Causal-R: A Causal-Reasoning Geometry Problem Solver for Optimized Solution Exploration
Authors: Wenjun Wu
Lingling Zhang
Bo Zhao
Muye Huang
QianYing Wang
Jun Liu
Venue: NeurIPS 2025 poster ()
Keywords: Geometry problem solving
Symbolic reasoning
Primary area: applications

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.

Causal-R: A Causal-Reasoning Geometry Problem Solver for Optimized Solution Exploration
Wenjun Wu Lingling Zhang 1.3g Zhao 1.3 Muye Huang!.4 Qianying Wang", Jun Liu
School of Computer Science and Technology, Xi' an Jiaotong University, Xi'an, 710049, China
Lenovv Research
Ministry of Education Key Laboratory of Intelligent Networks and Network Security, Xi an Jiaotong University, Xi Xi'an, an, 710049, China
Shaanxi Province Key Laboratory of Big Data Knowledge Engineering. Xi'an an Jiaotong University, Xi an, 710049, China
Abstract
The task of geometry problem solving has been long-standing focus in the automated mathematics community and is drawing 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 appli- cation in practical scenarios. First, the multi-step reasoning gap between the initial geometric conditions and ultimate problem goal leads a large search space for solution exploration. Second, obtaining multiple interpretable and shorter solu- tions remains an open problem. In this work, we introduce the Causal- Reasonnng 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 is iteratively conveyed by causal edges until reaching the target nodes, representing feasible causal deduction paths. In this way, the search space 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.
Introduction
Geometry Problem Solving (GPS) is long-standing task 30, in the automated mathematics problem solving community III 5 28 It aims obtain the final numerical answer with given geometric diagram and problem text which includes problem goal (e.g., "Find length of AD.") "). illustrated in example in fig. (a). In recent years, this task has attracted increasing attention due to its challenging requirements of multiple capabilities of the model, including multi-modd understanding. abstract geometric reasoning and mathematical computation.
Existing methods can categorized into two main branches according to their reasoning mechanisms. Neural-basee methods 2 3 35 extract the multi-modal features of original geometry problem and
Cooresponding author. Email: nickjumsorkel com, changl1 jigggxxijiiii tu. edu. en
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Geometric Diagram:
Solutions of other methods
DBC
B
BD - 32
Theorem
Problem:
in (a) the left
32
Solution:
AC =
Isosceles Triangle Theorem
LBDA - 54
17, 1, 2, 5, 8, 10...
BC
Problem Text:
Find length AD.
2 DCA 54
Isosceles Triangle Theorem
Moxure of anglle is I>, movore of anglc 1 Find miserr: of ofangle
Problem:
Candidate Choices:
B. 23.2 252 D.440
AB y
Pythagorean Theorem
BD 32
Solution:
Answer:
<BAD
C90 Bum
(a) Geometry Problem
(b) Partial Searching Space of Solution Path
(c) Solutions Cases
Figure 1: Illustration of solving typical geometry problem. The red part in solutions in (c) refers to redundant applied theorem indices in case I (19) and redundant program sequences in case 2 (35).
predict executable program sequences by neural models, which are then executed by external symbolic engine for the answer. Recent approaches 21 4 applying large language models primarily distinguish in the output content 9
Symbolic-basee methods transform the raw problem into formal language
representation 19 13 15 which is acceptable by symbolic systems, and continuously apply predefined
theorem rules 19. 24 25 to deduce new geometric conditions until the problem goal is solved. Both
approaches have been making efforts on the integration of multi-mode information 18 20 theorem prediction (39 14 and development of symbolic systems 23 resulting in significant improvements on problem solving capabilities of the model. However, the application of these models in real-world scenarios 31 such as educational use, is still impeded. We identify the following two overlooked challenges from a broader perspective, which need further exploration from the research community.
First, the multi-step reasoning gap between the initial geometric conditions and ultimate problem goal causes great search space for solution exploration. Different from general visual reasoning tasks 16 [ geometry problem solving requires multi-step logical and mathematical deduction, involving many geometric elements and intermediate values that are unknown in the problem. As a result, the search space of potential solution paths becomes extremely large when numerous combinations of deduction paths and geometric conditions are considered (as illustrated fig. (b)). For complex problems, the search space greatly challenges the pattern-fitting prediction mechanism of neural networks which are trained with only limited geometry problems requiring simple solutions. For symbolic-based approaches, some studies 19 23 employ pretrained neural models for step-wise theorem prediction to prune the search space each step, but with limited gains and unreliable performance. E- E-GPS [3 novelly proposes to solve the problem from the target in top-down manner, which effectively narrows the search space to a scope which finally leads to the problem goal being solved. Nevertheless, it ignores the constraints of "bottom" known conditions and still searches for paths that should be abandoned from the beginning.
Second, obtaining multiple interpretable and shorter solutions remains an open problem. On one hand, the correctness and interpretability of solutions are not well guaranteed previous works, impeding the application of models in situations that require solution accuracy (e.g., educational use). Despite leveraging the prediction power of neural models, most prior methods are inherently unable to prevent the introduction of redundant and incorrect steps, even if the final answer is accurate. We list some representative results given by these models in fig. (c). This issue becomes more serious for intricate geometry problems that require longer solution steps, not only increasing the difficulty of problem solving but also decreasing the reliability. On the other hand, obtaining shorter and diverse solutions remains unexplored. For majority of geometry problems, the solution path to problem goal is not unique. Obtaining shorter solution paths, as well as multiple solutions, is beneficial to providing optimized solutions to a problem from various perspectives, enabling the deployment of models in practical scenarios. Simultaneously. it could also address the limitations of existing manually annotated GPS datasets for neural training, using fixed and sub-optimal solution annotations [8 31
To overcome the aforementioned challenges, we take a step further into the common process of geometry problem solving, which can be roughly divided into problem understanding, problem

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reasoning and problem solving. Interestingly, most works utilize neural networks for the task of reasoning (e.g.. theorem prediction in tymbolic-based methods or sequence prediction in neural- based methods). Besides E-GPS 31 there is a notable absence of research grounded in symbolic reasoning approaches, where we have identified much potential. Take Pythagorean Theorem as an example, the lengths of two sides of the triangle are known and that one of its interior angles is right angle, the system could then calculate the length of the third side. Without performing detailed calculation, the process can be simplified as known A cond result, where known represents known attributes of primitives (e.g., lengths of two sides), cond represents the prerequisite conditions (e.g., right angle) and result refers to the conditions that acquirable (e.g., length of the third side). Such causal deduction reasoning process is challenging for neural models to learn, particularly under complicated theorem base and scarce GPS datasets, but is promising for symbolic systems. In this way. the model is able to perform symbolic reasoning by continuously applying the simplified and unified deduction process from the initial geometric conditions, excluding the complicated calculation. Furthermore, the deduction path explicitly records the intermediate outcomes between the start and the end, providing traceable and explainable reasoning paths.
To achieve this, we propose the Causal-Reasone we first follow the common strategy to parse the raw geometry problem into structural formal language representations. Then, we introduce the theory of Causal Graph Reasoning (CGR), where the causal relations between geometric primitives are built in causal graphs according to the predefined theorem base Each causal graph is a hypergraph consisting of primitive nodes, causal edges and prerequisite edge. Primitive nodes represent the attributes of unique geometric primitives in the parsed conditions, with initial status of positive (i.e., value is known) and negative (i.e., value is unknown). Causal edge builds the causal relations between head nodes and tail nodes, representing feasible deduction path. Prerequisite edge points from the prerequisite conditions to a causal edge, controlling its application. In this way, the model is able to perform cymbolic-reasoning before actually solving the problem by continuously applying the causal graph deduction until the problem goal is reached, It compresses the search space of solution paths not only from the established conditions the beginning, but also from the problem goal at the end and the intermediate reasoning paths. Based on the CGR theory, we further introduce the Forward Matrix Deduction (FMD) method which transforms the exhaustive application of causal deductions into matrix operation, enabling the model to perform faster reasoning within iterations. In each iteration, the value vector, representing the reachability of primitive nodes, is updated by matrix operations. When all target values are obtained, the solutions w.r.t. each target can be generated easily by tracking back from the target values to the initial conditions according to the recorded deduction path. These candidate solutions are then sorted in a enggh-increasinn order to verify the feasibility, after which we can obtain multiple solutions to solve the geometry problem. Our contributions can be summarized into the following three points:
Geometry Problem Solver (Causal-R). Specifically,
We propose the Causal Graph Reasoning theory which simplifies the theorem deduction unified causal deduction path for symbolic reasoning. It compresses the search space from the beginning, the end and intermediate process, and ensures the obtaining of optimized solutions.
We propose the Forward Matrix Deduction method to transform causal graph deduction into matrix operation, enabling the model to perform faster reasoning for solution exploration within iterations.
We propose the Causal ReasonnRe Geometry Problem Solver to verify the above methods. Ex- perimental results demonstrate the effectiveness of our method to obtain optimized solutions, iee. interpretable, shorter and multiple solutions, in geometry problem solving. Reference of codes is available at https//ggginu..cco/niccccee///ip
2 Related works
Automated mathematical problem solving has always been hot topic in the community II
5
28 where geometry problem solving has obtained increasing attention in recent years due to its complexity. requiring multiple crucial capabilities of machine. Earlier methods 7 8 32 such as Wu's Method 30 built fundamental approaches for understanding and proving geometry problems. Recent advancements in neural networks and symbolic systems have spurred increasing researches for breakthroughs problem solving performances of automated geometry problem solvers, which can be categorized into two mainlines.
3

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Neural-based methods predict executable or human-readable sequences based on the extracted multi-modal features of raw problem. For example, Chen al. [ made the first attempt and established baseline to uniformly process textual and visual data by encoder-decoder framework and generate program sequences, which are then executed by external symbolic engine. Since then, continuous improvements have been made on the problem types information extraction [35] 18] and cross-modal information alignment [20 [8] With the exploitation of large language models (LLMs),, many studies have evaluated and developed the performances of LLMs mathematic tasks 17 26 including geometry problem solving 33 Some works focus on improving the multi-modal problem understanding capabilities of multi-modal LLMs 21 especially
geometric diagrams 4 37 36
Other works try to enhance the problem reasoning and solving skills MLLMs by integration of external symbolic engines [22 6 such as ToRA[22am and MathCoder 29
However, these methods fall short of guaranteeing the accuracy and interpretability of generated solution paths due to the inherent shortcomings of autoregressive models, likely introducing redundant
and wrong steps.
symbolic-based methods continuously apply predefined theorem rules on parsed geometric con- ditions until the problem goal is solved, Inter-GPS 19 proposes to use a pretrained neural model to predict sequence of potential needed theorems, improving the efficiency based on brute-search strategy. Subsequently, several works aim at enhancing the capabilities of neural models to analyze the pattern and make improved predictions. For example, Peng al. (23] and Zou et al. [39] utilize reinforcement learning to enable the model with step-wise prediction power before each step of theorem application. Following them, Huang et al. [14] builds global hologram of geometric con- ditions and pattern holograms of theorems and uses neural models to predict appropriate theorem based pattern matching each step. Recent work Pi-GPS 38 mainly focuses resolving the textual ambiguities with diagrammatic information, significantly improving the applicability of symbolic-based methods. As an outstanding work, AlphaGeometry 28 highhlissh the potentials of symbolic models on mathematical proof problems even at Olympic-leve encouraging researchers to keep exploring. E-GPS is the first work that emphasizes the importance of explainability of solutions, advancing the practical application of geometry problem solvers.
However, most of these works mainly focus on solving the geometry problems instead of reasoning for optimized solutions, ignoring the importance of reasonable solutions in real application scenarios. Besides E-GPS, they are unable to ensure the interpretability and accuracy of solutions (e.g., without introducing redundant steps), leaving the exploration of shorter and multiple solutions an ongoing research problem.
3 Methodology
Given a geometry problem (D, T), where and represent the geometric diagram and problem text respectively, our model aims obtaining the answer of problem goal well as corresponding solutions S. The overall framework of our Causal-F - is illustrated in fig.
3.1 Problem understanding
Understanding the detailed content of multi-modal geometry problem is the basis of following stages. Unlike ordinary multi-moda. question answering tasks, solving geometry problems critically relies on the symbolic abstraction and precise mathematical expression. We follow the previous symbolic- based methods 19 31 to first parse the geometry problem into formalized geometric conditions (e.g., Equals (LengthOf (BD), , 32)), ), represented in structural clauses, with a commonly used diagram parser 34 and text parser 19] In this way, the problem is converted into a geometric condition set, describing the basic definitions of geometric primitives (e.g., line and angle), non-gcometric primitives (e.g. text and symbol), math expressions and their relations.
3.2
Causal graph reasoning
3.2.1
Causal graph construction
In this work, causal graph is used to represent the deduction path from known conditions to new acquirable conditions the prerequisites are met w.r.t. a specific theorem rule. Given the theorem rule base KB of K rules, K corresponding causal graphs & [G1, G2, ,,GK) } are constructed

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(a) Problem Understanding
(b) Causal Graph Reasoning
Uihauen Tooms Jepiralle
(d) Problem Solving
Hestiis
Gesmetric Conditions
Trianglert
Theoren Rube Base
Kode Staricn
Mede Status
TrangleeAA.. (.,0)
Program
TrinndesB....
ot 01.22
IneAA
0
Theorom Rule Base
Equarat AneeA,
Equa sthing 1o
Cantidatte
Linec, 2ip
Caasal Gragits
Prenttive Nodia
Teimiuve Nedia
LinerC,0y
Causal Graph Deduction
Fry uedd (n
13
(c) Fast Problem Reasoning
Resutiins
Vertfication
Prenquustta
Value Deduction
Geometry Problem length
SOOTA SACIEE
Final States
X
KKLL
Staras
Generation
Ferward Natta Deduction
Figure 2: The overall framework of our Causal-R. It mainly contains four parts: (a) in problem understanding stage, the raw geometry problem is parsed into formalized geometric conditions in section 3.I] (b) causal graph reasoning in section 3.2 provides the fundamental theory for causal deduction based causal graphs; (c) in fast problem reasoning stage in section 3.3 the causal graph deduction is implemented by forward matrix deduction to reason for solutions to final problem goal in iterations; (d) the solutions are generated by tracing back from the target node, and are then verified to obtain the final answer in section 3.4
according to the detailed theorem knowledge of rules. Each causal graph Gk is hypergraph consisting of primitive nodes N {1I1, 122," nMM,, }. causal edges Ej = [ef, e5, Ne and prerequisite edges EN {ef, eg- -.. N, which are defined as follows:
Primitive node: represents the attribute of a unique geometric primitive the condition set, which stands for visually identified primitive in the geometric diagram. Each node n, has initial status (i.e., positive and negative), where positive means the corresponding value of attribute is known or acquirable and negative means it is unknown. For example, /BDA and ZCDB allocated the same primitive node with negative status. For simplification, we use rū, to represent its status.
Causal edge: builds the causal relation between the primitive nodes that points from one more head nodes to a tail node, denoted as nī, < g(k; i) -f ni, i.e., g(k, i) positive, n, is considered reachable and ni, set positive all status of these head nodes are positive. Note that ni, will not be changed into negative once it has been changed to positive. Here we use n. fi, ne as placeholder for easier reference and understanding. Please refer to appendix B for detailed formulation of each theorem rule.
ri). (i,
When the prerequisite is satisfied,
Prerequisite edge: defines the prerequisite constraints to enable specific causal deduction, denoted as g(k: i) + c, rel(n,, n,)),, where the prerequisite conditions can be either status of primitive nodes or the relation status of nodes (e.g., length of equals length of BC). g(k, i) is positive only when all the prerequisite statuses are positive, otherwise negative. Please refer to Appendix C for detailed formulation of each theorem rule.
In this way, each G2 = (N, EC, F.?') records all the possible deduction paths between the geometric primitives in unified and simplified format according specific theorem rule without detailed calculation, which provides the basis for the following reasoning stage. To distinguish the problem goal f. we parse it into series of needed primitive nodes N {nf, niy ... nzz) n e N, where the problem goal is reached only if all status of N are positive.
3.2.2 Causal graph deduction
Based on the constructed causal graphs, we are able to perform symbolic reasoning from the initial status of primitive nodes N to a final status of target primitive nodes N Theoretically. for an a-step geometry problem, the model needs iterations of causal graph deduction update all status of N
5

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to positive, For an application of each causal graph G3 in an iteration, all prerequisite edges are first automatically applied to generate the controlling gates -) of causal edges. Subsequently, all the causal edges are applied based on the status of primitive nodes from last iteration and the controlling gates g(k::) .) to update the related node status (i.e. all the tail nodes in this causal graph). In each iteration, all causal graphs are applied separately to deduce new reachable node based on the previous status of N from last iteration, so the deduction process of causal graph is self-contained without interference. After each iteration, the status of N is updated according to all causal graph deduction results. Therefore, the causal graph deduction process is able to explore all acquirable primitive nodes after a iterations. The corresponding algorithm is given in appendix A
To be more specific, we provide a causal graph deduction example fig. to illustrate how the problem goal is reached regarding the given geometry problem. After two iterations of causal graph deduction, primitive node AD is considered acquirable. By tracing back from node AD, two different solution paths of causal deduction (i.e., 1-A->2-C and 1-C->2-B) can be generated, which corresponds to the actual solutions on the left. It can be seen that the causal graph deduction process includes all possible solution paths but narrows the solution space from the beginning, the end and the intermediate process. On one hand, it constrains the solution path to originate from the known conditions and terminate at the ultimate problem goal, with all intermediate steps adhering to the geometry theorem deduction. On the other hand, any solution path that is feasible to acquire the ultimate answer should be contained in such solution space because it should satisfy these three requirements. It imitates the process by which human experts reason for possible solutions based on the given conditions, but in a full exploration manner.
We highlight three advantages of our causal graph reasoning: (1) Interpretability: Since deduction paths are recorded specifically in the causal graphs, it is able for us to trace back for specific causal deduction path and head primitive nodes w.r.t. any acquirable tail node in each iteration, ensuring that the reasoning and solving solution is detailed and explainable. (2) Multiple solutions: For target primitive node, multiple solutions can be obtained by different combinations of feasible causal deduction paths and intermediate nodes. (3) Shorter solution: Benefiting from the fact that any feasible solution should be included within the space constrained by our method, it is possible to identify the shortest solution path by tracing back from the first iteration that acquires the target node.
Geametry Problem
Iteration
O
Uakeowe Mode
Known Kode
Acquirable Mede
Target Kode
Q: Find langth of AD?
Node
Isosceles Triangle Theorem
Congruent Triangle Thearem
Costnee Triangle
RC
BGA
ARo
Solution
BAD
SAO
1-G
1-B
tscacstes Triangle Theorem.
an nc i: LODAA _B04 =
Iteration
Precequtsites - Canael Edge
Prerequiaite Edge
Shep Contoe Triangle
AD 32% -
BAD
cost/ BDA)
Node
Pythagorean
Congruent Triangle Thearem
of Triangle
Solution
AC
AP
Cosino Triangle BNG J. B0 coal BCA)
A. AC 32 cmop44) 10.0
BBA
Congruent Triangle Thoorem.
A2D
RAD
RAD
BAD BBA
AAD i ABAC, AD - AC 10.0
Figure 3: Illustration of 2-iteration causal graph deduction (CGD). Two feasible solution paths of the problem are given the left, which be deduced by CGD on the right. For simplification, only the major content is presented in the figure. In iteration 1, the primitive node /BDA is acquirable by applying Isosceles Triangle Theorem (1-A) on node /BCA, because node /BCA is positive and prerequisite of rel (BD, BC) is satisfied in conditions (i.e., BD equals to BC). Similar situation applies to Cosine of Triangle (1-C). However, in graph I-B that represents Congruent Triangle Theorem, node AC is still negative, causing the failed deduction to obtain node AD. Next in iteration 2, AD is now acquirable due to the positive status of either node AC or BDA being just changed in the last iteration.

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It can be seen that causal graph reasoning lays the theoretical groundwork for optimized solution path exploration in geometry problem solving. However, the sequential and exhaustive application of causal graph deduction will diminish the advantages of reasoning-solving strategy. making the reasoning process inefficient and cumbersome. To overcome this, we introduce the forward matrix deduction method to realize faster problem reasoning before problem solving stage.
3.3 Faster problem reasoning
Moving beyond the aforementioned causal graph reasoning that sequentially applies the causal graph deduction, novelly propose a faster reasoning method for search space compression. Specifically, the causal graph deduction process is transformed into forward matrix deduction, changing from obtaining the acquirable positive status of target node to obtaining its status value.
3.3.1 Matrix definition
To achieve matrix deduction, several matrices are implemented according to the causal graphs G and primitive nodes N. First, an M-dimension status vector designed represent the status of 1 means the
primitive nodes N, denoted as v [U1, . UMr]", 2 E [-20, where n, positive, otherwise negative, and oo an infinite number. Second, the established relations between the primitive nodes given in the geometry problem are converted into an indicative condition matrix Rº {0, 10..1..xM, where R represents rel(n,, n1) is positive otherwise negative. Third, the prerequisites of causal graphs are transformed into another unchangeable prerequisite matrix P E 1) xxxxxMMM
where P.. indicates the relation rel(n;, ny) between primitive node n, and ny is required to perform corresponding causal graph deduction, and refers to number of possible deduction combinations within a causal graph. Fourth, the causal deduction path from head nodes to tail node is recorded by indicative causal matrix C E {0, 1] } x xM where C...j 1 means the value v2 is needed to obtain the value u4- To specify the target nodes, an indicative target vector = [of v/2 ... "M of E {0, used, where of = means n, E N
In this way, all the requirements of causal graph deduction are represented as the positional values in these matrices and vectors, enabling the model achieve fast reasoning by matrix operation.
3.3.2 Forward matrix deduction
Based on these matrices and status vectors, the causal graph deduction process is transformed into forward matrix deduction for problem reasoning. For an a-step geometry problem, the matrix deduction needs iterations to obtain all values of target primitive nodes. Specifically, in each iteration o, an intermediate relation matrix R is generated by combining the condition matrix R with value-relation matrix V that is calculated by vo-1;
R max R V.g). VL, max(v?" .000 O) soaxxv/-1.00
(1)
where max(-.) operation keeps the max value of two arguments and 0 < i.j < M. In this way, the relation matrix R records whether there comparable relation, either from the problem conditions or the values obtained from last iteration, between the primitive nodes in iteration o. Then, it is used check whether the existing conditions satisfy the prerequisites by comparing it with the prerequisite matrix P and generate the controlling gate signal g(k.
M M-1M-1
g(k. max
Σ
Σ
max (P..ss R2 0).0).
(2)
where 0 < A < K and 0 < l L. Subsequently, the gate signal g(k: () is utilized to control the following process value deduction, using causal matrix C' and previous status vector vo-1,
K-1L-1
M-1
= 7 1) 101 min
Σ
E g(k,))
max (0,
Σ 1=0
CRIIS x xy (3)
A=O
where min(..) returns the minimum of two arguments and 0 S < M. this way, the status of acquirable primitive node in this iteration is all recorded in the status vector v² and it is feasible to
7

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12
Opdars
A/R
X
Condition Matrix
Value-relation Matrix
Relation
Prerequisite Matrix
Matrix
Prerequlsits Chack
Value Oedection
Figure 4: The illustration of forward matrix deduction (FMD) for iteration 1-Isosceles Triangle Theorem (1-A) in fig.
check whether the problem goal reached by verifying the values between vo and v' using:
M
f(v', vo) = max 1
Σ 1=0
v min(y) .. 1)),
(4)
where f(v', v") indicates the problem goal is reached, otherwise f(v", v²) 0. We give an example the matrix deduction procedures fig. 4 illustrate how the primitive /BDA obtained by Isosceles Triangle Theorem in iteration in fig. 3 By applying forward matrix deduction, the update process of primitive node status can be transferred to GPU devices. Benefited from the matrix operation, this method is able to compress the solution search space with an approximate time complexity of ((a) for a-step geometry problem.
3.4 Problem solving
After applying forward matrix deduction, the model completes the process of problem reasoning and obtains solution space leading to the ultimate problem goal. It is noteworthy that although the forward matrix deduction may acquire many values of those primitive nodes that are not related to problem goal (e.g., new values obtained after a iterations). these values can be easily excluded from the end when tracing back. Theoretically, all feasible solutions are included in this search space, which is recorded in the intermediate results of matrix deduction in iterations. In order to generate shorter solutions, we start from the earliest iteration where each target node is acquired for the first time. Tracing back from the target node with the causal graphs and matrix deduction recordings, a candidate verified solution set = {81, 82, { can be obtained, where constraint the actual number of solutions that exist. Each solution path 8; is ordered sequence of theorem rule application that then executed by external symbolic engine for generating the final answer of geometry problem.
a manually set
4 Experiments
4.1 Experimental settings
Dataset. Following previous symbolic-based methods, we choose the popularly used Geometry3K 19 our benchmark, which includes 3,002 geometry problems that covers wide range of problem types such as triangles, circles and polygons. Each problem has a geometric diagram and problem text, and is annotated with explicit formal language representations of geometric conditions, providing friendly platform for symbolic- maasolic-based methods. For use of neural models, is split into 2,101, 300 and 601 samples for training, validation and testing respectively.
Metrics. For fair comparison, two main evaluation metrics are used: (1) Accuracy: A geometry problem is considered correctly solved only if the answer obtained by model is the closest to the ground-truth. (2) Solution Length: The average length of solution paths (i.e., the length of theorem application steps) for the solved geometry problems. Each step is vigorously defined in the theorem

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Table 1:
The main performance comparison results, Solution refers to the Solution Length metric. GT means using the annotated parsing results of the problem. A means using improved and disambiguated parsing results for reasoning. Methods with mark and mark use theorem rule base of 17 theorems and 24 theorems, respectively. Human (expert) performances are borrowed from Inter-GPST9
Question Type
Geometric Shape
Methods
Accuracy
Solution
Mcasure
Length
Area
Line
Triangle
Quad
Circle
Human
53.7
59.3
57.7
46.7
53.8
68.7
61.7
56.9
Human Expert
89.9
92.0
93.9
95.9
92.2
90.5
89.9
90.9
35 LANS
77.9
82.3
SeoGen-SFT-78]
58.4
"Inter-GPS]
59.1
61.7
30.2
59.3
66.0
52.4
45.5
57.5
Inter-GPS 19 (GT)
83.1
77.9
62.3
86.4
83.3
77.6
61.5
78.3
7.10
FE-GPS 31
76.8
62.6
24.5
72.8
73.0
55.7
51.5
64.7
(3.49) 4.19
EE-GPS 31 GTT)
83.8
80.0
66.7
87.7
85.3
79.2
65.9
79.8
(3.35) 3.99
GeoDRL 23
75.5
70.5
22.6
77.8
76.0
62.9
53.8
68.4
"GeoDRL 23 (GT)
86.5
93.7
75.5
87.7
93.1
78.3
89.4
2.34
"E-GPS 31
78.3
67.2
27.7
76.1
75.6
59.4
55.0
67.9
(1.63) 2.28
"E-GPS (GT)
90.4
92.2
73.6
91.4
93.1
87.9
81.1
89.8
(1,57) 2.18
Pi-GGPS
38
83.9
81.4
59.0
79.6
83.9
76.4
73.0
77.8
(2.31) 4.12
2 Ours
79.3
69.5
22.6
76.5
76.4
55.2
60.8
69.2
(1.83) 1.98
2 OOrs (GT)
90.7
93.7
77.4
91.4
94.8
86.7
82.5
91.2
(1.68) 1.89
rule base K.B, so that the comparison is reasonable. For comprehensiveness we follow previous works 31 38 to report the average used theorem rules in the parenthesis under Solution Length. However, we contend that this metric lacks significance in identifying shorter solutions, because it merges two steps of the same theorem rule into one.
Baselines. For symbolic- -ased methods, Inter-GPS 19 is the first and representative work that sequentially and continuously applies predefined theorem rules to deduction new geometric conditions. GeoDRL 23 presents typical collaboration of neural networks and symbolic systems, predicting the theorem be used each step by neural model trained with reinforcement learning. E-GPS introduces top-down decomposition mechanism for explainable solution exploration. Pi-GPS 38 adopts LLMs to refine the parsed formal language content for disambiguation and predict appropriate theorems. Additionally, several neural-based methods are selected for in-depth comparison, namely PGPSNet 35 LANS I8 and BeeGen-SSFT-77 22] AlphaGeometry 28 is not considered as our competitor because we handle different kinds of tasks.
we use the same theorem rule base provided by Peng et al. [23]. which includes 24 predefined rules that represent basic geometry theorems. The maximum constraints of deduction iterations and number of candidate solution paths w.r.f. each target are set to 7 and ( (1,2,)), respectively. The early stopping mechanism used terminate the iterative deduction process once all target nodes have been reached and the number of iterations exceeds 4. The machine is 24GB NVIDIA GeForce RTX 3090.
Implementation details. consistency with previous symbolic-based methods 31
4.2 Performance comparison
Quantitative analysis. Since our method is designed particularly for optimized solution exploration, we mainly focus on the solutions generated by the models, then the problem solving performances. The detailed experimental results are recorded in table We mainly make the following observations:
(1) Our method is able to obtain the best Solution Length score. i.e., the shortest average solution lengths. compared with all previous methods using the same theorem rule base. Specifically, our Causal-R further shortens the average length of feasible solution steps to 1.89 which is the only one lower than indicates that Causal-R not only generates more interpretable solutions with less redundancy, but also succeeds in finding shorter solutions at global-level. Although E-GPS also ensures the explainability solution, fails distinguish the shorter solution path when searching from the problem goal. (2) Causal-R achieves the best overall problem solving performances, with

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Table 2: The problem solving performances and solution length comparison with different can- didate solution constraint. À means the set con- straint number of candidate solutions w.r.t. each target. Best results are in bold.
Table 3: Comparison of solution quality from five aspects. Inter, Rea. and Solv. refer to interpretability, reasoning and solving process, respectively. Mul. means multiple solutions.
v means possessing the quality.
Solution Length
Methods
Inter.
Rea.
Solv.
Shorter Mul.
A
Accuracy
Step
Theorem
Inter-GPS T9
GcoDRL 23
91.2
1.89
1.68
Pi-GPS 38
90.7
1.97
1.71
E-GPS 31
3
91.0
1.92
1.71
Causal-R (ours)
1.4% gain on Accuracy. Our method is validated effective across all tested question types and geometric shapes, presenting promising applicability for future extension of symbolic systems. It even surpasses the accuracy of human experts, showing great potentials of symbolic-based methods for GPS. (3) For all symbolic-based methods, the performance is sensitive to the changes of theorem rule base KB and parsing results of geometric conditions. It not only affects the accuracy of solving geometry problems, but also leads to longer solutions if KBB is smaller. Since developing KCB is easy and a comprehensive KBB is promising for problem solving performance, the key challenge lies in the parsing performance and optimized solution exploration, which is the focus of this work. (4) Neural-based methods are trained with extra GPS data sets and present limited problem solving performances. Furthermore, they usually lack feasible mechanisms to ensure the interpretability of solutions, as well as to find shorter solutions.
To validate the effectiveness of the proposed Forward Matrix Deduction (FMD), we have implemented a basic strategy based on Python dictionaries for Causal Graph Reasoning (CGR). is noteworthy that CGR is introduced the theoretical foundation to achieve causal reasoning for solution exploration, and FMD is one of possible detailed strategies that we propose to implement it. Results and according analysis are recorded in appendix C We conduct further experiments on different number of candidate solution constraint w.r.t. each target node in table 2 Overall, Causal-R -R presents stable performances when Ả changes, remaining the best among previous methods in table [ Surprisingly, the highest accuracy and shortest solution length is achieved when only one solution is needed. We analyze that, on one hand, the symbolic system still exhibits certain randomness such symbolic assignment order in parsing stage and searching order in back tracing. On the other hand, there is possibility that one target node is the head node of another one. Therefore, acquiring multiple solution has a chance to change the shortest combination of solutions of the two.
Qualitative analysis. We further provide a comprehensive comparison between previous methods on the quality of generated solutions from five aspects, as shown in table Both Causal-R and E-GPS are able to ensure the interpretability of solutions, including the explainable reasoning and solving process. However, Causal-R is the only method that is able to generate multiple solutions as short as possible, benefiting from the mechanism of global-leved reasoning that acquires the first iteration when target node is achieved. Therefore, our Causal-B - possesses the key factors for application in practical scenarios that requires interpretable and accurate solutions. To discuss more profoundly, we have presented several typical cases in appendix D accordingly with case analysis.
5 Conclusion
In this work, we propose novel model Causal-R to obtain optimized solutions of problem goal from the symbolic-reasome perspective based on causal graph reasoning theory, which is implemented by the proposed forward matrix deduction method for faster reasoning iterations. On one hand, Causal- R compresses the search space of solutions from the beginning, the end and intermediate process, alleviating the difficulties for solution exploration. On the other hand, it is capable of obtaining shorter and varied solution paths, while ensuring the interpretability. It presents promising framework for scenarios where symbolic-reasoning is applicable to. We suggest that our method has potential for generality and applicability to specific problems whose reasoning processes can be approximated into a causal deduction format, beyond the scope of geometry problem solving. However, there are also limitations, which discussed carefully with feasible solutions in appendix E
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Acknowledgments and Disclosure of Funding
This work was supported by National Key Research and Development Program of China (2022FF33303600), National Natural Science Foundation of China (No.62137002, 62293550, 62293553, 62293554, 62437002, 62477036), "LENOVO-XITU" Intelligent Industry Joint Labora- tory Project, the Natural Science Basic Research Program of Shaanxi (2023-JC- (B-59C--Y-9))) the Youth Al Talents Fund of China Association Automation (Grant No. HBRC-JKYZD-2024-33 and the China Scholarship Council Program.
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16. Declaration of LLM usage
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A
Algorithm
Algorithm 1 Causal Graph Deduction
Input: G, N * 0, N' + 0. a 0,
/* Initialize
causal graphs, primitive nodes, target nodes maximum iteration constraint and current iteration index.
*/
Output: Nssignal
** Return the primitive nodes with their updated status and signal indicating whether the problem solvable.
signal + False
while o S a and signal is False do
signal + True
No N:
for G {N.
EN.ER. Eg. Ex in G do
for eg in Ex do
g(k; i) < ( rel(ma, nc)) // DGenerate the deduction control signal.
for ef in Eg do
no + g(k, i) ( my. me) // Apply the causal deduction.
N No // DUpdate the primitive node status. for n in N do
if n is negative then signal False // oThe problem target nodes are not fully achieved.
0
B Theorem rule simplification
We provide detailed illustration of how simplify the theorem rule into unified causal deduction representation forms in and table Note that we only list partial transformations w.r.t each theorem rule for clarity and brevity. For example, in Angle Sum Triangle in table we only list one deduction path, while the deduction path remains valid after the alteration of Ras ni and n Similar situations exist in Law of Sines in table too. These deduction paths are extracted from the equations between these values of nodes, where any one of them can be obtained if the other values known. More detailed implementation of causal graph construction can be found in codes.
21

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Table 4: The simplified causal deduction paths of original theorem rule base. For clarity and brevity, only partial deduction paths w.r.t. each rule are presented. A is used placeholder to represent the same content within each rule.
Geometry Theorem/Deffnition
Simplified Causal Deduction
For radius OA (n,), OB (nx) of circle O:
Circle Definition
nig f (m6)
Point A, B, C on circle O, for ZABC (m²). ZAOC (no):
Thales Theorem
Tùb + g(A) e+(() g( ((A) -- e (mo)
Point A, B, C on circle O, for ZABC (n6), ZAOC (n,):
Inscribed Angle Theorem
rig + (
AB // CD, Po on AB, Q on CD, for LAPQ (mp)) and ZDQP (n₂):
Parallel Lines Theorem
np s e, (ng)
For ZABC (nt), ZACB (n.), ZBAC (na) in AABC:
Angle Sum of Triangle
Ri s c, mi)
For AB (m61). AC (n.1), LABC (n12), ZACB (ne2) in AABC:
Isosceles Triangle Theorem (Side)
nbb1 - A) ef (Tee))
g(A) <- e (rel (162 n(2))
For AB (161), AC (n,)), LABC (112), ZACB (n22) in ABBC: nia g(A) (((22)
Isosceles Triangle Theorem (Angle)
g (44) - c/ (rel (1b1 nel))
For AB (na1), AC (161), BC (no1) LABC (n62), LACB (ne2), ZBAC (n₂2) in AABC:
Equilateral Triangle Theorem
Th22 + g(A) (.))), ni2 < g(A) e-_i(-),
(+),na, nia + g(A) (4+((
g( ( e [rel [a1: 161), rel nel))
I AB, M BC, N AC, CQ.AM.N intersects P. for AQ (n.)), BQ (n.2), BM (na1). CM (naz), AN (161). CN (122), CP (f11), PQ (ngz), AP (Rm1), PM (nm2). BP (nn1), PN (N,2)): n2 - g(A) (f 7m2 + g(A) ) E (RmL),
Triangle's Center of Gravity
nn2 g(A) (
(n
g(A (A) + (rel (1 1222), (11b1 T222 rel (22))
For AB (n.), AC (1t), ZBAC (no1). 1. BC (na2) in AABC and DE (nf), DF (ne), LEDF (na1), EF (n.z) in ADEF (SAS):
Congruent Triangle Theorem (Proving)
na t (( of (Ra2)
g( ( + oy (rel(n.: n2 rell (n6, me rel (Ma1 na1)
For AB (n.1). ) AC (1o1), BC (n.1). ). LACB (n LABC (nuz). BBC (na2) ) in AABC and DE (1 DF 1, EF (na1), DFF (n (2). ZDEF (no2). LEDF (n.2) in ADEF:
Congruent Triangle Theorem
n²z + g(A) (1i2). n222 < g( B(A) e (ni2). na2 g(A) (1+2(
( + (rel 1f11, rel(1111, ne1), rel(na.. 7d11)
AB and circle O tangent at point B, AM intersects circle N, for AB (n²), AN (n,,) and AM (n,)):
Tangent Secant Theorem
ni + (nm, nī,))
A,B,C,D on circle O, AB and CD intersects at point M, for AM (na), BM (πo), CM (n.), DM (na): Ni x- ef (f(,,, Ney rīa)
Chord Theorem
In AABC, M on BC, for AB (141), BM (162), AC (ne1), CM (ne2), ZBAM (n.,),, 2CAM (na2): nb1 < g( A) -ee nea ney)
Angle Bisector Theorem
g( A (rel( na2))
22

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Table 5: The rest simplified causal deduction paths of original theorem rule base, following table + For clarity and brevity, only partial deduction paths w.r.t. each rule are presented. A used as placeholder to represent the same content within each rule.
Geometry Theorem/Definition
Simplified Causal Deduction
For AB (n.), AC (mb), BC (na) in AABC, ZACB-90";
Pythagoras Theorem
Re < e, (Na, 166)
For AB (na),, ZACB (n.2), AC (m61), ZABC (162) in AABC: Ne1 e, (nic.. 161, n62)
Law of Sines
For AB (Ne1). AC (mp). BC (na): ZACB (7e2) AABC:
Law of Cosines
ne1 e (Tag fb. R222)
For LABC (m²), ZACB (n²)) ZBAC (ma) in AABC and ZDEF
Similar Triangle Theorem (Proving)
(ne), ZDFE (n ZEDF (na) in ADEF:
f1 f ((A) ef((aa)
g(A) < cf (rel (nb. ne) (elennn)))
For ZABC (na1), LACB (n.1), ZBAC (na). AB (n.2), AC (122) in AABC and ZDEF (n.1), <DFE (Rz1), ZEDF (n.)), DE (m12),
DF (n.2) in DEFF
Similar Triangle Theorem
nia2 - g )
ef (162)
g (A)
of (rel na1), rell [Ac1; n,))) rel na), (n n₂2))
For AB (na), BC (n²). EF (nz), FG (n,) in similar polygons
ABCD and EFGH:
Similar Polygon Theorem
n < g g(A) + e (rel nf))
(
Point M on AB and point on AC, for AM (1mm)), ), BM (1m2),
AN (nma), CN (Mm2). MN (na1). BC (naz) in AABC: Tho f g(A) ef (T(22)
Median Line Theorem
ef (Tel((mm1 72m2 ), (el Rnn))
In ABBC, point M on BC, AMLBC, for BC (na), AM (Rm).
Area Equation Theorem
Area Arca_AABC ABCC (n,):
nīg f ef nin)
For LABC (ng), LBCD (nc). ZCDA (na). ZDAB (na) in
Angle Sum of Polygon
Quadrilateral ABCD:
nug + ( ne, ria)
Table 6: Comparison of time, grouped by number of all deduction paths.
< 100
100
1000 1000
2000
2000
5000
5000
10000
10000
Basic 1.57 x 10-3
1.26
10-2 2
9.62
10-3
1.31 x 10-2
2.60 10-2
1.86
10-2
FMD 1.19 10-3 1.70 10-3 2.35
10-3
6.46 x 10-3
2.87
10-2
7.76
10-2
C Ablation studies of FMD
To provide fair and detailed comparison, we record the average reasoning time per iteration of these methods, grouped by the number of deduction paths (i.e., sum of all deduction paths in all causal graphs) in table and the number of unique primitive nodes in table [ The other parts and settings remain the same. From the results, we mainly have the following observations: (1) FMD consistently performs faster than using Basic strategy when the number of deduction paths is smaller than 5,000 in table and when the number of primitive nodes is smaller than in table Within this range, the time of FMD generally exhibits steady increasing trend with the increasing of deduction paths and primitive nodes. (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
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Table 7: Comparison of time, grouped by number of unique primitive nodes.
S 20
20 - 30
30 - 50
50 - 70
> 70
Basic 1.23 10-3 3
4.55 10-3
1.02 10-2
1.93 10 2
4.41 x 10-2
FMD 1.19 x 10-1 1.88 x 10-3 4.11 x 10 1.65 x 10-2 6.86 x 10
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. While for FMD, these paths are still recorded in the matrices and involved in matrix manipulation, resulting in more resource demands. In practice. we simply adopt sequential theorem application of a smaller theorem base that corresponds to feasible causal graphs each iteration on this problem or the original theorem base, when resources run short. 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.
Moreover, we suggest that FMD can be further optimized by matrix manipulation refinement tech- niques (we did not adopt any optimization techniques to matrix manipulations in current version). FMD presents great potential for future development in complex scenarios (i.e., scenarios with higher degree of node involvement), and provides certain insights for similar scenarios in other fields. (Note: A higher degree of node involvement refers to more head nodes and prerequisites required within one causal deduction path. FMD leverages the matrix manipulation and consolidates the sequential condition check into one single operation. This means, regardless of the number of head nodes and prerequisites in a causal deduction path, FMD is able to produce the deducted results at a similar speed. While for the Basic strategy, the needed time increases with the increasing of node involvement degree.)
D Case analysis
In order to provide more detailed qualitative analysis, conduct some cases from Causal-R and present them in fig. [ In case (a), the model is able to generate two solutions, where the order of theorem application consistent but the detailed involved primitives are different. Both solutions are reasonable and feasible and lead to correct answer. In case (b), two different solutions with inverted order of theorem applications of the problem seems complex. On one hand, it shows the superiority of Causal-R to explore shorter solutions. On the other hand, it also benefits from the design of symbolic- strategy which can precisely match the needed conditions within a large set of complicated and redundant geometric conditions. Case (c) is typical case that our FMD fails to perform matrix manipulation due to resource constraint. The number of deduction paths becomes extremely large when these triangles form different combinations causal graphs, especially for theorems such as Congruent Triangle Theorem. Therefore, the model only performs sequential application of theorems in KBB and obtains the final answer. Case (d) is a failure case that our model fails to obtain the solutions and answer, which mainly attributes to the incorrect parsing of the geometric problem and incomplete design of symbolic system. It remains a difficult task to parse the content with irregular representations such as shaded area. These two cases both indicate the necessity of further developing the symbolic system for adapting to more question types and more theorems.
obtained. The solutions are simple even if the geometric conditions
E
Limitations
While certain advantages and performance improvements have been obtained by our Causal-R, we also identify some limitations, along with potential solutions, from the following aspects: (1) The effectiveness of causal graph reasoning is influenced by the design of theorem rule base KBB It not only limits the upper bound of problem solving performances (i.e., the geometry problem can not be solved if the needed theorem rule is not contained in KBB), but also affects the comprehensive outcome of transformation from theorem rules to causal graphs. This can resolved by future development and refinement of K.B. providing more standard and larger theorem rule base. Note that once the KB is determined, the deduction logic for causal graph construction is also determined (such presented in table 4 and table 5) and does not change for different geometry problems. (2)
24

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Find CD- BE 16, and CD 2+5.
ARSV to ATVS, Find
Inroguo hessgan WWXY2 2. The skle cartimebera leng. Fixl W.Y.
Fasl the anee Iwe snded
Diagram:
Diagram
Diagram
Diagram
1- 245
240
Solutions:
Solutioes:
Solutions
Solutions None
Theorem
Stup1 Congruent Triangle
Stapa:
"BCHDE :. T.ZABC- AED
RLA u .TV C.,TBV> FV3-78
3, 10.
Stund Bimilar Triangle Theorem
TVS- RBN-00
14,
/ADG-/AED /GALL/DAE
Step2 Angla Sum Triangle
-ABBC-CADD -BBIVE-BE-1CWWC-000 CD-400
7B6-778 <TVS-00 10
Stapt Parallel Lines Theorem
Stop1 Angle Sum Triangle : ReN-ae', LPV8-77" :. 180 -70111111107
:eCVDD :-AACB= LADE Stup3 Simiar Trinngle
DAE
Step2 Congruent Triangle Theorem RBV sa a IVS
B.AGI--- -SAID
(AB COV CD-100
(c)
Figure 5: Four typical cases from Causal-R. (a), (b) successful cases and (c), (d) are failure cases.
For more intricate geometry problems, when the combinations of geometric primitives and theorem rules become extremely large, the temporary storage requirement for matrix-based deduction also increases. can be optimized through both refinement of KBB and application of more efficient matrix operation methodologies. (3) Currently, the method does not support theorem rules that involve constructing geometric primitives (e.g., connect point and point B as new line primitive AB). One possible solution is to apply such rules before the causal graph construction to augment the base geometric primitive nodes for a wider solution space that includes the action of constructing new primitives.
25


Reviews:
[{"review_id": "5caDJr0uKk", "reviewer": "Reviewer_8Ps7", "summary": "This 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.", "questions": "1. 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?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 2, "significance": 3, "originality": 4, "strengths_and_weaknesses": "#### 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.", "was_revised": true, "final_justification": "The authors have adequately addressed the questions I raised, so I will keep my positive score unchanged.", "rebuttal": "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.\n\nQ1: Suggestions on some professional term and position of overall framework.\n\nYes, 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.\n\nQ2: Explanation of the red content in figure 1 (c).\n\nIn 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.\n\nQ3: Some mistakes in the equations and reference citation format.\n\nThank 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. \nWe 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.\n\nQ4: The runtime performance compared with baselines and the ablation studies of the proposed Forward Matrix Deduction (FMD).\n\nWe 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.\n\nFor 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.\n\nGrouped by number of all deduction paths (/s):\n\n| | [0, 100]  | (100, 1000]  | (1000, 2000]  | (2000, 5000]  | (5000, 10000] | (10000, )  |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| **Basic** | 1.57e-3 | 1.26e-2 | 9.62e-3 | 1.31e-2 | 2.60e-2 | 1.86e-2 |\n| **FMD** | 1.19e-3 | 1.70e-3 | 2.35e-3 | 6.46e-3 | 2.87e-2 | 7.76e-2 |\n\nGrouped by number of unique primitive nodes (/s):\n\n| | ≤20 | 20~30 | 30~50 | 50~70 | ≥70 |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| **Basic** | 1.23e-3 | 4.55e-3 | 1.02e-2 | 1.93e-2 | 4.41e-2 |\n| **FMD** | 1.19e-3 | 1.88e-3 | 4.11e-3 | 1.65e-2 | 6.86e-2 |\n\nFrom the results, we mainly have the following observations: \n\n(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.\n\n(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.\nThese experiments and according analysis will be added in final version.\n\nQ5: 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\nNo. 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.\n\nFinally, we again appreciate your support for our work and it gives us much encouragement."}, {"review_id": "9oiHr18VSv", "reviewer": "Reviewer_kUXM", "summary": "The 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.", "questions": "See 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?", "limitations": "While 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.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "After considering the author rebuttal, I have updated my assessment. The authors responded constructively to most of the concerns raised. In particular:\n- The overclaim of generality has been acknowledged and will be revised.\n- The authors conducted an ablation study to compare FMD with a basic implementation.\n- The authors have committed to including a complete worked example and improved formatting, which will enhance interpretability.\n\nHowever, some issues remain unresolved:\n- There is still no empirical robustness analysis, and the evaluation remains limited to a single dataset (Geometry3K), with no experiments on other benchmarks or under noisy conditions.\n- The discussion of method-level failure modes (e.g., ambiguous or conflicting deduction paths) is still limited.\n\nOverall, I value their thoughtful engagement and planned improvements, and I have reflected that in my updated score. Nonetheless, the remaining issues limit the paper’s general significance at this stage, and I encourage the authors to fully incorporate the promised enhancements in the final version.", "rebuttal": "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?\n\nWe 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:\n\n— 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.\n\n— 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.\n\n— 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 .\n\n— 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.\n\nWe 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. \n\nAs 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).\n\nAs 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.\n\nQ2: Ablation of Causal Graph Reasoning (CGR) and Forward Matrix Deduction (FMD).\n\nThank 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. \n\nFirst, 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.\n\nGrouped by number of all deduction paths (/s):\n\n| | [0, 100]  | (100, 1000]  | (1000, 2000]  | (2000, 5000]  | (5000, 10000] | (10000, )  |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| **Basic** | 1.57e-3 | 1.26e-2 | 9.62e-3 | 1.31e-2 | 2.60e-2 | 1.86e-2 |\n| **FMD** | 1.19e-3 | 1.70e-3 | 2.35e-3 | 6.46e-3 | 2.87e-2 | 7.76e-2 |\n\nGrouped by number of unique primitive nodes (/s):\n\n| | ≤20 | 20~30 | 30~50 | 50~70 | ≥70 |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| **Basic** | 1.23e-3 | 4.55e-3 | 1.02e-2 | 1.93e-2 | 4.41e-2 |\n| **FMD** | 1.19e-3 | 1.88e-3 | 4.11e-3 | 1.65e-2 | 6.86e-2 |\n\nFrom the results, we mainly have the following observations: \n\n(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.\n\n(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. \n\nThese experiments and according analysis will be added in final version.\n\nQ3: 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.\nThanks 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.\n\nQ4: More case studies with deeper analysis is needed for more comprehensive discussion of limitations of the method itself.\n\nThanks 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.\n\nQ5: A complete worked example and a concrete walk-through this example could be included for better illustration.\n\nThank 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)\n\nQ6: The citation format in the paper looks strange and hard to follow.\n\nWe 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.\n\nFinally, 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."}, {"review_id": "tGTVyoLLor", "reviewer": "Reviewer_VP5W", "summary": "This 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.", "questions": "1. 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?", "limitations": "See weaknesses and questions.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "Authors address my main concerns during rebuttal and discussions, including:\n\n1. The ablation of CGR and FMD.\n\n2. The carification of parameters.\n\n3. The scalbility of the KB size.", "rebuttal": "Q1: Ablation of Causal Graph Reasoning (CGR) and Forward Matrix Deduction (FMD).\n\nThank 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. \n\nFirst, 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.\n\nGrouped by number of all deduction paths (/s):\n\n| | [0, 100]  | (100, 1000]  | (1000, 2000]  | (2000, 5000]  | (5000, 10000] | (10000, )  |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| **Basic** | 1.57e-3 | 1.26e-2 | 9.62e-3 | 1.31e-2 | 2.60e-2 | 1.86e-2 |\n| **FMD** | 1.19e-3 | 1.70e-3 | 2.35e-3 | 6.46e-3 | 2.87e-2 | 7.76e-2 |\n\nGrouped by number of unique primitive nodes (/s):\n\n| | ≤20 | 20~30 | 30~50 | 50~70 | ≥70 |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| **Basic** | 1.23e-3 | 4.55e-3 | 1.02e-2 | 1.93e-2 | 4.41e-2 |\n| **FMD** | 1.19e-3 | 1.88e-3 | 4.11e-3 | 1.65e-2 | 6.86e-2 |\n\nFrom the results, we mainly have the following observations: \n\n(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.\n\n(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. \n\nThese experiments and according analysis will be added in final version.\n\nQ2: The sensitivity analysis of hyper-parameters alpha and lambda and how these choices affect outcomes?\n\nIn 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.\n\nFor 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. \n\nFor 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.\n\nQ3: How initializations of the status vector in matrix deduction influence convergence speed (e.g., all negative, partial known node initialization) ?\n\nSorry 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.\n\nQ4: 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. \n\nThank 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:\n\n— 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.\n\n— 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.\n\nQ5: 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.\n\nWe 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.\n\nQ6: How does the current Reasoning LLM / VLM, such as o3, o4, and r1, perform on your task?\n\nThe 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.) \n\nQ7: How does your method perform on mathvista benchmark?\n\nMathVista 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.\n\nFinally, 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."}, {"review_id": "EwH5s4MbbG", "reviewer": "Reviewer_SAZz", "summary": "The 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.", "questions": "* 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.", "limitations": "Yes.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "After reading the rebuttal and the subsequent clarifications in the discussion, I believe there are a few concerns that remain unaddressed. \n\nWhile the authors provide their definition of causality (in the rebuttal phase), the framework still lacks a formal line to mainstream causal inference methods (e.g., d-separation, do-calculus, counterfactuals). This, in general, limits the generalizability and theoretical grounding of their \"causal\" terminology (and may be misleading for the causal literature community). It’s not a critical flaw for the current scope, but should be addressed in follow-up work or clarified more explicitly in the final version. Also, the lack of automated or scalable evaluation for solution diversity and optimality remains a minor gap. The authors acknowledged this in the rebuttal and plan to refine their claims. Given the exploratory nature of this claim, this concern is addressed. \n\nOverall, the primary concern can be addressed with more rigorous treatment of causality (terminologically and theoretically), and with minor revisions and more careful phrasing of certain claims, this work has the potential to be an important contribution to symbolic reasoning in structured domains.", "rebuttal": "Q1:  Is there causal graph reasoning formalization or step-by-step technical description of how this graph is constructed from parsed geometric inputs.\n\nThanks 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. \n\nFor 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. \n\nNext, 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).\n\nThen, 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).\n\nFinally, the target nodes are specified according to the existing primitive nodes and problem goal.\n\nWe hope the above information addresses your questions.\n\nQ2: Ablation of Forward Matrix Deduction (FMD) is needed.\n\nThank 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. \n\nFirst, 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.\n\nGrouped by number of all deduction paths (/s):\n\n| | [0, 100]  | (100, 1000]  | (1000, 2000]  | (2000, 5000]  | (5000, 10000] | (10000, )  |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| **Basic** | 1.57e-3 | 1.26e-2 | 9.62e-3 | 1.31e-2 | 2.60e-2 | 1.86e-2 |\n| **FMD** | 1.19e-3 | 1.70e-3 | 2.35e-3 | 6.46e-3 | 2.87e-2 | 7.76e-2 |\n\nGrouped by number of unique primitive nodes (/s):\n\n| | ≤20 | 20~30 | 30~50 | 50~70 | ≥70 |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| **Basic** | 1.23e-3 | 4.55e-3 | 1.02e-2 | 1.93e-2 | 4.41e-2 |\n| **FMD** | 1.19e-3 | 1.88e-3 | 4.11e-3 | 1.65e-2 | 6.86e-2 |\n\nFrom the results, we mainly have the following observations: \n\n(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.\n\n(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. \n\nThese experiments and according analysis will be added in final version.\n\nQ3: 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.\n\nThank 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.\n\nFor more comprehensive evidence, we have sampled 30 geometry problems to evaluate the results obtained by our method:\n\n— 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.\n\n— 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.\n\n— We manually checked that 5 out of 30 problems have 2 solutions obtained by our model, and the solutions are reasonable.\n\nAs 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.\n\nQ4: 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.\nThanks 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.\n\nQ5: 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.\n\nThank 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:\n\n1. 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).\n\n2. 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.\n\n3. The task goal is related to status of certain objects.\n\nWe 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.\n\nQ6: 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.\n\nWe 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."}, {"review_id": "oKDhPJ2qWS", "reviewer": "Reviewer_57ZD", "summary": "This 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.", "questions": "See weaknesses.", "limitations": "See weaknesses.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "I would like to maintain my score and a low confidence score as I am not an expert in the field of neural theorem proving.", "rebuttal": "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.\n\nQ1: 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)?\n\nFirst, 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. \n\nIn 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.\n\nQ2: 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.\n\nThank 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.  \n\n1. 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.\n\n2. 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.\n\n3. 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.\n\n4. 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.\n\nOur differences include but not limited to the above points. It is possible that we have some misunderstandings, and we welcome any corrections.\n\nQ3: 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\nOur work mainly distinguishes from theirs from but not limited to the following aspects:\n\n1. 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.\n\n2. 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.\n\n3. 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.\n\n4. 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.\n\nQ4: 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).\n\nThanks 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.\n\nFinally, we sincerely ask if you could kindly consider increasing the score if most of the concerns have been resolved. Thank you."}]

Author rebuttal:

Meta-review: **Claims and findings of the paper**

This paper introduces Causal-R, a novel symbolic reasoning framework for solving geometry problems. The core contributions are the "Causal Graph Reasoning" (CGR) theory, which models theorem-based deductions as causal paths, and an efficient implementation called "Forward Matrix Deduction" (FMD) that uses matrix operations to explore the solution space. The authors claim this approach effectively compresses the search space, enabling the generation of multiple, shorter, and more interpretable solutions. On the standard Geometry3K benchmark, the proposed method is shown to outperform existing symbolic and neural baselines in both accuracy and solution length.

**Strengths of the paper**

*   The paper's central idea of reformulating geometric reasoning as a causal deduction process on a graph is novel and well-motivated. This approach provides a new perspective on symbolic problem-solving that moves beyond simple forward/backward chaining. The reviewers unanimously praised the originality of the framework (8Ps7, kUXM, SAZz, 57ZD).
*  The method achieves state-of-the-art results on the Geometry3K dataset, demonstrating higher accuracy and producing significantly shorter solutions than prior methods and even human experts. This provides strong evidence for the effectiveness of the proposed approach (8Ps7, kUXM, VP5W).
*   By design, the solutions generated by Causal-R are interpretable, as each step corresponds to a specific geometric theorem. The ability to generate multiple, shorter solutions is a significant advantage, particularly for potential applications in education (8Ps7).
*  The paper is generally well-written and clearly presents its motivation, methodology, and results, making it accessible even to readers not deeply familiar with the subfield (8Ps7, SAZz, 57ZD).

**Weaknesses of the paper**

*  A primary concern raised by reviewers (kUXM, VP5W) is that the evaluation is confined to a single dataset (Geometry3K). The paper lacks experiments on other benchmarks (e.g., FormalGeo7k) or a robustness analysis, which limits claims of generalizability.
*   While FMD is presented as a key contribution for efficiency, the ablation study provided during the rebuttal showed that it can be slower than a basic implementation on problems with a very large number of deduction paths. The authors' justification focused on methodological potential (e.g., GPU acceleration, handling complex prerequisites) rather than demonstrating a consistent, practical speedup in the current experiments (8Ps7).
*   The use of the term "causal" was questioned (SAZz) as it does not align with the formalisms of mainstream causal inference literature (e.g., Pearlian frameworks). While the authors clarified their definition is specific to deductive necessity within theorems, this could be a source of confusion.
*   The initial submission lacked crucial ablation studies and contained some overstated claims regarding generality and solution "minimality." These were largely addressed during the rebuttal but were significant initial weaknesses.

**Key reasons to accept**

The paper presents a solid, novel, and impactful contribution to the field of automated mathematical reasoning. The proposed Causal-R framework is elegant and demonstrates impressive performance on a standard benchmark. The authors' diligent and effective engagement during the rebuttal period has addressed the most critical weaknesses of the initial submission and gives confidence that the final version will be of high quality. Despite the limited scope of the evaluation, the core ideas are strong and likely to inspire future work in symbolic reasoning. Therefore, the paper is a clear accept.
Decision: Accept (poster)