9
1.8
6,9+07
33.2+2.
1.0
3.4:14
20.6=211
F22
Chess Attack Task
Curriculum Design. The partitioning strategy for this task is also described in Appendix B. We again set the minimum phase size also in Table 4, where we evaluate т values ranging from 1 to 3 under the setting with concept labels. Results show performance advantage C-ABL over ABL and BL almost all tested values. As in the digit addition task, smaller introduces more fine-graines sub-bases and smaller abduction spaces, which may lead to improved accuracy or faster convergence.
A sensitivity analysis of this hyperparameter provided
F.3 Judicial Sentencing Tasks
Dataset Description. The dataset consists of 687 criminal judgment records for theft cases authored by judges in Guizhou, China, between 2017 and 2018 [Huang et al., 2020]. Each record included detailed descriptions of the defendant's background and criminal record, as well as some facts and legal basis of the ruling.
Knowledge Base. The legal domain offers rich structured prior knowledge that can be encoded as first-order logic rules, forming the knowledge base K.B. These include statutory law, attribute- matching rules, and general common sense constraints. Below is an example adapted from Huang al. [2020], which outlines how various attributes and the stolen amount of money determine the final sentence:
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Figure 6: Full training curves of three concept labels introduced in different phases (left) and zommed-in early-stage view for C-ABL (right).
penalty(X,7) base_penalty(X,2)) A weight(X, 22) A = (1 +
base bass_pennltyyle (eenaltyXXX)) +- money(X, m) A = 0.7m 5.7.
weight([]. f
weight((X(Xs],1) +- ellemen__veight 2,) weight(Xs, 22) A = 21 + Zz.
However, such symbolic knowledge is often incomplete- for instance, the rule "voluntary surrender reduces sentence" may exist, but the precise reduction is unspecified. To incorporate these partially defined patterns into a learnable system, we parametrize the knowledge base as:
(1 + w 2) + b),
(16)
where = a binary vector representing predicted case attributes (i.e., concept labels), m E the amount of money involved (a ground-truth numeric input), and RR+ is the predicted sentence length. This formulation mirrors the logical structure of first-order rules while allowing the unknown parameters w, a, and 6 be learned via a linear regression model. Like the original rule-based system, Eq. (16) supports abduction: given a known sentence length and monetary amount m, the model infer plausible case attributes z.
In summary, this task requires both the prediction of concept labels and the joint optimization of symbolic parameters to produce sentence predictions directly from judgment documents. The integration of symbolic reasoning and statistical learning makes this possible in a unified framework.
Curriculum Design. In C-ABL, the knowledge base is partitioned according the number of case attributes involved: earlier phases include rules with fewer attributes, while later phases expand to incorporate more. For example, the initial sub-knowledge bases contain rules for basic case attributes, such as whether the defendant is repeat offender. In contrast, later phases include rules incorporating additional factors, such the defendant's cooperation with law enforcement, whether the crime involves mitigating circumstances, etc. Specifically, we define the sub-knowledge base for phase as:
KBB {u (1 + w 2am b) 1 p).
(17)
ensuring that earlier phases focus on simpler cases, thereby facilitating smoother and more stable training process.
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Reviews:
[{"review_id": "IqHNsRRoVv", "reviewer": "Reviewer_7jdw", "summary": "This 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.", "questions": "- 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?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "#### **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.", "was_revised": true, "final_justification": "I recommend acceptance of this paper. The authors have not only addressed all major concerns but also provided substantial additional evidence through new experiments and theoretical elaborations that significantly strengthen the paper's contributions.\n\nHowever, there are still some very small points:\n\n1. While the probabilistic extension remains future work, the current deterministic framework already shows significant practical value in critical domains like judicial sentencing and medical diagnosis where rule-based certainty is paramount.\n\n2. Some implementation aspects of Algorithm 1, particularly regarding cycle handling in dependency graphs and the τ threshold selection strategy (currently tested with values 2-5 in Appendix F), could benefit from more detailed exposition in the final version to facilitate reproducibility.", "rebuttal": "Thank you for your supportive and insightful comment. We will address your concerns:\n\n---\n\n**Q1**  How can the robustness be validated in unstructured or noisy knowledge bases?\n\n**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. \n\nIn 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.\n\n---\n\n\n**Q2**  Has the impact of different thresholds on final performance been experimentally quantified?\n\n**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.\n\n---\n\n**Q3**  If new rules are added during training, does the knowledge base need to be repartitioned?\n\n**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.\n\n---\n\n**Q4**  Can C-ABL accommodate uncertain reasoning?\n\n**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.\n\nThe 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.\n\n---\n\nThank you again for recognizing the overall ideas and significance of our paper."}, {"review_id": "JjPXB2KdKU", "reviewer": "Reviewer_dk43", "summary": "The 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.", "questions": "**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.", "limitations": "Limitations are listed in the appendix.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "This is quite a solid paper with a nice contribution that can be used in several areas of Neurosymbolic AI. I had a clear discussion with the authors who seemed to have an open mind, and they resolved my main concerns (in particular, the use of validation concepts in phase transitions).", "rebuttal": "Thank you for your supportive and insightful comment. We will address your concerns as follows:\n\n---\n\n**Q1** Does the method use concept labels during training?\n\n**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. \n\nThank 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.\n\n|                                   | d = 1        | d = 2        | d = 3         |\n| --------------------------------- | ------------ | ------------ | ------------- |\n| ABL                               | 60.87 (20.8) | 61.75 (49.1) | 64.14 (99.7)  |\n| A³BL                              | 22.45 (31.2) | 60.75 (68.2) | 65.81 (109.8) |\n| C-ABL                             | 64.25 (16.4) | 64.25 (32.0) | 66.67 (55.5)  |\n| C-ABL (fixed-interval transition) | 64.68 (21.9) | 64.59 (45.5) | 66.70 (70.9)  |\n\nFinally, 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.\n\n---\n\n**Q2**  What assumptions underlie Theorem 4.2? How is the stability of previous phase predictions ensured in Theorem 4.2?\n\n**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. \n\nMoreover, 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 $z = \\texttt{three}$ 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. \n\n---\n\n**Q3**  What is the logical formalism used to represent domain knowledge?\n\n**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.\n\n---\n\n**Q4**   Details about the knowledge base partitioning algorithm.\n\n**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.\n\n- (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.\n\n- (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.\n\n- (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.\n\n---\n\n**Q5**  Connection between ABL and top-k inference.\n\n**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. \n\nWe 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.\n\n---\n\n**Q6**  Other minor points.\n\n**A6**  Thank you for the thoughtful comments and references—we appreciate the reviewer’s broad perspective and insightful connections across related work.\n\n- (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. \n\n- (Accuracy) Table 1 indeed reports concept prediction accuracy, as in ABL and A³BL—we will make this explicit in the caption to avoid ambiguity.\n\n- (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.\n\n---\n\nThank 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.\n\n---\n\n[1] Marconato et al. \"Neuro-Symbolic Continual Learning: Knowledge, Reasoning Shortcuts and Concept Rehearsal.\" ICML 2023.\n\n[2] Marconato et al. \"BEARS: Make Neuro-Symbolic Models Aware of their Reasoning Shortcuts.\" UAI 2024.\n\n[3] Huang et al. \"ABLkit: A Python Toolkit for Abductive Learning.\" Frontiers of Computer Science, 2024."}, {"review_id": "4MX6XDRkxY", "reviewer": "Reviewer_c9eA", "summary": "This 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.", "questions": "For 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.", "limitations": "N/A", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 1, "originality": 3, "strengths_and_weaknesses": "## 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.", "was_revised": true, "final_justification": "It is clear this paper is borderline, but it is hard to know whether it should be borderline accept or borderline reject. I do not have high confidence that I have the best perspective of the reviewers, but I will keep my score as the author rebuttal did not change my mind much about the significance of this line of research. If this concern were addressed, I would raise my score to a 4 or 5.", "rebuttal": "Thank you for your supportive and insightful comment. We will address your concerns as follows:\n\n---\n\n**Q1**   What is the significance of your paper?\n\n**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.\n\n---\n\n**Q2**   Why is ABL/NeSy research significant and relevant?\n\n**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**.\n\nABL 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.\n\nMoreover, 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.\n\nOur 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.\n\n---\n\n**Q3**  Further clarification on experiment of chess attack and judicial sentencing.\n\n**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.\n\nIn 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.\n\nFor 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.\n\n---\n\nThank you again for your high recognition of the quality, clarity and originality of our paper.\n\n---\n\n[1] Huang et al. \"Semi-Supervised Abductive Learning and Its Application to Theft Judicial Sentencing.\" ICDM 2020.\n\n[2] Gao et al. \"Knowledge-Enhanced Historical Document Segmentation and Recognition.\" AAAI 2024.\n\n[3] Wu et al. \"Abductive Multi-instance Multi-label Learning for Periodontal Disease Classification with Prior Domain Knowledge.\" Medical Image Analysis 2025.\n\n[4] Hu et al. \"Efficient Rectification of Neuro-Symbolic Reasoning Inconsistencies by Abductive Reflection.\" AAAI 2025 (AAAI outstanding paper).\n\n[5] Hoernle et al. \"MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks.\" AAAI 2022.\n\n[6] Belle. \"On the Relevance of Logic for AI, and the Promise of Neuro-Symbolic Learning\" Neurosymbolic Artificial Intelligence 2025."}, {"review_id": "05DjDIhAaq", "reviewer": "Reviewer_pZZ6", "summary": "This 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.", "questions": "I 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.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\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).", "was_revised": true, "final_justification": "The author's response has effectively alleviated my concerns about data leakage and the necessity of curriculum learning.", "rebuttal": "Thank you for your supportive and insightful comment. We will address your concerns as follows:\n\n---\n\n**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?\n\n**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.\n\nTo 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.\n\n|                        | d = 1 | d = 2 | d = 3 |\n| ---------------------- | ----- | ----- | ----- |\n| ABL (data scheduling)  | 17.58 | 15.62 | 10.30 |\n| A³BL (data scheduling) | 19.45 | 14.95 | 12.98 |\n| C-ABL                  | 64.25 | 64.25 | 66.67 |\n\nThe 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.\n\n---\n\n**Q2**  Is curriculum learning really necessary? What would happen if we train only in phase 1？\n\n**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).\n\nThe 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.\n\n---\n\nThank 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."}]

Author rebuttal:

Meta-review: The paper initially received mixed reviews in the initial review with 1 accept, 1 borderline accept and 2 borderline reject scores. Major concerns raised were centered around three key aspects: the fairness and potential information leakage in the curriculum design, uncertainties and gaps in the theoretical assumptions and algorithmic details, and the limited empirical evaluation and broader relevance of the proposed approach.

The authors have provided a rebuttal that addresses many of these concerns, as evident from the increased scores to 2 Accepts, 1 Borderline Accept, and 1 Borderline Reject. The remaining concern, highlighted by reviewer c9eA, centers on the broader significance of abductive learning and its applicability to real-world tasks. The AC acknowledges this point, as it speaks to a valid and ongoing discussion within the neurosymbolic AI community regarding the evaluation of new methods on "toy" problems versus large-scale, real-world applications. However, the AC believes that this line of research does have potential for providing a way towards interpretable, logic-consistent systems that can benefit from symbolic systems. For these reasons, I recommend acceptance. The authors should incorporate the clarifications from their rebuttal into the final version of the paper.
Decision: Accept (poster)