Review Reports

Comments 1:

[The title should be rewritten, say as “A Topological Parallel Algorithm for the Pure Literal Rule in SAT Problem Solving Using a Matrix-Based Approach” instead of “A Topological Parallel Algorithm for Pure Literal Rule in SAT Problem Solving Based on Matrix”.]

Response 1:

We appreciate the reviewer's appraisal and advice. According to the reviewer's suggestion, We have revised the paper's title. The new title is “A Topological Parallel Algorithm for the Pure Literal Rule in SAT Problem Solving Using a Matrix-Based Approach”.

This change can be found – line 2~4 in the revised manuscript.

Comments 2:

[Your use of the acronym compressed row storage (CRS) is correct; however, compressed sparse row (CSR) is generally more widely used.]

Response 2: 

We thank for the reviewer’s comment. We have replaced CRS with CSR in the revised manuscript.

 These changes can be found – lines 320,335,444,454,456,507~508 in the revised manuscript.

Comments 3:

[The authors might want to consider more recent references. For instance, “Jain, S., Neoh, T.Y., Stephan, F. (2024). An Improved Algorithm for Sparse Instances of SAT”, arXiv:2411.07389, to enrich their manuscript.]

Response 3: 

Thank you for pointing this out. We agree with this comment. We have added recent references to the manuscript and revised some of the content in the revised manuscript.

 These changes can be found – lines 82,389,395,402,521 in the revised manuscript.

Major:

Comments 1:

[The representation of CNF as an incidence matrix is well-known, it is not authors' invention. Incidence matrix and bipartite clause-term graph are two basic representations of the CNF. Nothing new here. ]

Response 1: 

Thank you for pointing this out. We agree with this comment.It may be that our expression was incorrect, leading you to misunderstand that we invented or proposed this representation method. A better way to put it would be that we adopted or drew on this method. We sincerely apologize. First, we acknowledge the fundamental role of existing CNF representation methods. Our research is premised on matrix representation, focusing on the design, function extension, and algorithm integration of customized incidence matrices for the implementation of parallel pure literal rules. Our overall direction is to solve automated reasoning problems using matrix algebra tools. There is also a corresponding matrix representation in predicate logic. In the manuscript, we provided example 1 to illustrate this point.

 These changes can be found in lines 196~199, 210~219 in the revised manuscript.

Comments 2:

[CVPLPTD algorithm is quite complex and involves building Literal Topological Graphs (LTG) and many other things. However it outputs simple list of literals, which have only positive-or-zero or only negative-or-zero entries in the appropriate column of the incidence matrix. This can be done by a plain summing up all negative values to one counter and all positive values to another counter and then checking if any of the counters is zero. Summation procedure can be easily parallelized in many ways. CVPLPTD with LTG might have sense if LTG are used somehow after reduction of the incidence matrix, so it can spare calculations, but the authors do not discuss this. So, CVPLPTD seems unnecessarily complex. ]

Response 2: 

We appreciate the reviewer’s comment. We agree with this comment. The CVPLPTD algorithm we designed is somewhat complex, involving components such as Literal Topological Graphs (LTG). Its output is a simple list of literals. We have also noted that other researchers have adopted the approach of summing up all positive and negative values to obtain pure literal results. We apologize for not elaborating on this point in detail in the manuscript. We will add relevant content in the manuscript to elaborate on this point in detail. The supplementary content is outlined as follows: Simplifying the incidence matrix in accordance with the pure literal rule requires obtaining both a list of pure literals and all rows (clauses) containing these pure literals. After other methods obtain pure literals, additional computational overhead may be needed to check whether all clauses contain the pure literals. In the CVPLPTD algorithm, the positive and negative literal matrices generated based on the Literal Topological Graph enable rapid identification of all clauses containing pure literals.

These changes can be found in lines 322~337 in the revised manuscript.

Comments 3:

[All methods have O(mn/p) algorithmic complexity. This is not informative. To compare them more research is required. There can be a theoretically estimation of the coefficient for this O(). In other words, not only state O(mn/p), but determine that complexity is in range [C_1*mn/p;C_2*mn/p], where C_1 and C_2 are some constants. Or perform practical solution of a task by various methods and collect the number of primitive operations (memory reads, memory writes, conditional jumps etc) or bluntly get the running time. The authors should present deeper investigation of complexity. ]

Response 3: 

We appreciate the reviewer’s comment. We have added content related to the theoretical estimation of the algorithm's time complexity coefficient and supplemented the coefficient range of the CVPLPTD algorithm in lines 272–280 of the manuscript. A column for the coefficient range has been added to Table 1, with corresponding explanations provided.

These changes can be found in lines 23, 275~283, 285~286, 452~455, 462~464 in the revised manuscript.

Minor:

Comments 1:

[ SAT should be disclosed in the title. ]

Response 1:

We appreciate the reviewer's appraisal and advice. According to the reviewer's suggestion, we have replaced the abbreviation “SAT” with the full name "the Satisfiability Problem" in the revised manuscript. The abbreviation "SAT" in the title is not our original creation; we noticed that in other articles, the authors used abbreviated forms, for instance《SAT-Based PAC Learning of Description Logic Concepts》,《SAT - Based Method for Finding Attractors in Asynchronous Multi - Valued Networks》,《On the power of clause - learning SAT solvers as resolution engines》,《A Machine Learning Approach to Predict SAT Solver Performance》.

This change can be found in line 3 in the revised manuscript.

Comments 2:

[Definition of negation is sometimes 'not' sign (line 111 and further), sometimes 'prime' sign (line 114 and further). Please use one way. ]

Response 2: 

Thank you for pointing this out. We agree with this comment. We have used the same representation form in the revised manuscript.

These changes can be found in lines 135, 138, 140, 141 in the revised manuscript.

Comments 3:

[Line 124. First V_1 -> V_2. ]

Response 3: 

We thank the reviewer for the comment. We have revised V1 -> V2. in the manuscript.

This change can be found in line 149 in the revised manuscript.

Comments 4:

[Lines 144, 145. 'l' clashed from line 144 to line 145.]

Response 4: 

Thank you for pointing this out. We agree with this comment. We have re-edited the content to ensure accurate expression.

These changes can be found in line 170 in the revised manuscript.

Comments 5:

[Definition 8. First, why F(), G(), H() stand in the top of matrix rather than x_1, x_2, x_n? Second, if the presented matrix is called 'general form', then it means all CNIMs should have last columns of zeros. Obviously this is not so. Third, all omegas are simple letters without any indices. It means they are all equal. w_ij should be used: w_11, w_12,..., w_nm. ]

Response 5: 

We thank the reviewer for the comment. We have revised Definition 8 in the manuscript. The matrices we define are not only applicable to propositional logic but also to predicate logic. Therefore, the top part of the matrices does not use literals. We have added Example 1 to provide a detailed explanation.

These changes can be found in lines 201~202 in the revised manuscript.

Comments 6:

[Equations (2) and (3) are incomplete. What is the condition for f and g to be set to 0 or 1? ]

Response 6: 

We appreciate the reviewer’s comment. We have added the judgment conditions in the manuscript.

These changes can be found in lines 219~220 in the revised manuscript.

Comments 7:

[Matrices in line 260 are better to be transposed to align with align with CNIM matrix in line 252. ]

Response 7: 

We thank the reviewer for the comment. We have transposed the matrix in the manuscript. 

These changes can be found in lines 298~299 in the revised manuscript.

Comments 1:

[Add information in the introduction explaining how the work is organized. What is considered in the individual chapters of the work?]

Response 1:

We appreciate the reviewer's appraisal and advice. According to the reviewer's suggestion, We have added an introduction to the organization of the paper in the manuscript.

These changes can be found – lines 99~122 in the revised manuscript.

Comments 2:

[The nature of IT work makes it difficult to precisely identify original solutions. For example, please try to determine the originality of the proposed code in Figure 1. Example 1 is original, illustrating IT considerations in a language understandable to specialists. The optimization problem (1), (2), and its subsequent solution are interesting and original.]

Response 2: 

We appreciate the reviewer’s comment. We have re-edited the pseudo-code of Algorithm 1 and added flowcharts for clarification in the manuscript.

These changes can be found – lines 232~237 in the revised manuscript.

Comments 1:

[Section:Abstract. Dense wording; some sentences are overly long. Recommendation: Simplify key points and highlight practical outcomes.]

Response 1:

We appreciate the reviewer's appraisal and advice. According to the reviewer's suggestion,We have simplified the content of the summary section to highlight the key points in the manuscript.

These changes can be found – lines 11~27 in the revised manuscript.

Comments 2:

[Section 2.4: Excellent formalization of CNIM. Recommendation: Add a simple illustrative example.]

Response 2: 

Thank you for pointing this out. We agree with this comment. We have added Example 1 to explain the representation of CNIM matrices in the manuscript.

These changes can be found – lines 206~215 in the revised manuscript.

Comments 3:

[Section 3: The pseudocode (Figure 1) should be visualized. Recommendation: Include real diagram or flowchart.]

Response 3: 

We thank for the reviewer’s comment. We have added flowcharts to illustrate the pseudo code in the manuscript.

These changes can be found – lines 232~235 in the revised manuscript.

Comments 4:

[Section 4: Table 1 is informative but purely theoretical. Recommendation: Add at least one empirical data row.]

Response 4: 

We sincerely appreciate your valuable comment. At the current stage, this study primarily focuses on the construction and exploratory analysis of algorithmic theoretical models. Due to the preliminary nature of the theoretical exploration, we are unable to reproduce and implement the operational results of all algorithms within a short timeframe. We would highly appreciate your understanding in this regard.

Comments 5:

[Section Conclusion: Well written, but overly general. Recommendation: Briefly restate quantitative gains and limitations.]

Response 5: 

We thank for the reviewer’s comment. We have reorganized the content of the conclusion in the manuscript.

These changes can be found – lines 482~491 in the revised manuscript.