Reviewer has chosen not to be Anonymous

Overall Impression: Good

Content:
Technical Quality of the paper: Excellent
Originality of the paper: Yes, but limited
Adequacy of the bibliography: Yes, but see detailed comments

Presentation:
Adequacy of the abstract: No
Introduction: background and motivation: Limited
Organization of the paper: Satisfactory
Level of English: Satisfactory
Overall presentation: Good

Detailed Comments:

Summary:
NeTheR revises imperfect propositional logic theories by incrementally inserting neural concepts and symbolic literals into an arithmetic circuit, guided by a modified Sharpe ratio heuristic. This is a well-written and technically clear paper. The core idea is useful, but the framing, scope, and some methodological aspects call for an honest revision.

Major Issues
----------------

1. Novelty is overclaimed:
The paper presents itself as broadly novel in handling imperfect symbolic knowledge and performing theory revision/predicate invention in NeSy - which it isn't at all (at that level of generality). Plenty of prior NeSy work covered that also - starting from old school KBANN work and alike with the NeSy learning/revision cycle (symbolic theory -> neural revision -> theory extraction), d-ILP and alike with differentiable structure learning over latent symbolic theories, or LRNNs with structure learning (particularly close with its greedy, accuracy-gain-driven theory revision via gradient descent, already in a FOL/relational setting)
- thus the actual, defensible novelty is much narrower - performing theory revision *within the ProbLog/arithmetic circuit paradigm* (DeepProbLog/Scallop style).
- that's arguably a solid contribution -- just scope it respectively, for which abstract and intro call for some honest rewriting...
- also related work should could mention some of the above mentioned (and alike) broader NeSy works (or scope them out by acknowleding the prob-prog focus right from the start)

1b. Better scoping needed:
On a related note, it could be argued that the ProbLog-style NeSy paradigm (narrow compositional integration) is a more clean/auditable NeSy methodology (compared to the tighter but "fuzzy" semantics-based integrations/works). Nevertheless, NeTheR's headline move of introducing "neural concepts with no predefined semantics" actually pushes it into that fuzzy NeSy territory that ProbLog-style works like to avoid/disdain.
- the cleanliness/rigor/interpretability motivation is then quite circular...
- and claims like "Gills = True ∧ neural concept" is meaningfully interpretable are largely overstated (authors lists SHAP etc. as remedies, but none is evaluated)

2. Propositional scope not stated clearly/upfront:
The paper works *entirely* with propositional theories, but this only becomes clear very subtly/slowly. It should be in the abstract, and very likely the title already! Afaik, "Predicate invention" is an ILP/FOL term -- using it for propositional literal insertion will confuse readers from the community.
- perhaps "Concept insertion" or "literal invention" could be more accurate, I think?
- mention that instead of examples like "autonomous driving" in the abstract, which just add to the confusion/overclaiming...

3. Experimental overclaims...
The experiments are more of an ablation study than a true benchmark comparison:
- M* is generated with a structure that matches what NeTheR is designed to search over. This implicitly favors NeTheR (i.e., structural bias).
- The And Or, Or And, MLP+NN are essentially constructed foils (i.e., weak baselines). Classical theory revision systems are cited but excluded -- the paper could perhaps at least explicitly say why (because they can't handle subsymbolic data? Perhaps discuss whether a hybrid pipeline would be a stronger baseline?)
- If I see correctly, Table 7 shows NeTheR actually hurts performance on some datasets (FMA/Motifs/StarLightCurves) in the uninformed setting; this is glossed over, but understanding when and why NeTheR fails matters (it's fine it does).
- the conclusion (abstract) that it "consistently achieves statistically significant improvements across all settings" is directly contradicted here, isn't it?
Overall, performance claims should thus be toned down, also...

Medium Issues
--------------

1. Neural training vs. symbolic modification interaction lacks detail:
The Sharpe ratio selection where to insert a neural concept before training it, using structural best/worst-case bounds is nice. The issue is that the greedy symbolic choice made pre-training may not be same post-training, since a trained neural concept changes the residual error landscape.
- The dynamic alpha update helps to calibrate this, which is also nice, but the interaction isn't too analyzed...
- under what conditions is greedy-before-training reliable? This could benefit from more discussion, I think.
Very similarly, freezing previously inserted neural concepts is empirically shown to not hurt F1 -- that's nice, but it means true end-to-end optimization across all neural concepts doesn't really happen, and the theoretical concern about suboptimality remains open.
- the ablation settles the empirical question, not the theoretical one.

2. The Sharpe Ratio choice:
The use of Sharpe ratio in this context is nice/fresh but, afaik, Sharpe uses symmetric variance as its risk measure -- penalizing large deviations in both directions (including large upside gains). In theory revision, penalizing a modification because it might improve performance a lot (but variably) seems counterproductive, no?
- I believe that this is a standard critique of Sharpe in finance and should at least be acknowledged (given its central role in the paper).
- there is e.g. Sortino ratio, which penalizes only downside variance, could that be a better fit here?

Minor Issues
---------------

If I see correctly, neural concepts seem to play a surprisingly small role. NeTheR trains on average only 1.5 neural networks during search? Given that neural predicate invention is the headline contribution, this seems somewhat weird, making it appear more like a neural theory "patching" rather than true revision...
- is the greedy search algorithm immediately collapsing? Could be connected to the (medium) issue 1. above...
- are neural concepts actually driving the gains, or are symbolic literal insertions doing most/all of the work?
- the paper could e.g. break down the contribution of each modification type.

I believe that d-DNNF compilation can be exponential in circuit size? Were any bottlenecks encountered? How does it scale with more modifications?
- relevant for potential actual applicability...

NeTheR / NeTHeR are used interchangeably throughout (e.g. Figure 1 vs. abstract).

Overall
------------
Solid, detailed (nice appendix), well-executed paper with a real contribution in the specific context of ProbLog-style NeSy model revision. The Sharpe ratio heuristic is creative, data efficiency results are genuinely strong, and the experimental scope is really nice in its breadth. However, there is quite a bit of *overclaiming* - just like DeepProblog doesn't really solve all of NeSy AI, this paper doesn't really solve "Neurosymbolic Theory Revision through Predicate Invention". Hence, I believe, the overall framing needs to be toned down - the novelty should be more honest/better positioned, the propositional scope should be very clearly acknowledged right from the start (title), and the experimental evaluation claims should reflect their actual nature...

I put major revision, nevertheless, all these issues can actually be solved with some careful/targetted rewriting hence, after the (major) toning down, this is acceptable, I think.

Reviewer has chosen to be Anonymous

Overall Impression: Excellent

Content:
Technical Quality of the paper: Excellent
Originality of the paper: Yes
Adequacy of the bibliography: Yes

Presentation:
Adequacy of the abstract: Yes
Introduction: background and motivation: Good
Organization of the paper: Satisfactory
Level of English: Satisfactory
Overall presentation: Excellent

Detailed Comments:

# Summary

This paper addresses an important scenario where initial logical background knowledge provided by domain experts is imperfect and incomplete. It proposes the NeTheR framework, which iteratively revises theories via neural predicate invention. It uses tractable probabilistic inference on arithmetic circuits, combined with a dynamic variant of the Sharpe ratio, to evaluate the risk and return (best- and worst-case bounds) of inserting neural concepts prior to actual training.

# Strengths

- I believe the problem scope is well-framed and critical. Many existing NeSy models (such as DeepProbLog) implicitly assume that the provided symbolic knowledge is perfect and that neural predicates have predefined semantics. In contrast, NeTheR tackles the practical challenge of directly learning unknown neural concepts from subsymbolic data and integrating them into an existing knowledge structure to correct it.

- The proposed framework resolves the dilemma of combinatorial optimization and numerical optimization. Training a neural network for every candidate structural modification is computationally prohibitive. The method introduces an elegant heuristic-based approach using intervention to calculate the upper and lower bounds of performance before training. I believe the finding can be extensively applied to many neuro-symbolic scenarios.

- The paper provides extensive empirical evaluations using diverse datasets encompassing image, audio, and time series data. It is highly commendable that the approach consistently and statistically significantly outperforms baselines such as CMR and MLP+NN in both structurally informed and uninformed settings.

# Areas for Improvement

To strengthen the paper for final publication, the authors should clarify and deepen the discussion on the following points:

- The current framework is restricted to propositional logic circuits. Since the true strength of relational representations is central to NeSy, the discussion in Section 7 should be expanded to provide a clearer outlook on how to handle the combinatorial explosion of the search space when creating new rules in FOL.

- To initialize the parameter $\alpha$ for the dynamic Sharpe ratio, Algorithm 1 requires pre-training on $k$ randomly selected modifications. This initialization step accounts for a significant portion of the computational overhead. A discussion exploring lighter initialization strategies would be beneficial.

- Because the invented neural concepts are not assigned predefined semantics, the resulting logic circuit may become harder to interpret, creating a trade-off between interpretability and predictive accuracy compared to a fully symbolic initial model. While post-hoc XAI methods are mentioned, it would be valuable to discuss the potential of incorporating semantic regularization directly into the training process.

Reviewer has chosen not to be Anonymous

Overall Impression: Good

Content:
Technical Quality of the paper: Good
Originality of the paper: Yes
Adequacy of the bibliography: Yes, but see detailed comments

Presentation:
Adequacy of the abstract: Yes
Introduction: background and motivation: Good
Organization of the paper: Satisfactory
Level of English: Satisfactory
Overall presentation: Good

Detailed Comments:

This paper introduces NeTheR, a neurosymbolic theory revision framework that refines imperfect symbolic knowledge by inserting symbolic literals and neurally learned concepts. The key novelty lies in combining theory revision with neural predicate invention, allowing the system to discover useful concepts directly from subsymbolic data and integrate them into an existing logical theory. To efficiently select revisions, the authors propose a dynamic Sharpe-ratio-based heuristic that balances the potential benefits and risks of candidate modifications. The experimental evaluation across multiple datasets demonstrates that NeTheR consistently improves upon the initial theory and outperforms several competing approaches in both structurally informed and uninformed settings.

The paper is generally well written and well structured. The background section provides a clear and comprehensive introduction to the relevant concepts, making the proposed methodology easy to follow. I also appreciate the use of both the Sharpe ratio and the dynamic Sharpe ratio as part of the modification-selection strategy, which is one of the most interesting aspects of the paper.

I have only a few comments that I believe would improve the manuscript:
1. Training objective. In the "Training the model" paragraph (Section 4.3), the authors describe the training procedure but do not explicitly state the loss function that is optimized. This should be clearly specified.
2. Figure 4. The performance metric used to construct the critical difference diagrams is not explicitly indicated in the figure caption or the main text. While it can be inferred that the experiments focus on F1-score, this should be stated clearly.
3. Distance function d(M, M’). How much does this approximation affect the results? Is there evidence that the algorithm frequently terminates before reaching a genuinely optimal revision? This should be discussed.
4. Discussion of results. The manuscript would benefit from a substantially more detailed discussion of the empirical results. The experiments show that NeTheR consistently outperforms competing methods across most datasets, both when the initial model (M') is structurally informed and when it is not. However, the paper largely reports these findings without providing sufficient insight into why the method performs better. A deeper analysis of the factors contributing to the observed improvements would strengthen the paper considerably. For example, it would be interesting to understand whether the gains primarily arise from the flexible placement of neural concepts, the theory-revision strategy itself, the Sharpe-ratio-based selection criterion, or the interaction among these components. In addition, for some datasets and structural information, the compared methods overcome NeTheR, this must be explained to understand under which settings NeTheR is more effective.

More generally, I believe the numerical results currently relegated to the appendix should be moved, at least in part, to the main paper and accompanied by a more thorough interpretation. The experimental section demonstrates the effectiveness of the proposed approach, but the paper would be significantly stronger if it provided a deeper explanation of the observed behavior and performance trends.

Regarding the bibliography, only NeSy methods based on probabilistic semantic losses are discussed/introduced. A more general overview including also methods based on fuzzy semantic losses would improve the scope of the paper.

A small typo at page 27: f_1 = 0.55 should be F_1 = 0.5

Reviewer has chosen to be Anonymous

Overall Impression: Average

Content:
Technical Quality of the paper: Average
Originality of the paper: Yes, but limited
Adequacy of the bibliography: Yes, but see detailed comments

Presentation:
Adequacy of the abstract: Yes
Introduction: background and motivation: Limited
Organization of the paper: Satisfactory
Level of English: Satisfactory
Overall presentation: Good

Detailed Comments:

The paper presents a method for neural-symbolic predicate invention guided by theory revision. While the approach of bridging neural computations with symbolic logic to introduce new concepts into the models is promising, the manuscript currently suffers from a lack of critical literature framing, unverified assumptions in the introduction, and a lack of clarity in both the methodology and evaluation setup.

Feedback by Section:
1. Abstract & Introduction
- * Change "achieving the advantages" to "aiming for the advantages", as it better reflects the paper's intent.
- Fix the typo "their their".
- The abstract mentions limiting the number of "high-impact modifications." Please clarify how the number of modifications is restricted and how "high-impact" is formally measured.
- The claims made in the second paragraph of the Introduction lack references. Please provide adequate literature citations to support these assertions.
- The authors state that neural concepts are represented by symbolic literals. Please clarify this mechanism.
- In Example 1.1, the claim that existing theory revision methods cannot correct the issue related to birds without hollow bones is too strong. While image data undoubtedly makes identification easier (depending on whether the knowledge is implicit or explicit), it is inaccurate to say existing methods are fundamentally incapable of inferring this knowledge. In case the method has been presented with examples of this case, they could modify the current knowledge to achieve the representation space of that case. Please tone down or better justify this claim.

2. Background & Related Work
- A significant body of modern theory revision literature is missing and must be integrated to properly contextualize this work. For example, Cerna & Cropper (AAAI 2024) focus on a limited number of examples, negation, and predicate invention to modify logical theories, while Morel & Cropper (MLJ, 2023) learn logic programs guided by failures, which is a core principle of theory revision. Concerning structural modifications, similar to the modifications implemented in the logic circutis, Azevedo et al. (MLJ, 2020) and Guimarães et al. (MLJ, 2019) propose modifications to tree-like structures supporting relational knowledge guided by examples. This is related to the logic circuit representation adopted in this study, as generalization and specialization operators are fundamentally related to adding or removing tree branches.
- The background section currently lacks foundational definitions regarding theory revision and predicate invention.

3. Problem SettingTerminology Alignment:
- Traditional theory revision explicitly distinguishes between Background Knowledge (assumed to be correct) and the Modifiable Theory (the incorrect or incomplete part open to modification). Please clarify whether the established setting defines $M'$ as encompassing both background knowledge and the modifiable theory, or just the latter.

4. Methodology
- Section 4.1 needs to more heavily emphasize and reflect the specific role of neural computations. As currently written, the framework heavily resembles classical, purely symbolic (logic-based) theory revision.
- Step 3 describes "neural concept insertion" via a neural network, but the description is vague enough that it implies any mechanism could be used (e.g., symbolic concepts, alternative ML models). What makes this uniquely neural-symbolic?
- It is unclear why the Sharpe Ratio is uniquely relevant or tied to neural concepts, given that it could easily be applied to purely symbolic methods.

5. Evaluation & Experiments
- The imperfect initial model $M'$ is artificially generated by removing variables to intentionally lower the F1-score. How are these variables identified for removal? How does this artificial degradation realistically simulate a naturally imperfect model?
- The evaluation lacks comparisons against existing Neural-Symbolic models. Please justify why these were omitted from the baselines, as they are the most direct competitors to the proposed method.