Research on Abstraction-Based Search Space Partitioning and Solving Satisfiability Problems

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper is interesting, especially for readers in formal methods fields. The formulation and description are detail. Meanwhile, there are certain circumstances that are should be handled.
1. There are a lot of techniques regarding to solve SAT problem published in 2025 (check Scopus). On the other hand, many of references in this paper is old. Please cite newer techniques at least 10 papers.
2. Make a short but concise review regarding the recent techniques, especially the approach, strength, and weakness.
3. A table that consists of the investigation of new techniques (2024-2025) should be presented.
4. What is the difference between the new existing technique with the proposed technique?
5. Scientific contributions should be written before paper organization in introduction.
6. All mathematic formulations should be numbered.
7. Despite its advantage, authors should provide the limitation or weakness of their proposed technique.
8. The recommendation for future studies should be presented in conclusion.
9. A block diagram to present step-by-step of the proposed technique should be presented.

Author Response

Comments 1: There are a lot of techniques regarding to solve SAT problem published in 2025 (check Scopus). On the other hand, many of references in this paper is old. Please cite newer techniques at least 10 papers.

Response 1: Thank you very much for pointing that out. We have added recent SAT solving techniques, including machine learning, quantum computing, and graph computing, in Section 2.2 on page 5. We have also supplemented this with some newer references. Additionally, we have replaced some outdated references with similar, more recent ones, and the replaced references are highlighted in yellow in the reference list.

Comments 2: Make a short but concise review regarding the recent techniques, especially the approach, strength, and weakness.

Response 2: We indeed overlooked this point, so we have revised the structure of Section 2. We now first discuss traditional algorithms and then provide additional information on popular and recent solvers, which is reflected in Section 2.2 on page 5.

Comments 3:A table that consists of the investigation of new techniques (2024-2025) should be presented.

Response 3: In the new Section 2.2, we have not only supplemented the latest techniques with additional text, but also summarized them in a table for clarity.

Comments 4: What is the difference between the new existing technique with the proposed technique?

Response 4: We provided a step-by-step explanation of abstraction-based solving but forgot to summarize it. Therefore, we added a small subsection, 4.3, at the end of Section 4 on page 17 to summarize the abstraction-based solving technique. Figure 14 is specifically used to highlight the differences compared to proposed technique.

Comments 5: Scientific contributions should be written before paper organization in introduction.

Response 5: Thank you very much for your reminder. We have added the main contributions and experimental results in the Introduction, and they are highlighted in yellow on line 54 of page 2.

Comments 6: All mathematic formulations should be numbered.

Response 6: Thank you very much for your reminder. We have now numbered all the formulas that were previously presented separately.

Comments 7: Despite its advantage, authors should provide the limitation or weakness of their proposed technique.

Response 7: Indeed, every method has its own limitations, especially considering that the approach we proposed is relatively new. Therefore, in the end of the newly added Section 4.3 on page 18, we have summarized the challenges encountered during the implementation process and the drawbacks that were reflected.

Comments 8:  The recommendation for future studies should be presented in conclusion.

Response 8: In the newly added Section 4, we presented the limitations and shortcomings of our approach. Therefore, in the summary, line 599, we have also included future work, which aligns with the ongoing work we are currently undertaking to address and overcome these deficiencies.

Comments 9: A block diagram to present step-by-step of the proposed technique should be presented.

Response 9: In Figure 17 of the newly added Section 4.3, not only do we compare the differences with proposed technique, but it also provides a clear visual representation of how abstraction works, specifically the "classification" mentioned in the text.

Reviewer 2 Report

Comments and Suggestions for Authors

please see the comments in the file

Comments for author File: Comments.pdf

Author Response

Comments 1: In the title and abstract change “Solving of SAT” into  “Solving Satisfiability Problem”. 

Response 1: Thank you very much for your correction. We have now standardized the terminology, changing "Solving of SAT" to "Solving Satisfiability Problem" in the title and abstract.

Comments 2: In Introduction on page 2, change respectively Chapter 2, Chapter 3, Chapter 4, Chapter 5, into Section 2, Section 3, Section 4, Section 5. 

Response 2: Thank you very much for pointing out the error. We have now standardized the term "Chapter" and changed it to "Section" throughout the manuscript.

Comments 3: Define the symbol that appears four times in the first formula in Section 2, on page 3, and in sveral 
subsequent formulas in the paper. 

Response 3: Your comment is absolutely correct. We mistakenly used symbols before providing their definitions, which was indeed an error. We have now revised Equations 1, 2, 3, and 4 on Section 2, Page 3, around line 90, and replaced them with standard mathematical symbols to avoid any ambiguity.

Comments 4: Replace “assignment -based” by “assignment-based” on page 19, in Section 6. 

Response 4:Thank you very much for pointing out the formatting error. We have corrected the formatting issue, and the update has been made on page 22, Section 6, line 739.

Additional comments :Some figures, for example  5-7 and 9 should be enlarged, references are appropriate and up to date

Response : Thank you very much for your suggestion. We have enlarged Figures 5–7 and 9 as you mentioned, and we have also updated the references, replacing some with more recent ones.

Reviewer 3 Report

Comments and Suggestions for Authors

The paper presents a novel abstraction method for search space partitioning in SAT problem solving. It has adopted distinct approaches like sequential, parallel, and hybrid solving algorithms. The search spaces and their partitioning are distinguishing from traditional assignment approach. The results show that the hybrid approach (abstraction and assignment based) performs better for solving most of the cases. The paper uses search space from both topological and logical outlook.

 

My comments are below:

1. The results may be compared with other latest solvers or parallel SAT solving techniques. It would help in better understanding of advantages of the proposed approach.
2. The evaluation is based on CNF formulas and a limited set of strategies. Expanding the experiments to include a wider range of problem instances and abstraction strategies would strengthen the validity of the results (variables clauses, length, etc.)
3. More details on time complexity analysis may be included.
4. The paper lacks the discussion on how the proposed method tackles the scalability requirements for bigger size problem with large number of variables, clauses, etc.
5. Discussion on how abstraction relates to other search space reduction techniques used in SAT solving can be elaborated.
6. Don’t use the word – Chapter for sections
7. English/grammar/typos need to be rechecked
8. SAT applications may be included so that increase the readability.

Comments on the Quality of English Language

English/grammar/typos - need to be relooked.

Author Response

Thank you very much for your suggestion regarding the experimental content. Indeed, well-conducted experiments and strong data can make a more compelling case and are more persuasive to readers. In response to the issues you raised, we have added additional details to the experiments.

Comments 1: The results may be compared with other latest solvers or parallel SAT solving techniques. It would help in better understanding of advantages of the proposed approach.

Response 1: Existing parallel solvers are already quite mature in areas such as shared clause learning and computation scheduling, so directly integrating and comparing our proposed abstraction method with them is challenging. This is a key focus of our future work: developing solvers that can incorporate abstraction. However, as a complement, we have tested the performance of abstraction in solvers with different orders, and the results are presented in Appendix A.2 in page 23.

Comments 2: The evaluation is based on CNF formulas and a limited set of strategies. Expanding the experiments to include a wider range of problem instances and abstraction strategies would strengthen the validity of the results (variables clauses, length, etc.)

Response 2: Thank you very much for your valuable suggestion. We have added new experiments where we control one variable, either the number of variables or the clause multiplier, while changing the other to observe its impact on solving time. The details are provided in Appendix A.1 on page 23.

Comments 3: More details on time complexity analysis may be included.

Response 3:Thank you very much for your guidance on the theoretical aspects. After reconsideration, we realized there is indeed a discrepancy in the time complexity, as noted on line 455 of page 13. However, since we have not yet developed a practical abstraction-based solver, we will defer a more detailed time complexity analysis to our future work.

Comments 4: The paper lacks the discussion on how the proposed method tackles the scalability requirements for bigger size problem with large number of variables, clauses, etc.

Response 4: Indeed, small-scale SAT problems do not provide conclusive insights, so we selected 20 instances from the SAT competition 2024 for a new round of experiments. These instances have variable counts ranging from a few hundred to as high as 2000. The experimental data can be found in Appendix A.2, page 23. We understand that 2000 variables are still not considered a large scale, but this is partly due to the immaturity of our method. In Section 4.3, around line 600 on page 18, we have added a discussion of the challenges we are currently facing and the limitations of our approach. In future work, we will focus on developing hybrid solvers based on abstraction and optimizing them for large-scale problems.

Comments 5: Discussion on how abstraction relates to other search space reduction techniques used in SAT solving can be elaborated

Response 5: We have added a new subsection 4.3 on page 18 to summarize and compare the relationship and differences between abstraction and existing methods. Additionally, we have visually represented how the two methods can operate independently or in collaboration, as shown in Figure 14, around line 587.

Comments 6: Don’t use the word – Chapter for sections

Response 6: Thank you very much for pointing out the incorrect terminology. We have now standardized the text by changing all instances of "Chapter" to "Section."

Comments 7:  English/grammar/typos need to be rechecked

Response 7: We have re-examined the paper and made changes to the terminology. We now consistently use "Solving" instead of "Computation" when referring to the solving process.

Comments 8: SAT applications may be included so that increase the readability
Response 8: We conducted a new round of experiments by selecting instances from the SAT Competition 2024, as shown in Appendix A.2, instead of solely using randomly generated clauses.

Reviewer 4 Report

Comments and Suggestions for Authors

This paper proposes a novel abstraction-based search space partitioning method for SAT solving, distinct from traditional assignment-based approaches. The authors introduce sequential, parallel, and hybrid algorithms combining abstraction and assignment, demonstrating through experiments that the hybrid approach accelerates solving in most cases. A unified framework for search space partitioning is also presented, rigorously defining independent and complete partitions. The work connects geometric interpretations of search spaces with logical operators, offering insights into improving SAT-solving efficiency. Here are some suggestions for modification.

1. Line 3 in Page 3, “x1=1”.
2. In Proposition 2, “is is”.
3. The experiments focus on randomly generated CNF formulas with fixed clause-to-variable ratios. Testing on industrial or competition benchmarks would better validate practical applicability.
4. Table 5, A clearer labeling scheme is recommended.

With these improvements, the work has the potential to advance research in SAT. The paper can be accepted after careful revision.

 

Comments on the Quality of English Language

No

Author Response

Comments 1: Line 3 in Page 3, “x1=1”.

Response 1: Thank you for your correction. We have revised all the subscripts of "x" in Example 1 on page 3.

Comments 2: In Proposition 2, “is is”.

Response 2:Thank you for your careful correction. We have removed the redundant "is."

Comments 3: The experiments focus on randomly generated CNF formulas with fixed clause-to-variable ratios. Testing on industrial or competition benchmarks would better validate practical applicability.

Response 3: You are absolutely right; using random data alone lacks persuasive power. Therefore, we selected 20 instances from the SAT Competition 2024 to conduct a new set of experiments, including comparisons of the performance of different solvers. The details have been added in Appendix A.2 on page 23.

Comments 4: Table 5, A clearer labeling scheme is recommended.

Response 4: Indeed, the labels in the table were too vague, so we have not only revised the table headers but also updated the description above the table. Since we added new tables earlier, Table 5 has been changed to Table 7. The changes were made on page 14, around line 475.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Overall, authors have made significant improvement. All my comments and suggestions have been addressed. Based on this improvement, this paper is now acceptable for publication.

Reviewer 3 Report

Comments and Suggestions for Authors

Nil