Title: Causal-R: A Causal-Reasoning Geometry Problem Solver for Optimized Solution Exploration
Venue: NeurIPS 2025
Accepted: yes
Keywords: Geometry problem solving

Symbolic reasoning
Abstract: The task of geometry problem solving has been a long-standing focus in the automated mathematics community and draws growing attention due to its complexity for both symbolic and neural models. Although prior studies have explored various effective approaches for enhancing problem solving performances, two fundamental challenges remain unaddressed, which are essential to the application in practical scenarios. First, the multi-step reasoning gap between the initial geometric conditions and ultimate problem goal leads to a great search space for solution exploration. Second, obtaining multiple interpretable and shorter solutions remains an open problem. In this work, we introduce the Causal-Reasoning Geometry Problem Solver to overcome these challenges. Specifically, the Causal Graph Reasoning theory is proposed to perform symbolic reasoning before problem solving. Several causal graphs are constructed according to predefined rule base, where each graph is composed of primitive nodes, causal edges and prerequisite edges. By applying causal graph deduction from initial conditions, the reachability status of nodes are iteratively conveyed by causal edges until reaching the target nodes, representing feasible causal deduction paths. In this way, the search space of solutions is compressed from the beginning, the end and intermediate reasoning paths, while ensuring the interpretability and variety of solutions. To achieve this, we further propose Forward Matrix Deduction which transforms the causal graphs into matrices and vectors, and applies matrix operations to update the status value of reachable nodes in iterations. Finally, multiple solutions can be generated by tracing back from the target nodes after validation. Experiments demonstrate the effectiveness of our method to obtain multiple shorter and interpretable solutions. Code is available after acceptance.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work introduces Causal-R, which solves symbolic geometry reasoning problems with causal graph theory. Based on the causal graph definition, this work also proposes a fast forward matrix deduction algorithm to explore the solution space more efficiently. The authors conducted experiments on the Geometry3K dataset, where the proposed causal graph-based reasoning technique outperforms existing methods in terms of accuracy and solution length.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength**\n- Building a connection between causal graph deduction and geometric reasoning is a promising direction and an important topic.\n- The proposed algorithm seems reasonable and new to me. (But I am not an expert in automated theorem proving)\n- The paper is well-written and not hard to follow.\n\n**Weaknesses**\n- I feel confused about the related works and existing literature, and have the following questions:\n1. 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)?\n2. 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)\n3. 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?\n\nIn 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).\n\n[1] Mao, Jiayuan, et al. \"What planning problems can A relational neural network solve?.\" Advances in Neural Information Processing Systems 36 (2023): 59522-59542.\n\nQUESTIONS:\nSee weaknesses.\n\nLIMITATIONS:\nSee weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a symbolic reasoning framework, Causal-R, designed to address the challenge of generating reasoning solutions for geometry problem solving (also termed as GPS in the paper). The authors mention key limitations in existing neural and symbolic methods, particularly regarding solution redundancy, lack of global reasoning, and limited interpretability. To overcome these challenges, Causal-R makes use of causal graph reasoning theory and introduces a forward matrix deduction method that helps in efficient and iterative deduction by reducing the search space, specifically in the problem goal and intermediate reasoning steps, resulting in faster and more accurate inference. Another feature highlighted by the author includes the ability to generate multiple valid and minimal-length solutions.\n\nThe authors validate their approach through a list of experiments, comparing Causal-R against both symbolic and neural baselines. The empirical results highlight the proposed method, Causal-R, achieving the highest accuracy (91.2%) and the shortest average solution length (1.89 steps) among all tested methods, outperforming even human experts in some cases. Furthermore, the model maintains stable performance under different constraints on the number of candidate solutions, indicating its robustness and deterministic reasoning capability. A qualitative evaluation is also provided in the main paper for solution interpretability, reasoning clarity, problem-solving effectiveness, minimal solution length, and the ability to generate multiple valid solutions. These results show the model’s practicality and generalizability, suggesting future works for broader application in symbolic reasoning tasks beyond the geometry domain.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n* The integration of causal graph reasoning into geometry problem solving seems to be a novel direction that distinguishes this work from prior methods, especially those relying on purely neural or symbolic pipelines (though I am not completely through with this line of work, particularly in geometry problem solving). Also, the design of the forward matrix deduction strategy adds another contribution, suggesting a new path for efficient symbolic computation in structured domains like geometry problem solving. \n\n* The presentation quality and clarity are quite good, I would say. The paper is mostly clear in its motivations and high-level design choices (making it easy to follow for someone not from the domain). The authors also do a good job in discussing the broader implications and challenges of symbolic-based systems for geometry problem solving. Tables and qualitative analyses are helpful for understanding the comparative advantages of Causal-R. Also, the empirical evaluation is thorough, showcasing performance gains in both accuracy and solution optimality over several compared baselines, which is helpful for the reliability of the work. \n\n\n**Weaknesses**\n\n* One of the major weaknesses comes from the causal graph reasoning formalization, since it is highlighted as a key novelty, the paper is expected to provide a formal or step-by-step technical description of how this graph is constructed from parsed geometric inputs. This not only weakens the reproducibility and clarity of one of the most important contributions but also makes it difficult to think whether the proposed method is generalizable to other domains where causal relationships are shaky and how one should construct them. \n\n\n* Another weakness comes from the lack of some important empirical experiments; for example, the impact of core components such as forward matrix deduction (also one of the primary contributions of this work) is not isolated in the experiments, i.e., it is difficult to validate looking at the current set of results. An ablation study (or even a basic controlled comparison) would significantly strengthen the empirical claims and clarify which elements are most responsible for the improvements.\n\n* At multiple places, there are claims made in the paper that remain open-ended (even after reading the appendix), for example, the Causal-R claims the ability to generate multiple minimal-length solutions; however, there is no quantitative evaluation or diversity metric included to assess this. Including a metric for solution diversity or optimality would make the claim more compelling/rigorous. Though it is understandable that in general these will be valid, stating it in the discussion section would be good rather than in the main contributions/claims/advantages of the listed method, in general, it is expected to justify all the claims made by the method, which, after reading this paper, remain open-ended. 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.\n\nQUESTIONS:\n* It seems the causal graph reasoning is the primary reason for improvements. The paper also emphasizes the importance of causal graph reasoning, but there seem to be very limited technical details on how the causal graph is derived from parsed geometric representations. It would be great if the authors could provide a formal or step-by-step description (or pseudocode) of this construction process. Also, if the authors could provide some lines on the accuracy/robustness of this formulation, it would help clarify the process more. \n\n* The forward matrix deduction is claimed to be one of the key innovations/novelty, however, its specific contribution to performance is not isolated in the experiments (which can be done using a set of ablation experiments). I was wondering if the authors tried any controlled comparison to evaluate performance with and without this component? This would help clarify its standalone value within the overall Causal-R framework.\n\n* Another claim regarding the advantage of Causal-R is its ability to generate multiple minimal-length solutions, but the evaluation lacks a metric that captures solution diversity or optimality across those candidates. Will it be possible to define or introduce a quantitative measure for multi-solution quality (e.g., minimality score, or maybe a diversity index in a controlled set of experiments), and report results under it? I guess this (even with a small set of experiments) would definitely provide a broader scope to the proposed approach. \n\n* 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 (some, e.g., that come to my mind include logical puzzles, procedural planning, etc., however, a direct application would need more effort, I guess)? I guess providing even a small case study or illustrative example of just the formulation would help support the generality claim and make it more applicable for future works on similar lines. Since the work seems to have broad potential, it would be good if they were explicitly stated for future work.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes Causal-R, a symbolic geometry problem solver based on Causal Graph Reasoning (CGR) and Forward Matrix Deduction (FMD). The method effectively compresses the solution space and generates interpretable, shorter, and multiple solution paths, addressing key limitations of existing approaches.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. Reformulates geometric reasoning as causal deduction, which is both elegant and practical.\n\n2. Outperforms baselines in both solution quality and accuracy on Geometry3K.\n\nWeaknesses:\n1. A critical limitation is the absence of ablation studies to validate the individual contributions of core components, namely Causal Graph Reasoning (CGR) and Forward Matrix Deduction (FMD). The current experiments only demonstrate the overall performance of the Causal-R framework but fail to isolate the specific impact of CGR on search space compression, FMD on reasoning efficiency, or their synergistic effects. For instance, it remains unproven whether CGR alone outperforms traditional symbolic reasoning in shortening solution paths or if FMD is genuinely responsible for accelerating the deduction process. For example, design ablation experiments comparing variants such as removing CGR and using only traditional symbolic reasoning (e.g., step-by-step rule application as in Inter-GPS), retaining CGR but removing FMD with pure symbolic iterative deduction, and the full method (CGR+FMD); through comparing accuracy, solution length, and reasoning time, the necessity of each module can be clarified.\n\n2. The sensitivity analysis of hyperparameters is insufficient. While the paper mentions setting the maximum number of iterations to 7 and candidate solution constraints (λ) to 1, 2, or 3, it does not investigate how these choices affect outcomes. There is no exploration of whether fewer iterations might miss optimal solutions for complex problems, whether larger λ values (e.g., 4 or 5) could enhance solution diversity without compromising accuracy, or how initializations of the status vector in matrix deduction influence convergence speed. To improve this, test different values of iteration count α (e.g., 3, 5, 7, 10) to observe their impact on solution length and accuracy, verifying the assumption that \"the shortest solution is found in early iterations\"; expand the range of λ (e.g., 1-5) to analyze the trade-off between solution diversity and redundancy; and test how the initialization of the status vector (e.g., all negative, partial known node initialization) affects convergence speed.\n\n3. Quantification of reasoning efficiency is inadequate. Although FMD is claimed to reduce time complexity to O(α), no concrete runtime data is provided, nor is there a direct comparison with existing methods like E-GPS in terms of inference speed. For practical applications such as real-time educational feedback, measuring efficiency across problem complexities (e.g., multi-step vs. simple deductions) is crucial but absent. To address this, record the average reasoning time of different methods on the same hardware, grouped by problem complexity; and compare the time consumption and memory usage of Causal-R and E-GPS on large-scale problem sets (e.g., 1000 complex problems) to verify the engineering value of FMD.\n\n4. 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. To fix this, randomly select some problems, invite geometry experts to provide the shortest solutions, and compare their length and correctness with those generated by Causal-R; and compare the length of solutions from Causal-R with the annotated solutions in the dataset to verify the assumption of \"surpassing manually annotated suboptimal solutions.\"\n\n5. Experiments on the scalability of the theorem knowledge base (KB) are missing. The paper notes that expanding KB improves performance but does not test how larger KB sizes (e.g., 30+ theorems) affect search space expansion or solution redundancy. It is unclear whether Causal-R’s causal graph mechanism can effectively suppress rule combination explosions when KB scales, a key claim for its practical applicability. To address this, gradually expand the number of theorems in KB (e.g., 17, 24, 30), test changes in accuracy and solution length, verify the robustness of the method to KB expansion, and analyze whether the \"causal graph’s role in inhibiting rule combination explosions\" holds.\n\nQUESTIONS:\n1. While Forward Matrix Deduction is proposed to accelerate reasoning, there is little quantitative evidence on runtime or memory usage. How does the reasoning efficiency compare to prior symbolic systems (e.g., Inter-GPS, E-GPS) on complex problems? Please include runtime benchmarks or complexity analysis, especially on problems requiring deeper reasoning chains.\n\n2. How does the current Reasoning LLM / VLM, such as o3, o4, and r1, perform on your task?\n\n3. How does your method perform on mathvista benchmark?\n\nLIMITATIONS:\nSee weaknesses and questions.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces Causal-R, a symbolic-based geometry problem solver that proposes a Causal Graph Reasoning (CGR) theory and a Forward Matrix Deduction (FMD) method to efficiently and interpretablely solve geometry problems. The approach systematically encodes theorem-based deduction paths as causal graphs and matrix operations to reduce the search space and extract interpretable solutions. The paper conduct experiments on Geometry3K, showing improvements over symbolic and neural baselines in both solution accuracy and length.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. I like the paper that provided a clear problem formulation and motivation. The paper identifies two critical issues in prior works: large multi-step reasoning space, and lack of short and interpretable multi-solution paths.\n2. The paper introduce a novel framework that combines the causal graph reasoning and faster forward matrix deduction. FMD is an efficient abstraction of symbolic deduction, making the reasoning process computationally tractable.\n3. By design, the solution step maps to a known theorem, aiding interpretability and potential educational application. \n4. The experiments demonstrate the good performance of the poposed method on the Geometry3K dataset and even outperforms human expert.\n\nWeakness:\n1. The proposed method only has been tested on a single dataset, Geometry3K, which was introduced in 2021, four years ago. There’s no exploration of robustness to noisy inputs, or evaluation on other datasets to test generalizability. For example, how does it perform on other popular datasets, such as the recent FormalGeo7k or geometry questions in the OpenAI MATH dataset?\n2. There is no ablation to disentangle the contribution of: causal graph reasoning vs. forward matrix deduction (e.g., performance and efficiency); Predefined theorem base vs. solution exploration strategy; Impact of number of iterations or candidate paths on final performance in more depth.\n3. The paper over-claims the method is general beyond geometry problem solving in the section 5 (e.g., “causal deduction format” can be applied broadly) but provides no empirical or theoretical support outside of geometry in this paper. This feels speculative and should be toned down unless supported.\n4. The case analysis in Appendix D attributes failure primarily to parsing errors or missing theorems in the rule base (e.g., absence of the Geometric Mean Theorem). However, these do not sufficiently probe fundamental limitations of the proposed causal reasoning framework itself. For instance, the method’s ability to handle ambiguous, non-deterministic, or cyclic reasoning paths is not evaluated. A more meaningful analysis should include failure cases that arise even when parsing is correct and the theorem exists, for example, when multiple conflicting causal paths are feasible or when causal deduction leads to incorrect intermediates due to compositional error propagation. Is the method robust to these intermediate error?\n5. While the paper emphasizes interpretability and optimality, it lacks a complete worked example that would substantiate this claim. A concrete walk-through of a single example, showing the causal graph traversal, matrix deduction steps, and resulting multiple (or shortest) solutions, would strengthen the claims. For comparison, the cited work E-GPS provides detailed solution reasoning trees and theorem-level step tracing, which make interpretability more transparent. I recommend including a similar case in the main paper or appendix.\n\nSubtle details:\n1. The citation format in the paper looks strange and hard to follow. First, the references are not numbered in the appendix, making it hard to search for the corresponding reference. Second, isn’t the reference number should be wrapped in a square bracket []?\n2. In section 3.2.1, line 151, the primitive node g is used before it got defined in the next paragraph.\n\nQUESTIONS:\nSee the weakness part for the questions. They are:\n1. For example, how does it perform on other popular datasets, such as the recent FormalGeo7k or geometry questions in the OpenAI MATH dataset?\n2. In the failure case analysis, Is the method robust to these intermediate error?\n\nLIMITATIONS:\nWhile the authors include a brief paragraph on limitations in Section 5 and reference Appendix E, the discussion is incomplete and overly superficial. The listed limitations primarily focus on external components such as diagram/text parsing or the lack of certain theorems in the rule base, but the core limitations of the proposed causal reasoning framework itself are not meaningfully examined. For example, is the 24 hand-crafted symbolic rules brittle in other datasets?\n\nAdditionally, the authors assert that their method has potential generality beyond geometry problem solving, but no limitations are discussed regarding this extrapolation, which may mislead readers about its broader applicability.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a graph-based framework for solving geometry problems, aiming to improve interpretability and generate shorter solutions. The method encodes the preconditions and reasoning rules of various theorems into a graph representation, and systematically applies iterative reasoning over this structure to derive solutions. To enhance efficiency, matrix operations are employed to accelerate the reasoning process. Empirical results demonstrate that the proposed approach achieves higher accuracy and produces shorter solutions compared to state-of-the-art methods.\n\nSTRENGTHS AND WEAKNESSES:\n#### Quality\nThe paper is of high quality. The proposed graph representation for solving geometry problems is both novel and well-grounded. Its transformation into matrix operations enables GPU acceleration, significantly improving reasoning speed. The methodology ensures interpretability of solutions, and the empirical evaluation demonstrates that the proposed approach produces shorter solutions.\n\n#### Clarity\n\nThe paper is generally easy to follow, though certain aspects could be improved for better clarity.\n\nI recommend bringing the overall framework into the main content to provide readers with a systematic understanding of the proposed method. Additionally, it should be clarified that the causal graph used here is not a simple graph, but a “hypergraph”, which supports edges from a set of vertices to another edge. A brief explanation distinguishing it from the causal graphs commonly used in causal inference would also help avoid confusion.\n\nSome minor comments: \n1. The citation format appears inconsistent. Papers are cited using indices, but the corresponding indices are missing in the reference list, and the brackets around the in-text citation numbers are also absent.\n2. In Figure 1(c), please explain the red redundant steps, particularly the meaning of the numbers, e.g., 17, 1, 2, etc. \n3. Consider briefly walking through Algorithm 1 to guide the reader.\n4. In Equation 1, $E_{i,j}$ should be $R*$.\n5. $g(k,l)$ is missing in Equation 3.\n6. In Table 1, it would be helpful to specify that the numbers in the first few columns represent accuracy. \n\n#### Significance\nThe proposed method is impactful. It not only produces shorter and interpretable solutions to geometry problems but also generates multiple solutions, which can be particularly valuable for educational purposes. Empirical results show that it outperforms both prior methods and human experts. \n\n#### Originality\nThe idea of representing theorem rules as a graph and accelerating the reasoning process via matrix operations is novel and original to the best of my knowledge.\n\nQUESTIONS:\n1. What is the runtime performance comparison between the proposed method and the baselines? Specifically, how much speedup is achieved by using matrix operations compared to the sequential reasoning process?\n2. 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?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-06"}