Evaluation Trees and Normalisation for Proposition Algebra^†

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript presents one of the most impressive recent contributions to propositional logic and algebraic approaches to conditional constructions. It is immediately evident that the work is the result of mature, rigorous, and methodologically sound research built upon solid theoretical foundations, while simultaneously offering clear and significant original contributions.

The introduction and motivation are well structured and allow the reader to enter the topic naturally and without difficulty. The results are presented in a concise, logically coherent, and technically precise manner. In particular, the systematic treatment of valuation congruences and the clarity with which the authors connect evaluation-tree transformations to normalisation functions deserve special praise. This part of the paper stands out for its mathematical precision and conceptual depth. To be entirely honest, I have not read a better paper in propositional logic in quite some time. The authors deserve sincere commendation.

The manuscript clearly satisfies all the criteria of a high-quality scientific contribution: well-defined motivation, precisely stated problems, novel technical results, well-organised proofs, and a coherent presentation of conclusions. It fully aligns with the high standards of Mathematics.

I have only one minor technical suggestion. In several places, the white square symbol (□) marking the end of a proof appears separated from the last line of the proof or pushed to the next line. I recommend that the authors ensure that the end-of-proof marker is consistently placed immediately after the final sentence of each proof, to maintain typographical uniformity.

Apart from this very minor point, I have no further comments. I strongly recommend the manuscript for publication.

Author Response

• Regarding your first four comments:
  
  ==> Thank you.
  
  
• Regarding your last comment, i.e.,
  - I have only one minor technical suggestion. In several places, the white square symbol (□) marking the end of a proof appears separated from the last line of the proof or pushed to the next line. I recommend that the authors ensure that the end-of-proof marker is consistently placed immediately after the final sentence of each proof, to maintain typographical uniformity.
  
  ==> This has been changed in the minor revision, thank you.

Reviewer 2 Report

Comments and Suggestions for Authors

Please have a look at the review in the attachment.

Comments for author File: Comments.pdf

Author Response

Below are our responses to each of your comments, all starting with ==>.

• The abstract is descriptive but lacks emphasis on the unifying contribution. Add a sentence at the end: “Our framework provides the first comprehensive tree-based semantics that unifies all major valuation congruences in proposition algebra, offering both conceptual clarity and practical decision procedures.”
  
  ==> We have added this sentence to the abstract, thank you.
  
  
• Page 2: The pedestrian example is too simplistic. Replace or supplement with a more il-
  lustrative example showing how different congruences matter in programming. “Consider (read() ◁ check() ▷ F) ◁ read() ▷ F where read() has side effects. Under free congruence, both reads execute; under repetition-proof, only one executes.”
  
  ==> We believe that the pedestrian example is a good example on free valuation congruence. On p.3, references have been added to Example 3 and (the new)
  Example 4 (both on p.32), in which side effects relating to repetition-proof and contractive valuation congruence are discussed.
  
  
• The dual notation ◁/⊴ is subtle and distracting. Authors are advised to add a brief explanation in Section 2: “We use ◁ for the conditional connective in terms and ⊴ for postconditional composition in trees to distinguish syntactic and semantic levels.”
  
  ==> We disagree and we trust that interested readers have no problen with these notations. Obviously, we intended to use related symbols to stress the relation between conditional statements and evaluation trees.
  
  
•  Many functions (se, rpse, cse, mse, sseσ , etc.) are introduced without quick reference. Add a small table after Section 2 introducing main functions and their meanings.
  
  ==> The definitions of the functions you mention all immediately state their main characteristic, and we believe that this is sufficiently clear. For example, "Definition 12. The unary repetition-proof evaluation function rpse : C_A --> T_A yields
  repetition-proof evaluation trees and is defined by [...]".
  
  
• Tree diagrams are crudely drawn. Use proper tree-drawing (TikZ/forest) or at least clearer ASCII art.
  
  ==> We disagree. All pictures of evaluation trees are drawn with TikZ, and we have strived to make these as simple and clear as possible.
  
  
• The conclusion section reads like a summary rather than a synthesis. It could be improved.
  
  ==> We disagree, in the synthesis part of this section, lines 581-627 (pages 30-31), we discuss our methodology to the best of our ability.
  
  
• Page 20(line 486): change “priciple” →“principle”, carefully check other typos and eliminate.
  
  ==> You mean line 403. This typo has been corrected (in the minor revision, p.20, line 404), thank you.

Reviewer 3 Report

Comments and Suggestions for Authors

- The abstract is highly technical and may be difficult to digest for readers outside the immediate subfield. Moreover, it does not clearly state why the results matter beyond internal completeness.

- The introduction emphasizes what is done, but less clearly why this extension is important now.

- The conceptual intuition behind each valuation congruence is introduced only implicitly.

- The review focuses almost exclusively on proposition algebra and its direct lineage. Connections to related areas are mentioned only briefly.

- The paper can benefit significantly from a short concluding subsection in the methodology that summarizes the methodological flowchart and clarifies its reuse potential.

- The manuscript presents a highly formal and rigorous methodological framework; however, it does not clearly articulate the methodological philosophy underlying the approach.

- The authors should clarify whether their methodology is best viewed as: semantic construction, normalization-based proof technique, unifying algebraic–semantic correspondence.

- While this strategy is effective, the methodological justification for choosing normalization as the central proof tool is not sufficiently discussed.

- The correspondence between term normalization and tree transformation is accidental or principled.

- The repetition of similar proof patterns may obscure conceptual differences between congruence.

- After each major completeness theorem, briefly state: 1) what new understanding this result provides about valuation congruence, 2) how it clarifies the nature of short-circuit evaluation, 3) and why the result is non-trivial beyond earlier work.

- Limitations of the framework are not explicitly stated.

- The discussion remains largely internal to proposition algebra.

Comments on the Quality of English Language

The language is highly dense and formal, which may reduce accessibility for readers outside the immediate research community. Several long sentences, particularly in the introduction and proof explanations, could be simplified or split to improve readability without sacrificing precision.

Author Response

Below are our responses to your comments, all beginning with ==>.

• The abstract is highly technical and may be difficult to digest for readers outside the
  immediate subfield. Moreover, it does not clearly state why the results matter beyond internal completeness.
  
  ==> We added an extra sentence to the abstract (highlighted in red).
  
  
• The introduction emphasizes what is done, but less clearly why this extension is important now.
  
  ==> On page 2, we have expanded on the sentence about "evaluation trees" (highlighted).
  
  
• The conceptual intuition behind each valuation congruence is introduced only implicitly.
  
  ==> we have added Example 4 on contractive valuation congruence (page 33) and now refer to Examples 3 and 4 on page 3 for conceptual intuition. Moreover, on page 33, we remark that the value of CPmem lies primarily in its being a stepping stone to some more complex results.
  
  
• The review focuses almost exclusively on proposition algebra and its direct lineage.
  Connections to related areas are mentioned only briefly.
  
  ==> We believe that the paper is well balanced in this respect.
  
  
• The paper can benefit significantly from a short concluding subsection in the methodology that summarizes the methodological flowchart and clarifies its reuse potential.
  
  ==> In lines 581-627 (pages 30-31) we discuss this methodology and its potential to the best of our ability. In lines 710-711 (p.33) we explicitly mention its reuse potential.
  
  
• - The manuscript presents a highly formal and rigorous methodological framework; however, it does not clearly articulate the methodological philosophy underlying the approach.
  - The authors should clarify whether their methodology is best viewed as: semantic construction, normalization-based proof technique, unifying algebraic–semantic correspondence.
  - While this strategy is effective, the methodological justification for choosing normalization as the central proof tool is not sufficiently discussed.
  - The correspondence between term normalization and tree transformation is accidental or principled.
  - The repetition of similar proof patterns may obscure conceptual differences between congruence.
  
  ==> We disagree with these four comments, please see our response above.
  
  
• After each major completeness theorem, briefly state: 1) what new understanding this result provides about valuation congruence, 2) how it clarifies the nature of short-circuit evaluation, 3) and why the result is non-trivial beyond earlier work.
  
  ==> We disagree, as this would disrupt the balance of each of Sections 2-6, which are technical in nature and entirely focused on the aforementioned completeness results.
  
  
• - Limitations of the framework are not explicitly stated.
  - The discussion remains largely internal to proposition algebra.
  
  ==> Yes, this is an article about proposition algebra, and we find it difficult to explicitly
  point out its limitations. Of course, extensions and generalisations are always possible. In [5], proposition algebra was introduced and various topics were discussed that were not used further or not elaborated on (yet), including other valuation congruences, and potentially infinite propositions and their recursive specifications. However, we believe that the current article is sufficiently rich in content.
  
  
• The language is highly dense and formal, which may reduce accessibility for readers outside the immediate research community. Several long sentences, particularly in the introduction and proof explanations, could be simplified or split to improve readability without sacrificing precision.
  
  ==> Both in the Introduction and in the "proof explanations", we are unable to rewrite the text into genuinely shorter sentences.