Reviewer has chosen to be Anonymous

Overall Impression: Good

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: Limited
Organization of the paper: Needs improvement
Level of English: Satisfactory
Overall presentation: Average

Detailed Comments:

Summary: The authors extend the ULLER language by considering a monadic interpretation of its semantics. This allows them to decouple the inductive definition of semantics from the operators used, and generalise the semantics to other categories and infinite domains (eg continuous distributions). This allows them to separate semantics and computation.

**Strengths**:
- The paper introduces many interesting new ideas, and is highly general. By changing the Agg 2Mon-BLat (a structure involving several operators), many different kinds of semantics can be immediately implemented. It is quite thorough at discussing a fairly wide array of existing semantics, and extending them to different contexts. For instance, the authors define extensions of probabilistic semantics and LTN semantics to infinite domains directly from the monadic interpretation of ULLER.
- The paper is mathematically very solid, and I could not (within the time I had) find any significant errors.
- The writing is also quite decent.
- The explanation of the implementation in haskell helps with understanding the main concepts (highlight this more in the writing though!)

**Weaknesses**:
- The paper might be hard to follow for many readers of this journal as it uses many advanced mathematical concepts without intuitive explanations.
- At several points, I wondered how some of the results can be practically used: The paper does not consider the computational complexity of mULLER, differentiability, or whether some equations are computable at all. The implementation only helps a bit.

**General comment**: Although I think this is a strong paper, I would like to leave the authors with this piece of feedback. It could be important to consider who the primary audience of this paper is. I would expect the results of this paper to be primarily interesting for people who build neurosymbolic tools. However, this community does not just consist of purely formal/ mathematical people. The speed at which many concepts are introduced is fast, and I believe should be adapted based on this audience. I will try to give some hints below on how I'd improve the readability.

**questions**
- Is there existing work on a monadic interpretation of probabilistic programming? It could be that there is some relevant literature to be found there.
- In the implementation, conj and disj are not restricted to being Monoids. Is this intentional?
- In ULLER, the semantics is not necessarily inductive (although all examples shown are). One could, for example, have more complicated semantics (called 'neurosymbolic systems' [[]]) that first compile the given formula into a different circuit representation, after which a recursive evaluation is used. Is it correct that mULLER does not support non-recursive semantics?

Points:
- Please use \citep's where relevant (in the intro I already noticed several improper cases)
- Section 2: It should be much clearer what the goal of each subsection is. A linear read is challenging: it starts off with the most complicated concept (at least to me), namely the Kleisli triples / monads, without any clear motivation as of yet how this will be used. I would recommend swapping 2.1 and 2.2, adding more motivation for how the Kleisli triples will be used, and adapting the first paragraphs of section 2 to outline how the background in 2.1 and 2.2 will be used to build up to section 2.3 where NeSy frameworks are introduced.
- Section 3: Make clearer why mULLER needs both computational and non-computational symbols (eg with an example)
- Section 3: The Tarskian semantics are quite invoved, and require going through the monad multiple times. It would be helpful to have a more in-depth explanation of some of the more complicated constructions, like the statements and aggregations.
- Section 4: It would be helpful for each semantics to have a short description of the 2Mon-BLat used (eg, in 4.2, Product BL-algebra is not defined, and it is not clear to me yet how equation 1 arises as a union over a set).
- Within categorical semantics, I was a bit lost how interpretations $\mathcal{I}$ are defined. Earlier, $\mathcal{I}(s)$ on some sort is defined as a set. So is $\text{id}_{\mathcal{I}(s)}$ then a morphism in Set, and not in $\mathcal{C}$? Or are interpretations redefined to pick objects in $\mathcal{C}$?
- Table 6: I had trouble following what the probability monads are (each column). Where are they defined? And how did the authors distinguish between $\circ$ and the checkmark? And what's the difference between Meas and MeasR, and what is QBS?
- Section 7.2: Several aggregation functions are discussed and introduced here. I did not understand exactly what the contribution of this is in the context of the larger paper, as the authors did not run experiments on them (which is ok!)
- Section 7.4: I had the a similar though in this section, it was unclear to me what the contribution of mULLER is to deriving the WMC/WMI interpretation. To me it sounded like a somewhat straightforward extension of the argument in ULLER when using probability measures, which doesn't require the monadic framework (same for the Bayesian factorisation)

Smaller points:
- Page 5: Several technical words are undefined, like bounded lattice, monoid, and right adjoint. You could have a glossary in the appendix for those to keep it more self-contained.
- In the definition 4, a NeSy framework uses a monad. Please note that up until that pint, mainly the notion of Kleisli triple is used rather than monad (this tripped me up). I'd suggest only using one of these two terms.
- Example 3 overloads $\mathcal{I}$ (it's also an interpretation)
- Page 20: Shouldn't the aggregator of $\exists$ be $1-\exp\circ...$ rather than $(1-\exp)\circ...$?
- Page 24, second half of page, page 25, eqs 14-19: The equations here use $:=$ to suggest a definition, but shouldn't these equations arise from the choice of Agg2Mon-BLat? (ie it's a regular $=$)
- I found footnote 23 a bit odd, LTN indeed never defined semantics for infinite domains, and infinitary LTN is more like an extension (at least to me)

Reviewer has chosen to be Anonymous

Overall Impression: Good

Content:
Technical Quality of the paper: Good
Originality of the paper: Yes
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: Average

Detailed Comments:

The work proposes mULLER, a framework for unifying the semantics of NeSy systems under ULLER, the recent work who proposed a unified syntax for NeSy systems. It does so using monads from category theory. In doing so, mULLER allows to more easily introduce new semantics, which the authors illustrate with the addition of infinite domains semantics. The paper contributions are both original and significant, but it needs improvement on clarity.

Strengths:
S1. Uniform definition of the semantics through monads: This is the main contribution of the paper. This seems quite powerful. I could not verify the soundness, but the main advantage of this seems to be that it allows to more easily derive new NeSy systems semantics, cf next strength.
S2. Extension to infinite domains: Infinite domain semantics, which were improperly defined in ULLER as shown by the authors, are now formally defined in mULLER.

Weaknesses:
W1. Presentation: My main criticism of the paper is its clarity, which an audience not familiar with category theory (or ULLER) would have a hard time to understand.
W1A. The paper uses examples, but they are still abstract and could themselves benefit from examples. e.g., When introducing the probability distributions monad, I would give an example of a concrete distribution which is relevant to NeSy and what their Kleisli extension would represent (can f*(\rho) be understood as computing the expected value of a distribution?). I would add a few words around each example for intuitions.
W1B. The structure goes from background to contribution here and there which can be hard to follow. For example, Section 2.1 uses the notation for identity morphism id_A but id_A is only introduced in Appendix A I think (I wonder if this Appendix could be moved to a background section instead? Together with Section 2.1).
W1C. A quick background section of useful concepts from ULLER would help.

Questions:
Q1. Am I right to understand that 2Mon-BLat has a slightly stronger semantic requirement (commutativity and neutral elements) on the NeSy system than the original ULLER paper ("classical behavior in the limit" when evaluated on 0 and 1 values)? These requirements are met by most (all?) existing systems, but this is good to mention for the sake of completeness.
Q2. I assume that the predicate True(x) from ULLER, to use the output of computational functions as logical atoms, is still usable in mULLER? If so, I would highlight this option when talking about the limitation of computational predicates in modeling dependence (in Section 3.2). I would also make this limitation more obvious in the text, maybe putting the sentence in bold "While computational function symbols enable the use of conditional probabilities, computational predicate symbols are always independent of each other.". This is easy to miss for someone not familiar with ULLER details.