Reviewer has chosen to be Anonymous

Overall Impression: Good

Content:
Technical Quality of the paper: Good
Originality of the paper: Yes, but limited
Adequacy of the bibliography: Yes

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

Detailed Comments:

This manuscript proposes Neural-Symbolic Energy-Based Models (NeSy-EBMs) as a unifying framing for neurosymbolic systems (prediction as energy minimization), introduces a DSVar/DSPar/DSPot taxonomy of neural-symbolic interfaces, and instantiates the framework via NeuPSL (deep HL-MRFs / PSL-style semantics). The strongest contributions are the NeuPSL inference formulation (LCQP + regularization enabling useful differentiability properties) and the learning suite, particularly the bilevel/value-function approach with smoothing, which avoids brittle KKT/implicit-differentiation behavior under active constraints and is supported by experiments across diverse tasks.

Strengths
- Concrete, useful system contribution (NeuPSL): Clear modeling pipeline (atoms/deep atoms, rules, grounding, Lukasiewicz/hinge(-squared) potentials) and inference-as-optimization that directly targets constraint satisfaction.
- Theory-to-learning connection: The regularized formulation provides a principled route to stable gradients in constrained inference-as-optimization settings, and the learning section ties solver quantities to learning signals rather than relying only on heuristic differentiation.
- Empirical evidence supports the core value proposition: On RoadR, NeuPSL increases constraint satisfaction from roughly 27.5% to 100%, with a small F1 increase. Across constraint-heavy tasks, NeuPSL consistently achieves 100% consistency where reported, while improving or maintaining task performance. The comparison against policy-gradient (IndeCateR) is informative and the reported timeouts are valuable negative results.

Major revision items
A. Streamline the unifying framework exposition (Section 3). Section 3 introduces multiple abstraction layers (e.g., E to g_sy to psi) that feel redundant in light of the later NeuPSL instantiation where the symbolic component largely is the energy. This front-loads notational complexity without a commensurate payoff in theorems or design guidance.
Requested change: Collapse the core definition to something like E_{theta, phi}(y, x) and introduce potentials only where they are required for later results/algorithms.

B. Clarify DSVar vs DSPar by inference topology, not argument placement. As written, DSVar/DSPar can read like syntactic reshuffling (and gradients can look similar by the chain rule). The real distinction is at inference time:
- DSVar: neural outputs are treated as fixed evidence (symbolic inference cannot revise them).
- DSPar: neural outputs parameterize the energy/prior so symbolic inference can correct weak neural signals.
Requested change: State this explicitly and give a short guideline on when to choose which (e.g., test-time correction under shift/safety constraints favors DSPar; fast inference or training-time regularization only favors DSVar).

C. Strengthen empirical positioning relative to external NeSy baselines (or narrow claims). The experiments support that NeuPSL helps vs neural-only and compare internal learning strategies, but the unifying framing invites at least one representative non-NeuPSL baseline per use-case family (constraint satisfaction, graph reasoning, LLM+symbolic pipeline).
Requested change: Add targeted external comparisons or explicitly narrow the empirical claim to the NeuPSL/HL-MRF instantiation.
D. Expand discussion of epsilon-regularization and scope of gradient applicability. Epsilon is central to the differentiability/continuity story; readers need typical values, sensitivity, and how much semantics are perturbed in practice. Also clarify when the value-function/bilevel gradients apply versus when discrete/integer constraints require modular or policy-gradient strategies.
E. Policy-gradient scalability discussion should match the observed timeouts. Since IndeCateR times out on multiple tasks, add discussion of whether this is expected (variance/credit assignment) and what mitigation might help (variance reduction, parameterization, curricula, alternative estimators).

Minor corrections / clarifications
- Theorem numbering: Ensure Smooth Formulation references match the final numbering (main-text Theorem 10 / appendix Theorem 11).
- Visual Sudoku metrics: Avoid conflating baseline digit accuracy (roughly 70.20%) with consistency. If baseline consistency is not reported, state that explicitly when contrasting with NeuPSL's 100% consistency.

Reviewer has chosen to be Anonymous

Overall Impression: Good

Content:
Technical Quality of the paper: Good
Originality of the paper: Yes, but limited
Adequacy of the bibliography: Yes

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

Detailed Comments:

The paper proposes NeSy-EBM, a unifying framework intended to organize NeSy systems. It categorizes systems into three paradigms (deep symbolic variables, parameters, and potentials). Then, the paper introduces NeuPSL as a framework to model a family of NeSy systems, together with different learning techniques.

While the paper has significant technical contributions regarding NeuPSL, its learning algorithms and experimental evaluation (sections 4, 5, 6), the initial presentation of the "unifying framework" (section 3) suffers from confusing notations and unclear contributions. I would suggest condensing this theoretical preamble, which doesn't bring much, and focus on the meat of the paper.

In particular:
1. On notation heaviness: Many definitions could be simplified.
1. The distinction between $x_{nn}$, $g_{nn}$, $x_{sy}$, $g_{sy}$ and their respective parameters is clear.
2. Then, the energy function $E$, symbolic function $g_{sy}$, and potential functions $\psi$ seem to all be different layers of abstraction of the same concept. Equation 3 explicitly defines $E$ as mapping directly to $g_{sy}$. The potential is also mostly defined directly by $g_{sy}$. These abstractions don't bring much to the reader's understanding.
3. DSVar and DSPar explicitly groups some arguments differently: $\psi([y,x_{sy},g_{nn}], w_{sy})$ vs $\psi([y, x_{sy}], [w_{sy}, g_{nn}])$. However, mathematically, this is just syntactic sugar. This seems to be for *conceptual* purposes, but it's not clear why it matters in a mathematical framing.

2. On the value of the mathematical framework: it is not clear what understanding it brings to the different existing NeSy frameworks.
1. In Appendix C, the paper maps three important NeSy frameworks (Semantic Loss, DeepProbLog, Logic Tensor Networks) to their framework. However, mapping these systems to the given framework essentially reduces them to a single equation for $g_{sy}$ (e.g., Equation 73 for Semantic Loss). The potential functions etc. do not help clarify any of the frameworks' internal mechanics. So I would simplify and condense much of the definitions of Section 3, focusing on $x_{nn}$, $g_{nn}$, $x_{sy}$, $g_{sy}$ .
2. The intuitive categorization of systems into the ones modifying neural predictions (DSPar) or not (DSVar) as detailed in Figure 2, is intuitive but is to nuance. Since a lot of frameworks in DSVar also modify predictions via gradient descent, wouldn't that give both the same purpose? Here I also have a question: how come Semantic Loss and LTN are categorized differently (DSPar and DSVar respectively)? They both define a differentiable loss function and act on the neural predictions in a similar same way (backpropagation)?

3. Technical contributions: Sections 4 and 5 on NeuPSL and Learning are in contrast very clear with significant contributions.
1. I would focus on these, perhaps highlighting more strongly what are the new contributions compared to the original NeuPSL paper.
2. The experimental evaluation in Section 6 is also thorough.

The paper has high-quality algorithmic work, but is prefaced by an overly complex theoretical framing with a unifying focus which, in my opinion, does not help much in the reader's understanding. I recommend streamlining Section 3 significantly and pivoting Sections 4 and 5 as the primary contributions.