Research on a General State Formalization Method from the Perspective of Logic

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I have the following comments for improvment: 
- Figures are minimal, more figures / diagrams / charts would improve readability. 

• the notations are too heavy!, summary table would help.
• Conclusions are too general
• Language is formal but at times repetitive; careful editing could reduce redundancy
• Some sections (e.g., higher-order logic definitions) overlap with standard textbooks;
• The introduction is well-written but too long; consider shortening the historical discussion of information theory and focusing on the problem statement.
• The sections on economics and sociology are promising but feel underdeveloped compared to mathematics and computer science. The authors introduce definitions and predicates but stop short of providing real example. 
• long sections of symbol definitions (e.g., L(1), L(k)) could be condensed or moved to an appendix.

• Definition 10 defines a state as "an interpretation of a set of well-formed formulas... The specific properties... are determined by the specific application scenario." While intended to be general, this definition risks being vacuously true.

•  What is the formal relationship between an "interpretation of a set of well-formed formulas" (Definition 10) and the "enabling mapping" that defines information in OIT?

• How does this logical definition of state enhance the "11 types of measurement systems" mentioned in the introduction? A concrete example would be highly beneficial.

• The proof of the central Theorem 1 relies directly on these axioms. However, the methodology for establishing these axioms is not explained.

• The method for moving from a specific domain (e.g., economics) to a general, universal formalization process is not specified. There is no algorithmic or systematic procedure described for how a researcher in a new field (e.g., biology, law, music theory) should apply this method to formalize their states. 

Good luck

Author Response

Comments 1 : Figures are minimal, more figures / diagrams / charts would improve readability. 

Response 1 : Thank you for pointing this out. I agree with this comment. Therefore, in this revision, we have added several figures/diagrams to improve readability, provide visual summaries of key ideas, and guide the reader through the technical results. Specifically:Figure 2 (new)-P14, Figure 3 (new)-P18, Table 3 (new)-P19, Figure 4 (new)-P21, Figure 5 (new)-P26, Figure A1 (new)-P31.

Comments 2 : the notations are too heavy!, summary table would help.

Response 2 : We thank the reviewer for pointing out that the notations are heavy and suggesting a summary table. In this revision, we streamlined the notation and added summary aids to improve readability. For example, we added a consolidated summary table in Appendix A.4.

Comments 3 :Conclusions are too general.

Response 3 : We thank the reviewer for this helpful comment. In response to the comment, we streamlined the Conclusions to cite the updated Theorems 8, 10, 12, 13, and 17, without elaborating on their statements.

Comments 4 :Language is formal but at times repetitive; careful editing could reduce redundancy.

Response 4 : We thank the reviewer for this observation. We have carefully edited the manuscript to reduce redundancy while retaining formal precision. As a result, several paragraphs with overlapping content were merged or removed (e.g.,Abstract, introduction, conclusion and outlook). 

Comments 5: Some sections (e.g., higher-order logic definitions) overlap with standard textbooks.

Response 5 : We appreciate the reviewer’s point. We kept a minimal self‑contained summary but cut tutorial exposition and duplicated proofs, adding pointers to textbooks for details and relocating standard lemmas to Appendix A.1.

Comments 6:The introduction is well-written but too long; consider shortening the historical discussion of information theory and focusing on the problem statement.

Response 6 : We thank the reviewer for this helpful suggestion. In the revision, we shortened the historical discussion of information theory and refocused the Introduction on the problem statement and our contributions. 

Comments 7:The sections on economics and sociology are promising but feel underdeveloped compared to mathematics and computer science. The authors introduce definitions and predicates but stop short of providing real example. 

Response 7 :We appreciate this suggestion. In the revision, we added concrete, reproducible examples to substantiate these sections: a worked Arrow–Debreu exchange example in Appendix A.3 (economics) and a predicate‑based counting/consistency example on p. 19 (sociology). These instances move beyond definitions to show end‑to‑end use (data → logical formulas → computed outputs). We hope this addresses the concern.

Comments 8:long sections of symbol definitions (e.g., L(1), L(k)) could be condensed or moved to an appendix.

Response 8 :Thank you for this suggestion. In the revision, we condensed the long blocks of symbol definitions and relocated non-essential material to an appendix. Specifically, the lists for L(1), L(k), and related notational families were reduced to a compact summary in the main text, with full definitions and variants moved to Appendix A.1 . 

Comments 9:Definition 10 defines a state as "an interpretation of a set of well-formed formulas... The specific properties... are determined by the specific application scenario." While intended to be general, this definition risks being vacuously true.

Response 9:The purpose of doing so is to make the state representation adaptable to diverse situations, ensuring both generality and simplicity, and to build a bridge between states and formal systems.

Comments 10:  What is the formal relationship between an "interpretation of a set of well-formed formulas" (Definition 10) and the "enabling mapping" that defines information in OIT?

Response 10: Thank you for this suggestion. OIT treats information as an enabling mapping between states without a rigorous and general definition of state itself.Defining state remedies the shortcomings of OIT, forms a logical closed loop, and completes the theory.

Comments 11:How does this logical definition of state enhance the "11 types of measurement systems" mentioned in the introduction? A concrete example would be highly beneficial.

Response 11: Thank you for this question. We now make the link explicit and provide a concrete example.We operationalize two of the 11 measurement system types (categorical/binary indicators) in sociology (p.19).

Comments 12:The proof of the central Theorem 1 relies directly on these axioms. However, the methodology for establishing these axioms is not explained.

Response 12:Thank you for raising this point. In the revised manuscript, we now explain—immediately before stating the axioms—the methodology by which these axioms are established. 

Comments 13:The method for moving from a specific domain (e.g., economics) to a general, universal formalization process is not specified. There is no algorithmic or systematic procedure described for how a researcher in a new field (e.g., biology, law, music theory) should apply this method to formalize their states. 

Response 13: We appreciate this important concern. In the revision, we clarify that the chosen domains (mathematics, computer science, economics, sociology) are representative cases that instantiate a common pipeline, and we make this connection explicit in the Conclusions. 

Reviewer 2 Report

Comments and Suggestions for Authors

This paper introduces a universal state representation framework that refines the state definition within Objective Information Theory and facilitates cross-disciplinary state-related research and communication. I find the core idea interesting and believe the paper is, on the whole, well-written and well-structured. I only have some minor suggestions for the authors to consider. After these refinements, I would recommend acceptance.

1. The paper introduces extensive notation. It would be helpful to thoroughly check and verify the consistency and proper definition of all symbols.
2. Please verify the placement of references and punctuation (e.g., periods) throughout the manuscript to ensure formatting consistency.
3. The research motivation could be further sharpened. It would be beneficial to more explicitly outline the limitations of existing studies and to clearly highlight the advantages offered by the proposed approach.
4. Including numerical examples or a step-by-step demonstration within the main text would be helpful to illustrate the specific application process of the proposed formulas.
5. To better contextualize the contribution of this work, it is suggested to cite the recent relevant study: DOI: 10.1016/j.inffus.2024.102902.

Author Response

Comments 1: The paper introduces extensive notation. It would be helpful to thoroughly check and verify the consistency and proper definition of all symbols.

Response 1: Thank you for this important suggestion. We conducted a thorough notation audit and revised the manuscript to ensure that all symbols are properly defined and used consistently.

Comments 2：Please verify the placement of references and punctuation (e.g., periods) throughout the manuscript to ensure formatting consistency.

Response 2: Thank you for this reminder. We performed a manuscript-wide consistency pass on references and punctuation. 

Comments 3：The research motivation could be further sharpened. It would be beneficial to more explicitly outline the limitations of existing studies and to clearly highlight the advantages offered by the proposed approach.

Response 3: We appreciate this suggestion. In the revision, we sharpened the motivation by explicitly contrasting limitations of prior work with the advantages of our approach. Concretely, the Introduction now includes a short “Problem and Gap” subsection and a bullet list of contributions with theorem pointers (Thms. 8, 10, 12, 13, 17). We highlight that existing studies typically (i) treat states in domain‑specific encodings with unclear cross‑domain transfer, (ii) lack verifiable artifacts (formal constraints/queries) tied to measurements, and (iii) provide background expositions that do not yield algorithmically checkable outputs. Our approach addresses these gaps by (a) defining a domain‑agnostic state as a logical structure, (b) providing end‑to‑end, verifiable instances in economics (App. A.3) and sociology (p. 19), and (c) establishing expressiveness boundaries with formal guarantees.

Comments 4：Including numerical examples or a step-by-step demonstration within the main text would be helpful to illustrate the specific application process of the proposed formulas.

Response 4: Thank you for the suggestion. In the revision, we added a compact, step-by-step numerical example to the main text to illustrate the application of our formulas, with pointers to the full worked instances (economics in App. A.3; sociology on p. 19).

Comments 5: To better contextualize the contribution of this work, it is suggested to cite the recent relevant study: DOI: 10.1016/j.inffus.2024.102902.

Response 5: Thank you for the pointer. We have added a citation to the recent study (DOI: 10.1016/j.inffus.2024.102902) in the Introduction and Related Work to better contextualize our contribution.

Reviewer 3 Report

Comments and Suggestions for Authors

Comments and Suggestions for Authors

The manuscript “Research on a General State Formalization Method from the Perspective of Logic” addresses the problem of formalizing “state” in a logical framework. The topic is relevant, but the paper requires substantial revision before it can be considered for publication.

1. The central research question is meaningful, but it is not sharply stated in the introduction. Authors should clearly define whether the work focuses on discrete, continuous, or hybrid states, and what kind of logic is being formalized.
2. The idea of a general state formalization is interesting, but the novelty is limited without explicit comparison to well-known frameworks such as modal logic (Kripke structures), temporal logics (LTL/CTL), dynamic logic, TLA+, or abstract state machines.
3. The formalization is described in a general way, but it lacks rigor. Definitions of syntax, semantics, and proof system must be precise, with soundness and completeness addressed.
4. Results are not well-structured, and theoretical claims are stated without sufficient proof detail or illustrative examples. A running example (e.g., a finite-state process) should be added to demonstrate applicability.
5. No clear figures or formal diagrams are provided. A schematic of the proposed framework would improve clarity.
6. The reference list is weak. Key foundational works (Kripke, Tarski, Pnueli, Lamport, Harel–Kozen–Tiuryn, Gurevich, Rutten, Goguen–Burstall) and recent developments in logical state systems should be included.
7. The paper requires significant English editing for grammar and readability.

Overall: The manuscript has potential but is currently underdeveloped. It requires major revisions to strengthen theoretical rigor, improve clarity of formal definitions, provide comparisons to established frameworks, and support claims with examples and stronger references.

Author Response

Comments 1: The central research question is meaningful, but it is not sharply stated in the introduction. Authors should clearly define whether the work focuses on discrete, continuous, or hybrid states, and what kind of logic is being formalized.

Response 1: Thank you for this valuable suggestion.This article addresses all types of states, with discrete states being more intuitive. I have clarified this in the introduction.

Comments 2: The idea of a general state formalization is interesting, but the novelty is limited without explicit comparison to well-known frameworks such as modal logic (Kripke structures), temporal logics (LTL/CTL), dynamic logic, TLA+, or abstract state machines.

Response 2: Thank you for this important suggestion. In the revision, we added an explicit comparison to well-known frameworks—Kripke semantics for modal logic, temporal logics (LTL/CTL), dynamic logic, TLA+, and Abstract State Machines (ASMs)—to clarify novelty and scope. Section 2.6 (Related Work and Comparison, pp. 6–7) now shows embeddings/compatibilities (e.g., Kripke frames as structures with a binary accessibility relation R; LTL/CTL as constraints on paths over such structures; PDL as program-indexed relations; TLA+ states as variable valuations with next-state relation; ASMs as transition rules over structures)

Comments 3:The formalization is described in a general way, but it lacks rigor. Definitions of syntax, semantics, and proof system must be precise, with soundness and completeness addressed.

Response 3: Thank you for raising this important point. We revised the formal definitions (syntax, semantics) to be fully explicit, aligning notation and removing ambiguities. Definitions and theorem statements were tightened throughout (see Thms. 8, 10, 12, 13, 17). For syntax and semantics, we adopt standard first‑order and higher‑order frameworks.

Comments 4:Results are not well-structured, and theoretical claims are stated without sufficient proof detail or illustrative examples. A running example (e.g., a finite-state process) should be added to demonstrate applicability.

Response 4: Thank you for the helpful feedback. We also incorporated additional illustrative examples and a running example to demonstrate applicability. In particular, Appendix A.2 and A.3 now provide worked examples with step-by-step derivations and solver-checkable artifacts, complementing the main-text results. 

Comments 5:No clear figures or formal diagrams are provided. A schematic of the proposed framework would improve clarity.

Response 5:  I have added more figures in the main text to help readers understand the theory and examples. Specifically:Figure 2 (new)-P14, Figure 3 (new)-P18, Table 3 (new)-P19, Figure 4 (new)-P21, Figure 5 (new)-P26, Figure A1 (new)-P31. In addition, a notation summary table has been included in Appendix A.4.

Comments 6: The reference list is weak. Key foundational works (Kripke, Tarski, Pnueli, Lamport, Harel–Kozen–Tiuryn, Gurevich, Rutten, Goguen–Burstall) and recent developments in logical state systems should be included.

Response 6:Thank you for pointing this out. We have substantially expanded the reference list to include key foundational works and recent developments relevant to logical state systems. Specifically, we added classic sources by Kripke (modal semantics), Tarski (model theory and truth), Pnueli (temporal logic), Lamport (TLA+), Harel–Kozen–Tiuryn (Dynamic Logic), Gurevich (Abstract State Machines), Rutten (coalgebraic systems), and Goguen–Burstall (institutions/algebraic specifications), along with recent work on logical state frameworks.

Comments 7: The paper requires significant English editing for grammar and readability.

Response 7: Thank you for the feedback regarding language quality. We have undertaken a thorough English edit to improve grammar, clarity, and readability. The revision includes (i) restructuring long sentences, (ii) standardizing terminology and notation, and (iii) smoothing transitions between sections. 

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors addressed my concerns good luck. 

Reviewer 3 Report

Comments and Suggestions for Authors

All the comments were addressed properly. There are no further comments.