Causal-R: A Causal-Reasoning Geometry Problem Solver for Optimized Solution Exploration

First of all, we sincerely appreciate your affirmation of our paper and novelty. It encourages us to continuously improve our work. Here are the responses to your questions and concerns.

Q1: What is the advantage of using the causal graph deduction against using the problem solver for one-step fact deduction (e.g., LEAN)? Is it that the graph deduction can be formulated as forward matrix deduction (efficiency)?

First, we want to clarify the differences between our work and works using LEAN such as AlphaGeometry. This is explained in details in our responses to your Q3, so please kindly refer to Q3.

In our work, the core idea of proposing causal graph deduction is that we can abbreviate the high-order geometry theorem rule into unified causal deduction, removing the detailed mathematical calculation in reasoning stage (discussed between L76-87). Each causal deduction path implies a reasoning path that represents a fake theorem application. During this stage, no actual fact deduction is performed and the values in the environment remain unchanged, enabling us to perform global reasoning for solution exploration without actually changing the environment. This process simulates the reasoning stage of neural networks when obtaining the solutions to a geometry problem, but is implemented as symbolic reasoning.

Q2: There has been some work [1] discussing converting symbolic rules into graph operations in the symbolic planning context, what is the key difference between this work and [1]? (especially, to me, the forward matrix deduction seems quite like the planning graph expansion process) [1] Mao, Jiayuan, et al. "What planning problems can A relational neural network solve?." Advances in Neural Information Processing Systems 36 (2023): 59522-59542.

Thank you for providing this work. So far as we know, this work is greatly different from ours from the task, the method and the contribution.

1. The task of the provided work is to analyze the circuit complexity of a Relational Neural Network (RelNN) when it is used as a policy for planning problems. They try to provide theoretical guidance of the circumstances under which a RelNN-based policy can be successfully trained. Our task is geometry problem solving, and we aim at providing a causal-reasoning geometry problem solver for exploration of explainable, shorter and multiple solutions.

2. The rules and states mentioned in their work also distinguish from ours. The state in their work is represented in the classic STRIPS form and is tied with the detailed scenarios. While we represent the status of node using only positive and negative, which means acquirable/known and unknown, respectively. All nodes have the same indicative status representation format. Similarly, each causal deduction path is abbreviated from the theorem rules in theorem rule base, and involves only the status of nodes and relations of nodes. We do not consider any sequence of involved nodes in a deduction path.

3. The graph in their work is used to represent a planning problem as input for RelNN, where the node refers to object and the edge refers to the relation between objects. However, in our work, each causal graph corresponds to a theorem rule, representing the deduction path between the nodes by edge connection. Additionally, the causal graph is used to perform symbolic reasoning.

4. We are confused that we haven’t found the mentioned planning graph expansion process in this work. However, in our Forward Matrix Deduction (FMD), the matrices are transformed from the geometric conditions and all the causal graphs. The status vector is generated from the geometric conditions, where 1 means known (positive) and -unlimited means unknown (negative). During the FMD process, only matrix operation is performed, without integration of neural networks.

Our differences include but not limited to the above points. It is possible that we have some misunderstandings, and we welcome any corrections.

Q3: I agree that AlphaGeometry is not a competitor to this work since you are training-free, so maybe Causal-R is a complementary work to AlphaGeometry (which is based on LEAN). But I would like to see some more explanations about "we handle different kinds of tasks", what kinds of tasks do you and AlphaGeometry consider, respectively?

Our work mainly distinguishes from theirs from but not limited to the following aspects:

1. Different Task: AlphaGeometry is an automated geometry problem proving system designed for Olympic-level geometry proving problems, which rarely involve calculation of detailed values of primitives. Their task is to prove certain geometric relation between primitives. Our task is to develop a geometry problem solver for solving geometry problems that mainly involve detailed mathematical calculation.

2. Different Scenarios: The rules in the symbolic deduction engine they adopt are fundamental and basic, particularly designed for each fine-grained step in Olympic-level proving problems. Our work utilizes the higher-order geometry theorems that are applicable in educational scenarios (e.g., Thales Theorem and Law of Cosines). A comprehensive list can be found in table 1 and table 2 in appendix C. These rules are more suitable for educational use.

3. Different Goal: AlphaGeometry aims at increasing the problem proving accuracy to demonstrate the unlimited potential power of machines on proving extreme difficult geometry problems. Our goal is to propose a geometry problem solver for optimized solution exploration, providing explainable, shorter and multiple solutions in problem solving.

4. Different Methodology: AlphaGeometry utilizes a symbolic deduction engine to exhaustively generate new geometric conditions until reaching a closure. Our Causal-R transforms the detailed mathematical deduction into unified causal deduction and performs parallel causal graph reasoning at global level for solution discovery.

Q4: In my opinion, the related works should be presented in the main paper for a complete understanding of your work and existing works (instead of putting to Appendix).

Thanks for your advice. We agree that the related work part is better to be placed in main content. Since we have discussed a lot of related methods in Introduction part, we chose to put it in appendix in the first submission. We will present it in the main paper in final version.

Finally, we sincerely ask if you could kindly consider increasing the score if most of the concerns have been resolved. Thank you.

Q1: Is there causal graph reasoning formalization or step-by-step technical description of how this graph is constructed from parsed geometric inputs.

Thanks for your advice. We have presented how the causal graph is constructed in section 3.2.1 (Causal graph construction) and formal description of explicit theorem rule simplification in appendix C. Based on these, here we provide a more clear guidelines.

For a geometry problem, the unique primitive nodes are generated from geometric conditions (e.g., Line AB and Angle ABC). Initially, all the node status are negative.

Next, for each causal graph, we match the causal deduction of causal graph with the existing primitive nodes for combinations that satisfy the constraints. In this step, the edges from head nodes to the tail node are specified, and if there are prerequisites, the prerequisite edges are also specified. Initially, all the edges are unactivated (negative status).

Then, we check the geometric conditions to change the initial status of primitive nodes. For those that are provided with detailed values, we change their status to positive. Consequently, the status of relations of nodes are also specified according to the status of primitive nodes and geometric conditions (e.g., Line AB equals to Line BC).

Finally, the target nodes are specified according to the existing primitive nodes and problem goal.

We hope the above information addresses your questions.

Q2: Ablation of Forward Matrix Deduction (FMD) is needed.

Thank you for pointing out this. We initially thought that it is not appropriate to compare the efficiency (e.g., time) of using and not using FMD, because the resource consumption is different. However, we find it still necessary to better support the introduction of FMD.

First, we want to re-emphasize that Causal Graph Reasoning (CGR) is proposed as the theoretical foundation to achieve causal reasoning for solution exploration. FMD is one of possible detailed strategies that we propose to implement CGR. So it’s not simply an A+B module-combining architecture that supports independent ablations. We implemented a basic strategy, leveraging Python dictionaries for causal graph reasoning and node status storage. To provide a fair and detailed comparison, we record the average reasoning time per iteration of these two methods, grouped by the number of unique primitive nodes and the number of deduction paths (i.e., sum of all deduction paths in all causal graphs). The other parts and settings remain the same. Experiments were run for 3 times and the average result is reported.

Grouped by number of all deduction paths (/s):

Grouped by number of unique primitive nodes (/s):

From the results, we mainly have the following observations:

(1) FMD consistently performs faster than using Basic when the number of deduction paths is smaller than 5,000 in table 1 and when the number of primitive nodes is smaller than 70 in table 2. Within this range, the time of FMD generally exhibits a steady increasing trend with the increasing of deduction paths and primitive nodes. When the sum of deduction paths increases to 10,000 and even greater, the dimensions of matrices will expand, resulting in more resource demands. Under such extreme and abnormal circumstances, which is not applicable to most geometry problems, it is recommended to simply use basic strategy of causal graph reasoning.

(2) When the number of deduction paths exceeds 10,000, the time of basic strategy is surprisingly lower than when the number of paths ranges between 5,000 and 10,000. This is because in this kind of extreme samples, there are only few feasible deduction paths (i.e., successfully activate the path because head nodes and prerequisites are activated) even if there are many connected deduction paths.

These experiments and according analysis will be added in final version.

Q3: There are some claims made in the paper that remain open-ended, for example, the Causal-R claims the ability to generate multiple minimal-length solutions.

Thank you for the helpful advice. We want to first clarify that we know “minimal” or “shortest” is a very definite term, so we only claim to obtain “shorter” solutions as presented in our paper. On one hand, obtaining the shortest solution is not critically ensured due to certain randomness of current symbolic system (which we already discuss in L323-330). On the other hand, we do obtain shorter solutions according to experiments in table 1, for both applied theorem length and solution step length. As for multiple solution, our Causal-R is designed to inherently possess the abilities for multiple solution exploration, because the causal reasoning is performed in a global manner that includes all feasible solution paths. Note that, this multiple solution is defined on the basis of explainability, so that the multiple solutions are meaningful. We use the parameter lambda to control the maximum number of feasible solutions obtained w.r.t. each target node; however, returning fewer solutions is also accepted, as not all problems have multiple solutions in practice.

For more comprehensive evidence, we have sampled 30 geometry problems to evaluate the results obtained by our method:

— We manually checked whether there is any shorter step to obtain the correct final answer, using the theorems that are contained in the theorem rule base. All solutions are verified to be the shortest.

— We exhaustively applied the theorem rules by all combinations (i.e., 1+2, 1+3, …, 2+1, 2+3, …, 1+2+3, ….) and obtain the shortest applied theorem length that could lead to correct answer. 2 out of 30 are obtained with shorter length, where the applied theorem lengths are the same. Our solution is longer because we consider applications of the same theorem rule on different primitives different.

— We manually checked that 5 out of 30 problems have 2 solutions obtained by our model, and the solutions are reasonable.

As for a new metric to evaluate the diversity, we are afraid that it is currently not feasible. The challenge arises not only from the difficulty of determining whether a problem actually has multiple solutions or not (ground-truth labeling), but also from the lack of related design in baselines.

Q4: A similar claim regarding the generalization to other domains without any specific direction or experiment should be toned down for better presentation quality of the paper. Thanks for your suggestion on this. Our motivation on this point is to inspire more researchers from domains where the task can be reformulated as such, especially for those relying on explicitly pre-defined action space and costly real action application. We have recognized that stating such conclusion might not be appropriate and we will revise the related content to present a more rational discussion in final version.

Q5: Another question is regarding the generalizability beyond geometry tasks, though the authors mention it clearly, there are no concrete illustrations or preliminary results outside the domain (which I understand can go beyond the scope of this work). However, if the authors could describe a specific non-geometry task where Causal-R has the potential to be directly applied/tested? Since the work seems to have broad potential, it would be good if they were explicitly stated for future work.

Thank you for your interest in our method and suggestions. We understand that providing related experiments or illustrative examples (prohibited in rebuttal) could better support the broader potential use of our method. As in our responses to Q4, we decide to refine the claim we made in the original submission into a more rational discussion, so it wouldn’t mislead the readers. Currently, we are not able to give detailed application examples because it needs vigorous technical analysis of feasibility and expertise knowledge of a wide range of research domains, which is really out of the scope. However, here we list some critical principles to evaluate its feasibility in tasks in other domains:

1. There is a set of independent objects that are specified and determined. These objects should also associate unified indicative status representations (e.g., on and off).

2. There is a pre-defined action space, where each action is expected to change the status of objects directly based on status of other objects. If there are restrictions of performing this action, the restrictions should be able to be transformed from the status of objects.

3. The task goal is related to status of certain objects.

We find that using causal reasoning prior to the actual execution of an action is helpful if performing the action is costly. Hope this can assist your understanding.

Q6: Formatting concerns, the style of references used in the paper. The paper refers to the related works using numbers; however, there are no numbers in the references section. It is recommended to use a style where the citations are easily findable. It would be good if the authors could take care of those in the updated version of the paper.

We are truly sorry for the inconvenience caused to your reading experience. We have also noted that the current reference format is not appropriate, as it uses the default plain style. It will be corrected and aligned with the official requirements in final version.

Q1: Ablation of Causal Graph Reasoning (CGR) and Forward Matrix Deduction (FMD).

Thank you for pointing out this. We initially thought that it is not appropriate to compare the efficiency (e.g., time) of using and not using FMD, because the resource consumption is different. However, we find it still necessary to better support the introduction of FMD.

First, we want to re-emphasize that Causal Graph Reasoning (CGR) is proposed as the theoretical foundation to achieve causal reasoning for solution exploration. FMD is one of possible detailed strategies that we propose to implement CGR. So it’s not simply an A+B module-combining architecture that supports independent ablations. We implemented a basic strategy, leveraging Python dictionaries for causal graph reasoning and node status storage. To provide a fair and detailed comparison, we record the average reasoning time per iteration of these two methods, grouped by the number of unique primitive nodes and the number of deduction paths (i.e., sum of all deduction paths in all causal graphs). The other parts and settings remain the same. Experiments were run for 3 times and the average result is reported.

Grouped by number of all deduction paths (/s):

Grouped by number of unique primitive nodes (/s):

From the results, we mainly have the following observations:

(1) FMD consistently performs faster than using Basic when the number of deduction paths is smaller than 5,000 in table 1 and when the number of primitive nodes is smaller than 70 in table 2. Within this range, the time of FMD generally exhibits a steady increasing trend with the increasing of deduction paths and primitive nodes. When the sum of deduction paths increases to 10,000 and even greater, the dimensions of matrices will expand, resulting in more resource demands. Under such extreme and abnormal circumstances, which is not applicable to most geometry problems, it is recommended to simply use basic strategy of causal graph reasoning.

(2) When the number of deduction paths exceeds 10,000, the time of basic strategy is surprisingly lower than when the number of paths ranges between 5,000 and 10,000. This is because in this kind of extreme samples, there are only few feasible deduction paths (i.e., successfully activate the path because head nodes and prerequisites are activated) even if there are many connected deduction paths.

These experiments and according analysis will be added in final version.

Q2: The sensitivity analysis of hyper-parameters alpha and lambda and how these choices affect outcomes?

In our work, the iteration constraint alpha and candidate solution number lambda are both system parameters that do not affect the method. These values can be settled according to detailed application scenarios and demands.

For the iteration constraint alpha, it is only used to set a maximum iteration for reasoning in case the model keeps running without an end. For majority of the geometry problems, 7 steps is enough to obtain the final answer, while the rest are often unsolvable by the design of symbolic system at this stage (e.g., parsing mistakes and lack of theorem rules). It is a definite result that decreasing the iteration constraint will affect the solving performance, and the number of possible solutions.

For different lambda, the model is signaled to try to obtain lambda solutions to obtain the final answer. So, if multiple solutions are needed, the lambda can be set to greater than 1. In situations that a geometry problem does not have multiple solutions, the model will only return one. Since we generate solutions from the first iteration that all target nodes are reached, there is very small possibility of existing shorter solutions at longer iteration, combing our experiments when lambda is set to 2 and 3 in table 2.

Q3: How initializations of the status vector in matrix deduction influence convergence speed (e.g., all negative, partial known node initialization) ?

Sorry that we do not quite understand this requirement. The initialization of the status vector depends on the given information in the problem, where the value of status vector is 1 means the value of corresponding primitive is given (known) in the original problem. Setting the status vector to all negative indicates no value is provided in the problem, which means the problem is unsolvable (e.g., you can’t get the value of side of triangle without any detailed value). Similarly, setting the status vector to all positive means the problem is already solved. In all cases, changing the status vector from its initialization leads to inconsistency to the original problem and directly affects the results.

Q4: The "optimality" of generated solutions is not rigorously verified. While Causal-R is said to produce "shortest solutions," there is no comparison with human experts’ optimal solutions or the manually annotated "standard solutions" in the Geometry3K dataset. It remains untested whether Causal-R’s solutions are truly shorter than these benchmarks or if they merely avoid redundancy in existing annotations.

Thank you for the helpful advice. We want to first clarify that we know “shortest” is a very definite term, so we only claim to obtain “shorter” solutions in our paper. On one hand, obtaining the shortest solution is not critically ensured due to certain randomness of current symbolic system (which we already discuss in L323-330). On the other hand, we do obtain shorter solutions according to experiments in table 1, for both applied theorem length and solution step length. This is compared using the same predicted input and annotated input without any pre-processing. Further, Inter-GPS didn’t provide any annotation of standard solutions. For more comprehensive evidence, we have sampled 30 geometry problems to evaluate the results obtained by our method:

— We manually checked whether there is any shorter step to obtain the correct final answer, using the theorems that are contained in the theorem rule base. All solutions are verified to be the shortest.

— We exhaustively applied the theorem rules by all combinations (i.e., 1+2, 1+3, …, 2+1, 2+3, …, 1+2+3, ….) and obtain the shortest applied theorem length that could lead to correct answer. 2 out of 30 are obtained with shorter length, where the applied theorem lengths are the same. Our solution is longer because we consider applications of the same theorem rule on different primitives different.

Q5: Experiments on the scalability of the theorem knowledge base (KB), such as gradually expand the number of theorems in KB (e.g., 17, 24, 30), test changes in accuracy and solution length.

We are sorry that we are unable to carry out such experiments because its beyond our scope. Increasing the theorem rule base requires expertise knowledge on geometry theorems and explicit definition of theorems that should be orthogonal to the existing ones. Addition of theorem rules that are able to solve the problems that cannot be solved by old theorem rules is promising to increase the accuracy. Addition of higher-order theorem rules that solve problems with one step rather than combination of several theorems is promising to decrease the solution length. We contend that developing theorem rule base is beyond our scope.

Q6: How does the current Reasoning LLM / VLM, such as o3, o4, and r1, perform on your task?

The existing top-tier large language models do perform well on simple geometry problems, but still present a performance gap on more difficult ones compared to symbolic-based methods. Compared with accuracy performance, there are more severe problems of these LLMs to solve geometry problems. On one hand, the solutions generated are often unreliable (e.g., including incorrect calculated intermediate result and inconsistency of contents) even if the final answer is correct. On the other hand, it is not sure whether there is data exposure issue. We use the edited diagram and problem text (e.g., change of value) to test Gemini-2.5 Pro, and it still generates the similar solution content or different solutions but the same answer as the original one. (We have cleared the history and tested in different windows.)

Q7: How does your method perform on mathvista benchmark?

MathVista is not applicable for the evaluation of symbolic-based GPS methods. It is a dataset mainly to test the math capabilities of LLMs/VLMs. It includes many topics such as numerical commonsense, statistical reasoning and scientific reasoning. There is a small portion of geometry problems but do not provide any formal language annotation. Geometry3K is a widely recognized and well-annotated data set that has provided both predicted and ground-truth formal language parsings, covering a wide range of problem types, many of which are also sampled as composition of other datasets. Therefore, it has been considered as a more suitable benchmark for comparison of symbolic-based methods by previous works.

Finally, we want to emphasize that our main goal and contribution is to provide a symbolic-reasoning method using causal graphs for optimized solution exploration (i.e., explainable, shorter and multiple solutions). We sincerely hope if you can kindly consider the limitations of existing development of the geometry problem solving domain, and increase the score if most of the concerns have been resolved.

Q1: How does the proposed method perform on other popular datasets, such as FormalGeo7K and MATH? There should be exploration of robustness to noisy inputs, or evaluation on other datasets to test generalizability. Is the 24 hand-crafted symbolic rules brittle in other datasets?

We understand that in most scenarios (e.g., involving neural networks), evaluation on more datasets would make the results more supportive (e.g., better generalizability). Here are the reasons why we only report results on Geometry3K in the first submission:

— Geometry3K is a widely recognized and used data set that has provided both predicted and ground-truth formal language parsings. All problems in this dataset are well annotated and standard, covering a wide range of problem types, many of which are also selected as composition of other datasets (e.g., FormalGeo7K). Since previous baselines are all established and developed based on the same symbolic system (i.e., Inter-GPS style), it is a common sense for fair comparison on this dataset.

— For symbolic-based methods, one of the intrinsic advantages is stableness. That means, if the input is sent in the pre-defined format, each step of symbolic deduction is promising and vigorous. Therefore, the generalizability mainly depends on geometry problem parsing, the design of symbolic system and theorem rule base. As the performances on Geometry3K shows, the theorem rule base has been developed as applicable to most of the geometry problem scenarios. This indicates its applicability to a wide range of standard geometry problems. Note that we do not augment the theorem rule base for fair comparison on solving performance.

— FormalGeo7K is another widely used data set with formal language annotations, but in a different symbolic system design. It is now mainly used for neural-based methods and methods developed on its system. We have been making efforts during the rebuttal time to translate the system language, and sadly it is too much workload for us to implement a correct projection. Further, we have observed that FormalGeo7K test set includes a large portion of geometry problems selected from the Geometry3K train. Considering that they both are standard geometry problem type, we believe the results on Geometry3K has strong .

— MATH dataset includes problems include but not limited to the topics of algebra, probability, calculus and a small portion of geometry. It is designed for LLM-integrated methods and evaluation of LLM’s math capabilities. It does not provide any parsing annotations in formal language. Therefore, it is not quite applicable for our task and its beyond our scope.

We sincerely hope that you could reconsider the necessity of evaluation on these datasets for symbolic methods, taking into account the existing limitations of this task and public data.

As for the robustness to noisy inputs, we have listed both results of ground-truth input and predicted input (i.e., with incorrect parsing results) in last two lines of Table 1. For all symbolic-based methods, the performance is sensitive to the correctness of input content. This has been analyzed and discussed in the third point of our quantitative analysis in section 4.2 (Line 314).

As for the 24 hand-crafted symbolic rules, these theorem rules were meticulously designed, encompassing most geometry theorems required for common geometry problems in educational scenarios. A list of these theorems can be found in appendix C. Moreover, these rules have been validated to be effective for problem solving in previous works and our work (e.g., surpassing the problem solving accuracy obtained human experts). Therefore, for geometry problems in other datasets that are similar to those widely adopted in education, these rules are applicable.

Q2: Ablation of Causal Graph Reasoning (CGR) and Forward Matrix Deduction (FMD).

Thank you for pointing out this. We initially thought that it is not appropriate to compare the efficiency (e.g., time) of using and not using FMD, because the resource consumption is different. However, we find it still necessary to better support the introduction of FMD.

First, we want to re-emphasize that Causal Graph Reasoning (CGR) is proposed as the theoretical foundation to achieve causal reasoning for solution exploration. FMD is one of possible detailed strategies that we propose to implement CGR. So it’s not simply an A+B module-combining architecture that supports independent ablations. We implemented a basic strategy, leveraging Python dictionaries for causal graph reasoning and node status storage. To provide a fair and detailed comparison, we record the average reasoning time per iteration of these two methods, grouped by the number of unique primitive nodes and the number of deduction paths (i.e., sum of all deduction paths in all causal graphs). The other parts and settings remain the same. Experiments were run for 3 times and the average result is reported.

Grouped by number of all deduction paths (/s):

Grouped by number of unique primitive nodes (/s):

From the results, we mainly have the following observations:

(1) FMD consistently performs faster than using Basic when the number of deduction paths is smaller than 5,000 in table 1 and when the number of primitive nodes is smaller than 70 in table 2. Within this range, the time of FMD generally exhibits a steady increasing trend with the increasing of deduction paths and primitive nodes. When the sum of deduction paths increases to 10,000 and even greater, the dimensions of matrices will expand, resulting in more resource demands. Under such extreme and abnormal circumstances, which is not applicable to most geometry problems, it is recommended to simply use basic strategy of causal graph reasoning.

(2) When the number of deduction paths exceeds 10,000, the time of basic strategy is surprisingly lower than when the number of paths ranges between 5,000 and 10,000. This is because in this kind of extreme samples, there are only few feasible deduction paths (i.e., successfully activate the path because head nodes and prerequisites are activated) even if there are many connected deduction paths.

These experiments and according analysis will be added in final version.

Q3: It is over-claimed that the method is general beyond geometry problem solving in the section 5, because no empirical or theoretical support outside the scope of GPS is discussed. This should be toned down. Thanks for your suggestion on this. Our motivation on this point is to inspire more researchers from domains where the task can be reformulated as such, especially for those relying on explicitly pre-defined action space and costly real action application. We have recognized that stating such conclusion might not be appropriate and we will revise the related content to present a more rational discussion in final version.

Q4: More case studies with deeper analysis is needed for more comprehensive discussion of limitations of the method itself.

Thanks for your precious advice. We will further add more cases and enrich related discussion in the final version from the potential dimension explosion aspect: in some extreme geometry problem sample, the diagram includes hundreds of unique primitives and thus largely increasing the composition of primitives and dimensions of matrices. In this situation, there is high risk for resource being exhaustive. We will try to find more cases to have deeper analysis.

Q5: A complete worked example and a concrete walk-through this example could be included for better illustration.

Thank you for this useful advice. We also believe that a walk-through example which is used to explain in details how the data is processed is beneficial for deeper understanding. In this submission, we simplified the matrix and depicted only partial matrix deduction details because the actual dimension of matrix is usually not appropriate for illustration. We will find a more suitable case for illustration of all the detailed procedures, e.g., step-by-step causal graph reasoning and forward matrix deduction. Please provide us this chance to present it in the final version. (Inclusion of figures in rebuttal is not allowed)

Q6: The citation format in the paper looks strange and hard to follow.

We are truly sorry for the inconvenience caused to your reading experience. We have also noted that the current reference format is not appropriate, as it uses the default plain style. It will be corrected and aligned with the official requirements in final version.

Finally, we sincerely appreciate the invaluable suggestions you provided, as they really contribute to refining the presentation of our work. We also respectfully hope that you could kindly increase the score if most of your concerns are resolved.

We sincerely thank you for your support of our work. Your affirmation of our paper and novelty of the method provides much encouragement for us. Here are our responses to your concerns and questions.

Q1: Suggestions on some professional term and position of overall framework.

Yes, we agree that the overall framework is better to be placed in the methodology part in main content. In our initial submission, we utilized two separate, detailed figures to better illustrate our methods, consequently placing the framework in appendix due to page limit. Also, the term ‘hypergraph’ should be more exact than ‘graph’ and we will change it in our final version.

Q2: Explanation of the red content in figure 1 (c).

In case 1 of figure 1 (c), the numbers shown refer to the indices of theorem rules in the theorem rule base. The solution records the application sequence of theorem rules to get final answer, where the red numbers indicate redundant rules applied but yielded no benefit in answer acquisition. In case 2 of figure 1 (c), the tokens represent the operators and operations of an executable program sequence, where the red tokens represent redundant steps. For the problem in case 2, only one step is needed to get the final answer. Thanks for your question and we will add detailed explanations in the figure description in final version.

Q3: Some mistakes in the equations and reference citation format.

Thank you for pointing out our mistakes in the equations. We will thoroughly review the paper and ensure the correctness of equations, figures and other content in final version. We have also noted that the current reference format is not appropriate, as it uses the default plain style. It will be corrected and aligned with the official requirements in final version.

Q4: The runtime performance compared with baselines and the ablation studies of the proposed Forward Matrix Deduction (FMD).

We want to specify that a direct and thorough comparison of runtime with baselines may not be fair and applicable. On one hand, the goal and contribution is different, where our method aimed at exploration of explainable, shorter and multiple solutions. On the other hand, some baselines are not open-sourced and the resource consumption is varying. Therefore, we default to keep the time within acceptable range. As our method needs extra causal graph establishment and reasoning time, it does take longer time than the basic method Inter-GPS. Specifically, nearly half of problems can be solved within 1s, most problems can be solved within 5s, and only a small fraction of problems need more than 20s, where the method usually fails to obtain the answer.

For more comprehensive analysis, we add the ablation studies of the proposed Forward Matrix Deduction (FMD). We want to re-emphasize that Causal Graph Reasoning (CGR) is proposed as the theoretical foundation to achieve causal reasoning for solution exploration. FMD is one of possible detailed strategies that we propose to implement CGR. So it’s not simply an A+B module-combining architecture that supports independent ablations. We implemented a basic strategy, leveraging Python dictionaries for causal graph reasoning and node status storage. To provide a fair and detailed comparison, we record the average reasoning time per iteration of these two methods, grouped by the number of unique primitive nodes and the number of deduction paths (i.e., sum of all deduction paths in all causal graphs). The other parts and settings remain the same. Experiments were run for 3 times and the average result is reported.

Grouped by number of all deduction paths (/s):

Grouped by number of unique primitive nodes (/s):

From the results, we mainly have the following observations:

(1) FMD consistently performs faster than using Basic when the number of deduction paths is smaller than 5,000 in table 1 and when the number of primitive nodes is smaller than 70 in table 2. Within this range, the time of FMD generally exhibits a steady increasing trend with the increasing of deduction paths and primitive nodes. When the sum of deduction paths increases to 10,000 and even greater, the dimensions of matrices will expand, resulting in more resource demands. Under such extreme and abnormal circumstances, which is not applicable to most geometry problems, it is recommended to simply use basic strategy of causal graph reasoning.

(2) When the number of deduction paths exceeds 10,000, the time of basic strategy is surprisingly lower than when the number of paths ranges between 5,000 and 10,000. This is because in this kind of extreme samples, there are only few feasible deduction paths (i.e., successfully activate the path because head nodes and prerequisites are activated) even if there are many connected deduction paths. These experiments and according analysis will be added in final version.

Q5: Could you clarify the evaluation metric: a problem is considered solved only if the answer obtained is the closest to the ground-truth? Does this imply that longer or substantially different, but still correct, solutions are not counted as successful?

No. Under this metric setting, all solutions of all methods that finally lead to the correct answers are counted as correct, regardless of the lengths. We consider these solutions correct in order to calculate the final solving performance.

Finally, we again appreciate your support for our work and it gives us much encouragement.