AutoGPS: Automated Geometry Problem Solving via Multimodal Formalization and Deductive Reasoning
Authors: Bowen Ping
Minnan Luo
Zhuohang Dang
Chenxi Wang
Chengyou Jia
Venue: ICLR 2026 Poster (2026)
Keywords: Neural-symbolic
geometry problem solving
interpretable and reliable reasoning
Primary area: interpretability and explainable AI

Abstract: Geometry problem solving presents distinctive challenges in artificial intelligence,
requiring exceptional multimodal comprehension and rigorous mathematical reasoning capabilities.
Existing approaches typically fall into two categories: neural-based and symbolic-based methods,
both of which exhibit limitations in reliability and interpretability. To address this challenge, we propose AutoGPS, a neuro-symbolic collaborative framework that solves geometry problems with concise, reliable, and human-interpretable reasoning processes.
Specifically, AutoGPS employs a Multimodal Problem Formalizer (MPF) and a Deductive Symbolic Reasoner (DSR).
The MPF utilizes neural cross-modal comprehension to translate geometry problems into structured formal language representations,
with feedback from DSR collaboratively.
The DSR takes the formalization as input and formulates geometry problem solving as a hypergraph expansion task,
executing mathematically rigorous and reliable derivation to produce minimal and human-readable stepwise solutions.
Extensive experimental evaluations demonstrate that AutoGPS achieves state-of-the-art performance on benchmark datasets.
Furthermore, human stepwise-reasoning evaluation confirms AutoGPS's impressive reliability and interpretability,
with 99\% stepwise logical coherence.

Published as a conference paper at ICLR 2026
AUTOGPS: AUTOMATED GEOMETRY PROBLEM SOLV- ING VIA MULTIMODAL FORMALIZATION AND DEDUC- TIVE REASONING
Bowen Ping, Minnan Luo', Zhuohang Dang, Chenxi Wang, Chengyou Jia Xi' an Jiaotong University
31 512227706sttu. xjtu edu. en, mimluu@xjttu.ddu.com
ABSTRACT
Geometry problem solving presents distinctive challenges in artificial intelligence, requiring exceptional multimodal comprehension and rigorous mathematical rea- soning capabilities. Existing approaches typically fall into two categories: neural- based and symbolic-based methods, both which exhibit limitations reliability and interpretability. To address this challenge, propose AutoGPS, a neuro- symbolic collaborative framework that solves geometry problems with concise, reliable, and numan-interprettin reasoning processes. Specifically, AutoGPS employs a Multimodal Problem Formalizer (MPF) and a Deductive Symbolic Reasoner (DSR). The MPF utilizes neural cross-modal comprehension to translate geometry problems into structured formal language representations, with feedback from DSR collaboratively. The DSR takes the formalization as input and formulates geometry problem solving as a hypergraph expansion task, executing mathemat- ically rigorous and reliable derivation to produce minimal and readabll stepwise solutions. Extensive experimental evaluations demonstrate that AutoGPS achieves state-of-the-art performance on benchmark datasets. Furthermore, hu- man stepwise-reasoning evaluation confirms AutoGPS's impressive reliability and interpretability, with 99% stepwise logical coherence.
INTRODUCTION
Mathematical reasoning constitutes fundamental component of human intelligence. Among various mathematical domains, geometry problem solving (GPS) has attracted significant attention due to its intrinsic elegance and deductive completeness (Tarski, 1998; Trinh al., 2024). Distinet from other mathematical disciplines such as algebra (Hendrycks et al., 2021; Cobbe et al., 2021; Yu et al., 2024), geometric problem solving inherently involves the analysis and comprehension of multimodal information, including visual diagrams and textual descriptions (Chen et al., 2022b; Sco et al., 2015; Lu et al., 2021; Wu et al., 2024; Peng et al., 2023; Huang al., 2024). This process necessitates not only the understanding of multimodal problem formulation but also rigorous mathematical deduction to derive solutions (Lu al., 2023; Gao al., 2025; Huang et al., 2025b; Zhang al., 2024a; Chen et al., 2022b; Liu al., 2024a).
Recently, many methods have been proposed for this challenging task, which are primarily fall into two categories: (1) Neural-bused methods, including specialized neural geometry solvers (Zhang ct al., 2023; Li et al., 2024c) and multimodal large language models (MLLMs) (Gao et al., 2025; OpenAI, 2024; OpenAl et al., 2024; Qwen et al., 2025; Huang al., 2025b; Chen et al., 2024c;b;
Gao et al., 2024; Wang et al., 2024b; Chen et al., 2024a; Zhang et al., 2025; Sun et al., 2023; 2024), demonstrate superior multimodal comprehension capabilities. However, shown in Figure 1 (left), such methods are unreliable and prone to generating plausible but logically flawed reasoning steps due to hallucination (Huang al., 2025a), leading erroneous conclusions.
(2) Symbolic
approaches (Lu et al., 2021; Peng et al., 2023; Wu et al., 2024) operate on formal language inputs through predefined rules and algebraic computations with basic mathematical rigor. However, as
"Corresponding Author.

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Published as a conference paper at ICLR 2026
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Figure I: Failure cases of current methods. Left: Awen2.5-VL-322 --3V--3---ntttuucttttt exhibits hallucination induced errors during reasoning, producing an incorrect conclusion. The symbolic method (Inter-GPS) also fails and lacks traceable steps for error diagnosis. Right: Inadequate cross-modal understanding leads to incomplete formalization, further obstructing symbolic solving. Blue/red annotations indicate correct/erroneous reasoning answers.
exemplified in Figure 1 (right), they struggle to completely formalize given multimodal problem input, leading to unreliable problem-solving results. Furthermore, neural methods inherently lack interpretability (Zhang et al., 2021; Li et al., 2024c), and existing symbolic methods fail to provide explicit human-interpretable reasoning steps (Lu et al., 2021; Peng et al., 2023; Wu et al., 2024) as shown in Figure (left). This naturally leads us pose significant research challenge: Can develop an automated geometry problem solving methodology that demonstrates reliable and interpretable reasoning capabilities?
To address this challenge, we propose AutoGPS, a neuro-symbolic collaborative framework that solves geometry problems with reasoning processes.
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concise, reliable, and human-interpretaul Specifically, AutoGPS comprises two principal components: the Multimodal Problem Formalizer (MPF) and the Deductive Sym- bolic Reasoner (DSR). The MPF formalization agent that processes image-text pairs of geometric problems. By enhancing multimodal large language models with domain-specific parsing tools, it executes rreffommalization and multimodal alignment op- erations to transform geometric problems into formal language representations. The DSR is symbolic computation engine based on predefined rules, accepting formal problem descriptions as in- put. It validates the formalization and provides feedback to the MPF for refinement until the formalization is error-free. Subse- quently, the DSR formulates the problem-solving process as hypergraph expansion task, where the formal language literals are treated as nodes and derivation steps are treated as hyperedges. iteratively applies two complemen- tary strategies to expand the hypergraph until the solution node is reached. Upon solution attainment, identifies the minimal sub-hypergraph for solution derivation and generates a syllogistie-struccured reasoning process. Therefore, the proposed AutoGPS effectively solves the geometry problem with rigorous, concise, and human-rcadable derivation, maintaining both reliability and interpretability.
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Figure 2: Performance compari- son among existing methods.
Experimental results two GPS benchmarks (Lu et al., 2021; Zhang et al., 2023) demonstrate that our AutoGPS outperforms the existing methods by margins of 4.1% 1% and 9.2% and the state- of-the art mathematical reasoning MILMs by 18.0% and 26.4%. Human stepwise reasoning eval- uations validate that AutoGPS achieves 99% stepwise accuracy, surpassing the 71% accuracy of the perf-merg MILMM (Zhu al., 2025), demonstrating absolute superiority in reliability and interpretability over existing methods.
RELATED WORK
2.1
MATHEMATICAL REASONING WITH MLLMS
Recent advancements in MLLMs have demonstrated significant potential for mathematical reasoning in visual contexts, evidenced by multiple benchmark studies. The research community has
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developed specialized datasets to systematically assess these capabilities: Math Vista (Lu et al., 2023) serves as the pioneering benchmark for evaluating visual mathematical reasoning, while MATH- Vision (Wang et al., 2024a) provides comprehensive assessments across core mathematical domains, including algebra, statistics, and geometry. Math Verse (Zhang et al., 2024b) contributes 2,612 human- annotated visual mathematical problems spanning planar geometry, solid geometry, and functional analysis. Complementing these efforts, GeoEval (Zhang et al., 2024a) specifically targets geometric reasoning evaluation through plane and solid geometry problems. Building upon these foundational works, our research involves a comprehensive evaluation of MLLMs' performance in plane geometry contexts. We utilize two larger-scale datasets: Geometry3K (Lu et al., 2021) and PGPS9K (Zhang et al., 2023), with particular emphasis on both formalization and solution-generation.
2.2 GEOMETRY PROBLEM SOLVING
Geometry problem solving remains a critical research focus (Lu et al., 2021; Peng et al., 2023; Wu et al., 2024; Li et al., 2024c; Gao al., 2025; Huang et al., 2025b; Seo et al., 2015; Hao et al., 2022; Zhang et al., 2022; 2023; Sun et al., 2025), with existing approaches falling into three main categories: neural-based, symbolic-based, and neural-symbolic methods. Neural methods leverage trained networks (PGPSNet (Zhang et al., 2023) and LANS (Li et al., 2024c))
fine-tuned multimodal
LLMs (LLaVA (Liu et al., 2023; 2024a;b), G-LLaVA (Gao et al., 2025), Vision-RI (Huang et al., 2025b)) for geometric reasoning. while symbolic systems (Inter-GPS (Lu et al., 2021), E-GPS (Wu et al., 2024)) employ formal language (Xu et al., 2024b;a) and deductive rules. Neural-symbolic approaches such as GeoDRL (Peng et al., 2023) and FGeo-HyperGNet (Zhang al., 2024c) integrate neural networks with symbolic solvers for heuristic search, reducing the search space to enhance efficiency. These methods, however, exhibit limitations in reliability and interpretability. Neural methods suffer from hallucination issues, leading to incorrect reasoning and answers, while symbolic methods produce algebraic solutions lacking procedural transparency. Our framework first translates multimodal problems into formal language representations, then executes symbolic deduction to generate human-readable reasoning steps, ensuring reliability and interpretability.
2.3 NEURAL-SYMBOLL METHODS FOR MATHEMATICS
The limitations of purely neural (unreliable) and purely symbolic (non-scalable) methods have driven the development of neuro-symbolic frameworks for mathematical reasoning (Trinh et al., 2024; Li et al., 2025; 2024b; Wu et al., 2025; Shang et al., 2025; Singh et al., 2025; Wu & Yu, 2025). Specifically, these approaches are utilized for complex problem-solving by leveraging neural methods for heuristic guidance in tasks like geometry proof and inequality deduction (Trinh et al., 2024; Li et al., 2025). They are also employed for reliable dataset generation, ensuring the logical rigor of mathematical problem-proof pairs via symbolic verification (Li et al., 2024b; Wu et al., 2025). Another key direction involves agent-style methods, which empower LLMs with planning and self-correction to optimally interact with external symbolic tools for enhanced precision and robustness (Yang et al., 2023; Shang et al., 2025; Singh et al., 2025; Wu & Yu, 2025). These methods influence our approach. The proposed AutoGPS utilizes an agent-style mechanism in its MPF coupled with a feedback and refinement loop involving the DSR. This collaboration ensures precise formalization and guarantees that the DSR's 's output is highly reliable and interpretable, providing minimal, syllogistic-sttrueuued human-readable reasoning steps.
METHODOLOGY
3.1 PROBLEM FORMULATION
Following (Lu et al., 2021), given a predefined theorem set T, we aim to solve geometry problems formalized as D= (DD,, N.). Here, (D,, is the i-th multimodal geometry problem descrip- tion, where D; represents the original geometric diagram and T; denotes the corresponding textual description, specifying the problem objective (e.g., "Find length of line AB"). In the following, we omit the subscript to represent T.) as (D,T) for brevity. To ensure the reliability and inter- pretability, given (D, T). our AutoGPS aims to generate the correct solution a" with corresponding stepwise reasoning process S {81, 82,.... ... ). where 8; is the i-th reasoning step.
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3.2 MODEL OVERVIEW
The proposed AutoGPS framework, as illustrated in Figure 3, comprises two core components: the Multimodal Problem Formalizer (MPF) and the Deductive Symbolic Reasoner (DSR). Initially, given formal language space defined in Appendix the MPF formalizes input geometrie problems (D.T) into structured formal language representations L = 22.22..... .......... 1k. Find C L. Here each 1; is formal language literal (e.g.. Line(A, (A, B)). Find(t) a statement indicating the problem-solving goal t (e.g., BerughhOf(Lingg and provides feedback to the MPF for refinement iteratively until the formalization is error-free. Then, the DSR formulates the mooblem-solving process hypergraph expansion task, where the formal language literals are treated as nodes and derivation steps are treated as hyperedges. Subsequently, DSR iteratively expands the hypergraph through complementary strategies, Deductive Reasoming and Algebraic Reasoning, until the solution literal a" is reached. Here a" is of form Equals(t, v), B)), 2)).
B))). Given formalization L, the DSR validates the self-consistency of L
where é is the target literal and the solution value (e.g.. Lennuls(eenghOO((Liing Finally, DSR generates a human-readable and concise solution consisting of sequence of reasoning steps S {51, $2 ... Jn based on the minimal reasoning sub-hypergraph leading a". The details of each component are described in the following sections.
Multimodal Problem Formalization
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3.3 MULTIMODAL PROBLEM FORMALIZER
To transform geometry problems into formal language representations, we propose the Multimodal Problem Formalizer (MPF). Specifically, the MPF is a multimodal agent equipped with a tool library and executes a predefined sequence tasks (Annotation, Pre-formalization, and Multimodal Aligament) to generate a complete formal language representation L.
Annotation. Formal language specifications depend on point labels, while real-world geometric diagrams often lack annotations of critical points. To address this issue, we introduce pre-trained model M,, based on the FPN architecture (Lin et al., 2017) to detect key points and explicitly annotate them in the diagram. For given geometry diagram D, the model M, detects key points such as intersection points and circle centers, and assigns labels to these points, formulated as Pt. Y Ma where P; represents the detected key points (e.g., triangle vertices in Figure 3) and Y refers to the corresponding label annotations (e.g., ABCD Figure 3). Then, a new diagram D' P,,YY is constructed with all key points annotated for subsequent processing.
Fre-formalization. Given the annotated diagram D', we introduce a pre-formalization process to extract geometric relations. However, Figure (left) shows that existing MI.LMs still struggle with
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extracting precise geometric relations. To address these, we introduce a Diagram Parser and Text Parser to parse the information in the diagram D' and the problem text 'T' respectively.
To preciscly obtain the information in the diagram, introduced pre-trained neural model Ma based on PGDP-Net (Zhang et al., 2022) as the Diagram Parser. For a given annotated diagram D', the model Ma extracts geometrie primitives (e.g.. segments, circles) and relations (e.g., parallelism, perpendicularity), formulated as Pg. R = (D)) where Pa represents geometric primitives (e.g.. lines shown in Figure 3) and R represents geometric relations (e.g., BD 1 BC shown in Figure 3).
Inspired by Lu et al. (2021), for better efficiency, we utilize a rule-based Text Parser Mr to translate the natural language text T into formal language literals 7 = 12 2.......... : .'g. (()d where each 1 is pscudo formal language literal and Find(t') is a pscudo problem solving goal. Here, "pscudo" indicates that the literals are not yet in the formal language space C, which may include incomplete expressions (e.g., Shape($) in Figure 3 with the unknown represented as 8). Unlike neural sqquence-t--squeeee approaches (Vaswani et al., 2017; Gan & Yu, 2018), rule-based methods empirically show superior efficiency and better performance without relying on large-scale datasets and computational resources (Seo et al., 2015; Bansal et al., 2014; Lu et al., 2021). However, previous rule-based methods (Seo et al., 2015; Lu et al., 2021) struggle with handling complex mathematical expressions (e.g., chain of equalities like 21 = I2 = = In). ). To address this, we extend the mattern-matching rules of the Text Parser to transform complex expressions into formal language literals. Finally, the pre-formalization F is formulated as F = (Pg. R, T') for further processing.
Multimodal Alignment. Since the global geometric information may not be fully captured in the pre- formalization stage (e.g.. shaded region and diameters in Figure (right)), we further introduce the multimodal alignment phase to enhance the completeness of the formalization. Given the annotated diagram D' and the pre-formalization F, we leverage MLLMs' multimodal understanding capabilities to generate: a complete formalization set L by aligning F with D'. Specifically, the multimodal agent: (1) takes D' and F as input, then (2) analyzes multimodal information to clarify ambiguities and fill in missing information, and (3) generates a complete formalization set L {4 12. ... lg. (t). Finally, the MPF outputs the complete formalization set I. to the DSR for further processing.
Compared to existing methods which only parse elementary geometric relationships, the proposed MPF effectively comprehensively extracts fine- imeeggained and global geometric relationships from mul- timodal information, thereby enhancing the completeness and accuracy of the problem formalization.
3.4 DEDUCTIVE SYMBOLIC REASONER
Given the geometry problem formalization I. = (21, 12.....4. Find(t)). to derive the solution a" with a reasoning process S {si, 52,
{2,,,2,........ ..., }. we propose the Deductive Symbolic Reasoner (DSR), Specifically, the DSR is symbolic engine that formulates the problem-solving process as hypergraph expansion task, and generates human-readable reasoming steps S leading to a*.
Geometry Validation. Due the input sensitivity of rule-based symbolic solvers (Lu et al., 2021), we propose a geometry validation step to ensure the consistency and completeness of the formalization L. Specifically, the DSR constructs symbolic representation d of the formalization I., and verifies whether these relations are self-consistent and complete. the formal descriptions are incomplete but can be logically completed using existing knowledge. the symbolic engine automatically supplements the missing relations. For instance, given PH 1 AB with point H on AB, the DSR infers other two missing relations, i.e., PH 1 AH, and PH T BH. Moreover, contradictory expressions, such as asserting collinearity of points A, B, C while simultaneously including Triangle (A, B, C) E L, will be flagged as inconsistent. Consistent and complete formalization should provide equivalent information given in the original diagram D. Inconsistent formalizations with error messages will be sent back to the MPF for refinement and resubmission iteratively, until the DSR receives self- consistent formalization. If the maximum iteration threshold is reached, the DSR will terminate the process, and the problem will be marked as unsolvable. Finally, a complete and consistent symbolic representation d (7f, 1d. .., 1,)) is constructed for the next step, hypergraph reasoning.
......,,)
Hypergraph Construction. Prior symbolic solvers (Lu et al., 2021; Peng et al., 2023; Wu et al., 2024) derive solutions by constructing algebraic equation systems between known and unknown quantities, expanding these systems via theorems, and solving them algebraically. However, this approach exhibits limited interpretability due the lack of traceability in the algebraic solving process.
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Inspired by AlphaGeometry (Trinh et al., 2024), we propose typergraph-basee deductive reasoning Specifically, we enforce each reasoning step s; E S following syllogistic structure: (1) theorem E T. (2) premise set P = {p1, P2. .... . Pm
framework that preserves full reasoning traceability. Specificcal
including m premise literals pi, and (3) a conclusion set Q = {q), 92, (mm') including m'
conclusion literals qi- Each reasoning step is formulated as P 1 Q. For example, applying the
Pythagorean theorem can be represented as: {4ABC, AB 1 AC} Pythagorean Theorem
(AB² + AC
BC*). Then, we model each literal node and each derivation step as a hyperedge. The reasoning process is represented as a hypergraph G = E), where V is the node set (all known literals) and E the hyperedge set (all derivation steps). Given symbolic representation and problem formalization L, the initial hypergraph is constructed with hyperedge labeled "Known Facts" connecting the start node (a trivial root) to all literals 1, E L and 1g E d.
Hypergraph Expansion. While AlphaGeometry (Trinh et al., 2024) excels at proving geometric theorems, its core framework strictly confined to deductive reasoning (DD) and geometric relation transitions (AR), leaving general algebraic equation solving outside of its scope. To address this limitation, we propose unified hypergraph expansion framework that integrates both deductive and algebraic reasoning. To derive the solution literal a" from the initial hypergraph G, we expand G through the following two complementary strategies: (1) Deductive Reasoning matches each theorem t T across node set V to expand G with new hyperedges (reasoning steps) and conclusion nodes. (2) Algebraic Reasoning solves algebraic equations through stepwise transformations instead of bulk solving (Lu et al., 2021; Peng et al., 2023; Wu et al., 2024), ensure that each computation step is intuitively human-interpretable. This strategy implements four atomic operations: Equivalent Substitution, Constant Evaluation, Univariate Non-Linear Equation Solving, and Linear Equation System Solving. Each algebraic transformation is formulated as a special deductive operation. For example: Equivalent substitution deriving a sin(r) from a b and b sin(r) is represented as:
{a b, sin(x)} Equivalent Substitution
= sin(x))
This uniform hyperedge representation ensures coherent graph expansion. To eliminate redundant premise nodes in hyperedges, we enforce minimal sufficient equation sets for each atomic operation. This is straightforward for the first three operations. For Linear Equation System Solving, which yields multiple new equations, we solve a mixed-integer linear programming problem to identify the minimal sufficient equation set required to derive each new equation.
The DSR iteratively applies Deductive Reasoning and Algebraic Reasoning alternately to expand the hypergraph until either the solution node a" is identified threshold is reached. Additionally, to prevent cyclic reasoning during the process, the DSR tracks all predecessors for each node (including all its premise nodes and their recursive predecessors) to block backward reasoning toward ancestral nodes. This mechanism ensures the entire reasoning hypergraph remains a directed acyclic hypergraph. This unified hypergraph-basee framework guarantees consistent formulation of diverse reasoning processes while maintaining full traceability and interpretability throughout the problem-solving trajectory.
predefined maximum iteration
Minimal Solution Generation. Once identifying the solution node a", the DSR generates the most concise solution steps by extracting the minimal reasoning sub- aubbhyerggrphh that connects the start node to the a" node. This subgraph must satisfy the following criteria:
(I) Single Source: Contains exactly one source node (in-degree of zero), i.e., the initial node start;
(2) Single Sink: Contains exactly one sink node (out-degree of zero), i.e., the solution node a":
(3) Minimality: Among all subgraphs satisfying (1) and (2), it minimizes the number of hyperedges.
This optimization problem, formerly known as the hypergraph shortest path problem, is generally NP-hard but solvable in polynomial time for directed acyclic hypergraphs like ours (Gallo et al., 1993; Gao et al., 2014). Once the minimal sub-hypergraph Gmin is obtained, the DSR first performs topological sorting on Gmin to establish dependency order, then translates the sorted hyperedge sequence into a human-readable stepwise solution coherence and preserves interpretability throughout the solution trajectory. Appendix E provides some examples of minimal reasoning hypergraphs. The complete algorithm is detailed in Algorithm 1.
S {s1, $2, .2......55 S..) This process ensures logical

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Algorithm Deductive Symbolic Reasoning Algorithm
Input: problem formalization L == {41 .2, 4e Find(t)), max iteration Output: solution literal a", minimal stepwise solution S
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Check if solution is found
8:
i < ++1
9: end while
10: Gmin -indMMiniinn/eeerriiiiiiiboroooa
11: S + TooploogicaSSor((G miinn)
B Minimal Solution Generation
12: return S
EXPERIMENTS
4.1 EXPERIMENTAL SETUP
Datasets and Evaluation. Experiments were con- ducted on two specialized GPS benchmarks: Ge- ometry3K (Lu et al., 2021) and PGPS9K (Zhang al., 2023), which contain 3,001 and 9,000 image- text pairs of plane geometric problems, respectively, both providing four candidate options and corre- sponding ground-truth truth answers. To enable compre- hensive performance comparison, we adopted both Choice and Completion evaluation formats. Addi- tionally, we utilized Stepwise Accuracy evaluated by human experts for stepwise reasoning quality. More details are provided Appendix
Table Performance comparison among state- of-the-art geometry problem solvers.
Geomerry K
PGPS9K
Method
Choice
Completion
Choice
Completion
MLLMs
G-LLaVA- 13B
29.0
0.3
27.0
0.0
Owen2.5.VL-322
67.6
41.9
56.1
43.5
InternV1.2.5522
60.9
36.1
51.3
28.1
GPT-4o
57.1
46.3
46.0
37.2
Neuzal Sodvers
Geofommer
50.3
36.8
47.3
35.6
POPSNet
77.9
65.0
70.4
62.7
LANS
82.3
71.3
73.8
66.1
Implementation Details. We employed several state-of-the-an the-off-tt--rttt MILMs as agents in our framework.
Symbolic Solvers
InterGPS
63.5
50.6
66.2
57.4
E-GPS
67.9
Open-source: Qwen2.5-VL-322B -nntt--ttttttt et al., 2025) (Qwen2.5), Intern VL3-78B (Zhu et al.,
Instruct (Qwen
GeaDRL
68.4
Ouns (IntemVV1.3
77.6
70.2
79.2
72.3
Ours (Qwen2.5)
78.2
71.0
78.0
70.7
2025) (Intern Closed-source: GPT-4o (Ope-
(GPT-44)
nAL, 2024). See Appendix A.3 for more details.
Baselines. (1) MLLMs: G-LLaVA-13B (Gao et al., 2025), Vision-Rl-7B (Huang et al., 2025b), Dowen2.5-VL332B-Innttee
(L2.5-7L2.5775 (Chen
al., 2024c), Intern VL33-7BB and GPT-4o. (2) Specialized neural solvers: NGS (Chen al., 2022b), Geoformer (Chen et al., 2022a), PGPSNet (Zhang et al., 2023) and LANS (Li et al., 2024c). (3) Symbolic solvers: InterGPS (Lu et al., 2021), GeoDRL (Peng et al., 2023) and E-GPS (Wu al., 2024).
4.2 EXPERIMENTAL RESULTS
This section presents the main experimental results of the proposed AutoGPS framework. We provide additional experiments and analyses in Appendix B, including: (1) answer reliability evaluation, (2)
symbolic solver comparison, (3) failed cases analysis and (4) efficiency analysis.
Performance Comparison with State-of-the-Art Methods. As shown in Table our AutoGPS achieves competitive performance across task formats and datasets. For Choice tasks, it attains comparable accuracy to SOTA methods on Geometry3K (81.6% 82.3%), while demonstrating a 7.7% improvement on the more complex PGPS9K dataset. In Completion tasks, our framework outperforms SOTA methods by 4.1 1% and 9.2% Geometry3K and PGPS9K, respectively.
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Notably, most of the neural methods exhibit significant performance degradation in open-form Com- pletion tasks (c.g.. Qwen2.5-VL dropping by 25.7% on Geometry3K and 12.6% on PGPS9K), as their probabilistic guessing mechanisms struggle to constrain solution spaces without predefined options. In contrast, our AutoGPS demonstrates superior stability across task formats and dataset complexities, maintaining minimal accuracy gaps between Choice and Completion tasks (A=6.2% on both datasets), attributed to its rigorous symbolic reasoning that derives solutions through explicitly defined rules rather than data-driven predications.
Performance Comparision with Specialized MILM. G-LLaVA-133 and Vision-R RI-BB are special- ized MLLMs fine-tuned for geometry problem solving. While Vision-RI ---7B- achieves comparable accuracy with small-scale model, 1-LLaVA-13B struggles across all tasks 22.00299.05 on Choice, near-7ero on Completion), revealing critical limitation: such methods that rely on exhaustive textual descriptions fail when text-diagram alignment is not perfect, which is common in real-world problems. In contrast, our AutoGPS framework fundamentally addresses this gap by effectively extracting geo- metric relations from diagrams and leveraging texts supplementary information. This methodology enables a thorough understanding of geometry problems through multimodal integration, thereby attaining enhanced robustness in geometric comprehension.
Reasoning Reliability Evaluation. Generative large models suffer from hallucination phenomena, leading to cascading errors in reasoning chains. In contrast, AutoGPS employs symbolic syllogistic reasoning to enforce rigorous derivations, inherently avoiding such issues. To quantify this advantage, conducted a human evaluation experiment and utilized the Stepwise Accuracy metric to assess the reasoning reliability of different methods, as shown in Figure 4. From the subset of problems consistently and correctly solved by all methods, we randomly selected 100 instances to exclusively focus on the reliability of their reasoning processes. Despite equivalent final accuracy, MLLMs exhibited substantial logical flaws (Qwen2.5: 33.0%, InternVL3: 29.0%, GPT-4o: 32.0%), whereas AutoGPS achieved almost perfect logical coherence (99%) across all cases. This demonstrates AutoGPS's capability in simultaneously guaranteeing answer correctness and reasoning validity, thereby establishing both interpretability and reliability. Figure 6 illustrates a reasoning process comparison example. Additional comparative examples are provided in Appendix D.2.
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Figure 5: Formalization quality comparison.
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Formally Grounded Reasoning Enhances Noise Robustness.
The inherent rigor of syllogistic reasoning endows AutoGPS with exceptional resilience against inconsistent problem statements. As demonstrated in Figure (left), the framework successfully solves problems with missing valid options through autonomous derivation of ground truth solutions. The complete reasoning process is detailed in Appendix D.1, with corresponding hypergraph representations visualized in Figure 20. In Figure 7 (right), it detects geometric contradictions (e.g., right triangle leg exceeding hypotenuse
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19 27 = 46 41) during the reasoning process. These results validate the robustness of DSR against noisy inputs, demonstrating its appealing potential for real-world applications such as in education and tutoring systems, where robustness and reliability important.
length:
Formalization Quality Comparison. Symbolic solvers are highly sensitive to the formalization quality. We evaluate our AutoGPS against InterGPS (Lu et al., 2021) Geometry3K through two metrics: (1) Jaccard Similarity measures overlap between generated and ground-truth formalizations, (2) Completion Accuracy assesses DSR's problem-solving success using formalizations as input. As demonstrated in Figure 5, AutoGPS achieves a Jaccard similarity of 0.868, significantly higher than InterGPS's s 0.395, indicating substantially more accurate formal representations. This improvement translates to a 27.6% increase in completion accuracy, showcasing that AutoGPS's formalization captures more complete and accurate geometrie relations by Pre-formulization's guidance and Multimodal Alignment's s refinement, thereby enhancing the overall problem-solving performance.
4.3 ABLATION STUDY
To systematically evaluate component efficacy within our framework, we performed ablation studies targeting both the MPF and DSR. Specifically, we analyzed the impact of Pre-formalization and Multimodal Alignment in the MPF module, and the effectiveness of Minimal Solution Generation in the DSR module. Additional ablation studies on the DSR module are provided in Appendix B.
Pre-formalization We compared the per- formance of AutoGPS with and without Pre- formalizution on the Completion task. When pre- formalization is not applied, the model generates formalizations directly from annotated diagrams and problem texts. The results are shown in Fig- ure 8, where the Completion accuracy is signifi- cantly improved by 50.9%-62.44
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Figure 8: Completion performance with/without pre- pre-formalization.
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Multimodal Alignment. We evaluated problem- solving accuracy across alignment configurations, comparing no alignment with three distinct mul- timodal alignment methods (Figure 9). Experi- ments show that applying multimodal alignment rrefformalization achieved performance improve- ments ranging from 7.8% to 13.0%, when eval- uated with our symbolic solver. Notably, even for solvers such as InterGPS that demonstrated primary capability in handling ambiguous for- malizations, multimodal alignment maintained a 4.2% improvement in problem pooblem-solving accuracy. This demonstrates the effectiveness of our multi- modal alignment phase to address semantic ambi- guities and capture global geometric relationships, thereby providing more complete formalizations.
InterGPS
AutoCPS
Figure 9: Completion performance with different alignment models.
50
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Figure 10: Steps distribution analysis for solu- tion generation with versus without minimal so- lution generation (MSG).
Minimal Solution Generation. With Geome- try3K ground-truth formalization, we analyzed the reasoning step distribution with and without minimal solution generation, as visualized in Figure 10. The framework achieves a substantial step reduction, decreasing the average number of reasoning steps from 237.15 to 16.69 (93% reduction). demonstrates that minimal solution generation systematically identifies the optimal deduction trajectory, resulting in more concise solution without sacrificing mathematical rigor.

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5 CONCLUSION
This study introduces AutoGPS, a neural-symbolic framework that addresses key challenges in automated geometry problem-solving through multimodal formalization and deductive reasoning. By aligning diagram-text semantics and generating verifiable stepwise proofs, AutoGPS outperforms existing approaches on benchmark datasets while ensuring reliability and interpretability.
ACKNOWLEDGEMENTS
This work is supported by the Fundamental and Interdisciplinary Disciplines Breakthrough Plan of the Ministry of Education of China (No. JYB2025XDXMI01) the National Natural Science Foundation of China (No. 62272374, No. 62192781), the Natural Science Foundation of Shaanxi Province (No. 2002CC--CCNN-62) the State Key Laboratory of Communication Content Cognition under Grant No. A202502, the Key Research and Development Project in Shaanxi Province (No. 2023GXLH-024).
ETHICS STATEMENT
This research aims to advance multimodal geometry problem solving for academic and educational purposes. All experiments conducted publicly available datasets without sensitive information. We see minimal risk of misuse, but recommend responsible use in practice.
REPRODUCIBILITY STATEMENT
We provide all code and data necessary to reproduce our main experiments in the supplementary materials. The detailed experimental setup is described in Section 4.1 and Appendix A. Our codebase will be made publicly available to facilitate further research.
LLM USAGE STATEMENT
LLMs were used for (1) language polishing and (2) multimodal agents inside AutoGPS pipeline. Specific models, versions, typer-parameters are provided in the Section 4.1 and Appendix A.3. The prompt templates are provided in Appendix G. All LLM outputs incorporated into the manuscript were reviewed and verified by the authors; authors accept full responsibility for the final content. LLMs were listed authors.
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A MORE EXPERIMENTAL SETUP DETAILS
A.1 DATASET
We evaluated our AutoGPS on two datasets: Geometry3K (Lu al., 2021) and PGPS9K (Zhang et al., 2023). Geometry3K contains 3,001 geometry problems with ground-truth formalization labeled and double-checked by our human experts. The ground-truth formalization was used for two purposes: (1) to evaluate the quality of generated formalization results, (2) as the input of symbolic solvers to evaluate the performance of symbolic solvers.
A.2 EVALUATION METRICS
Choice and Completion Accuracy. To enable comprehensive performance comparison, we adopted both Choice and Completion evaluation formats. For Choice tasks, the option answered by the model or numerically closest the solver's output selected, with random selection from four options if resolution fails. For Completion tasks, answers are strictly validated through numerical equivalence to ground-truth values, with unresolved cases automatically classified incorrect. For all tasks involving MLLMs, performance was evaluated using the @ass@3 metric, which deems task successful at least one out of three independent sampling attempts produces a correct solution.
Formalization Quality. To evaluate the quality of formalization, we employed Jaccard similar- ity (Jaccard, 1901) as one of metries. defined as the size of the intersection divided by the size of the union of two sets, specifically:
J(PYY) PnY PUY
where P is the set of predicted formalization results and Y is the set of ground-truth formalization results. The order literals the formalization does not matter.
Human Evaluation of Stepwise Reasoning. To evaluate the quality of stepwise reasoning, we employed Stepwise Accuracy. When evaluated by Stepwise Accurucy, reasoning process correct if and only if all reasoning steps logically valid and lead to correct answer. Our evaluation conducted on the subset of problems that are consistently and correctly solved by all methods, to isolate the impact of final answer correctness. These problems with corresponding reasoning processes were distributed to three human experts for evaluation. The detailed criteria of stepwise reasoning evaluation are provided as follows:
Accuracy of Geometric Information Comprehension: Whether the textual and graphical information in the problem has been properly captured, including geometric elements (e.g.. points, lines, circles), geometric relationships (e.g., perpendicularity. intersections), and text labels (whether they are correctly associated with the corresponding geometric elements).
2. Correctness of Theorem Application: Whether each theorem is applied to appropriate geometric entities and derives valid conclusions.
Validity of Algebraic Transformations: Whether algebraic operations are correctly implemented and yield accurate equations.
4. Logical Coherence and Consistency: consistency. Intermediate conclusions must align with the final answer, with no critical steps omitted.
Whether the reasoning exhibits logical coherence and
All reasoning steps were independently examined by three human experts. For contested items (i.e., those with divergent evaluations), the correctness of the reasoning process was ultimately determined through collective deliberation and majority voting.
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A.3 IMPLEMENTATION DETAILS
The symbolic method InterGPS was reproduced using the authors' open-source code. while results for GeoDRL (Peng al., 2023) and E-GPS (Wu et al., 2024) extracted from original publications due to code unavailability. All experiments were executed on an Intel(R) Xeon(R) Gold 6226R CPU 2. platform in conjunction with eight NVIDIA GeForce RTX 3090 GPUs. strict timeout threshold of 1800 seconds was imposed on symbolic solvers, where any computation exceeding this duration was systematically categorized as resolution failure. For MILMs hyperparameters, we set the max tax_ns to 3096, temperature to 0.1, and top-p to
B ADDITIONAL EXPERIMENTAL RESULTS
Answer Reliability Analysis. To further evaluate the reliability of answers generated by different methods, we computed the Answer Reliability Rate (ARR) on the Completion task, defined follows:
Neoorect
ARR
Nvalid
where Neorvect denotes the number of correct answers and Noali refers to the number of valid answers produced by the model. This metric reflects the trustworthiness of given answer, i... the probability that a returned answer is correct. For MILMs, valid answer is defined as one that numerically well-formed and interpretable. For symbolic solvers, a valid answer is one derived within the time limit. We calculated ARR as the proportion of correct answers among all valid responses; the results illustrated in Figure 11.
Our findings indicate that symbolic solvers (including InterGPS and our method) achieve substantially higher ARR than MLLMs, underscoring the inherent reliability of symbolic reasoning. Furthermore, our AutoGPS attains an ARR exceeding 90% across all experimental settings, outperforming InterGPS by 10.3% to 11.8%, which highlights the enhanced reliability of our framework. This improvement can be attributed to more rigorous formalization achieved by MPF, which reduce formalization inaccuracies and establish more dependable foundation for symbolic reasoning.
Geometry
95.5
01.9
95.9
96.0
91.8
PGPS9K
83.7 81.5
Answer Reliabiiity Rate (9%)
55.7
50.1
47.3
46.9
36.4
Vilom-t1-1
Interi Pab
Internts hea
towen.
Gut Intenti.
Obes (PP
Figure I1: Answer reliability rate comparison among different methods.
Additional Evaluation.
To further demonstrate the generalization and scalability of proposed AutoGPS framework, we evalu- ated its performance on two additional datasets: Vath (GPS) (Lu et al., 2023) and GeoQA (Chen al., 2022b), as shown in Table 2 and Table 3. The proposed AutoGPS (InterVVL3) achieves 86.2%, ranking 1st among existing open-source methods and competitive with several close-source models on Vista (GPS) and achieves 86.2%, outperforming NGS, SCA-GPS, and DualGeoSolver, and is comparable to FGeo- Hieo-HyyerGNet (85.6%) on GeoQA. These results further validate the effectiveness and robustness of our AutoGPS framework in solving diverse geometry problems.
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Table 2: Math Vista (GPS) Performance Comparison
Table 3: GeoQA Performance Comparison
Method
Choice Acc.
Method
Choice Acc.
Human
48.4
Human (Chen et all, 2022h)
92.3
NGS (Chen al., 20226)
57.4
Close-Source
NGS-Auxiliry (Chen et 20226)
60.0
DreamPRM (o4-mini) (Cao al., 2025)
95.7
SCA GPS (Ning et al., 2023)
64.1
Step RI-V-Mini
89,9
DualCeeSolvvr (Xiao
2024)
65.2
Kimi-k1. momi-k1..6ppoe 1omm-k1.--preview-222 df-revveew-22250300
91.8
FGeo- Helo--HyerGeeeeee (Zhang et al,, 2024c)
85.6
Doubao-pro-1.5
88.9
AutoGPS (InternVL3)
86.2
Open-Source
Vision-R1-7B (Huang et al., 2025b)
82.7
Ovis2_344 34B (Lu al, 2025)
84.6
AutoGPS (1.t)
86.2
Performance Comparison of Symbolic Solvers. We benchmarked the solving accuracy across different solvers using the Geometry3K ground-truth formalization. This approach effectively isolates the impact of formalization inaccuracies on problem-solving outcomes. As shown in Table 4, our approach achieves superior performance most tasks, even surpassing human experts in average accuracy. This advantage of DSR originates from its enhanced capability for handling complex geometric configurations and an advanced symbolic reasoning algorithm with a larger theorem set.
Table 4: Symbolic solvers comparison using the Geometry3K ground-truth formalization by Choice.
Question Type
Geometric Shape
Methed
Average
Angle
Length
Area
Ratio
Line
Triangle
Quad
Circle
Other
Human (Iin et al., 2021)
53.7
59.3
$7.7
42.9
46.7
53.8
68.7
61.7
58.3
56.9
Human Expert (Lu et al., 2021)
89.9
92.0
93.9
66.7
95.9
92.2
90.5
89.9
92.3
90.9
InterGPS (Lu et al. 2021)
83.1
77.9
62.3
75.0
86.4
83,3
77.6
61.5
70.4
78.3
E-GPS (Wu et al,, 2024)
90.4
92.2
73.6
100.0
91.4
93.1
87.9
81.1
75.3
89.8
GeoDRL. (Peng
2023)
86.5
93.7
75.5
100.0
87.7
93.1
90.2
78.3
77.8
89.4
HyperCiNet(Z et al., 2024c)
92.0
Ours
Failed Cases Analysis. In the human evaluation of reasoning processes, AutoGPS achieved 99%
stepwise accuracy, with only one failed case. We provide this case in Figure 12. Although the final answer correct, the formalization is not fully faithful the original problem statement, violating the requirement for accurate comprehension of geometric information (Criterion in Appendix A.2).
The underlying causes are as follows:
(1) Ambiguous Labeling. The labels are spatially ambiguous. The label "13" could refer to either segment DC or CE, which may mislead both MLLMs and even human experts incorrectly associate it with the nearer segment DC.
(2) Training Corpus Bias. The polygon "ABCD" appears far more frequently than "ABCE" in the LLM's training corpus, resulting in bias, especially when low decoding temperature is used.
Nevertheless, the formalization remains logically consistent and produces both sound reasoning steps and the correct answer. This case raises more challenging question for future work: How can we ensure faithful formalization that accurately reflects the original problem, even when the formulization is logically consistent and yields a correct answer?
For MILMMs, we observed that stepwise accuracy was surprisingly higher than anticipated. We therefore further examined several failed cases and found that the correctly solved problems were generally simpler, requiring only straightforward reasoning without complex theorem applications or deep geometric understanding. Most complex problems, however, MILMs often failed to provide valid reasoning steps, leading to incorrect answers, and thus such cases excluded from this category. Even when correct answers were produced, the following reasoning flaws were evident:
not solved correctly, since
(1) Incorrect theorem application (e.g., Qwen2.5 in Figure 15).
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A
B
5 cm
13 cm
5 cm
13 cm
12 cm
E
D 12 cm
C
(a) Original problem diagram.
(b) MPF labeled diagram.
Problem: Perimeter parallelogram.
Formulization:
Point ine,,,, Line(E, C))
EguallsLonghhhfLLinnee,,, D)).
Lggall(LengttOO(iiine C)). 13)
Equals([Lengln CegtiOLLL, 12)
Find TomiPerrinteOOOPPallllleee B, C, D)))
Answer: 50
(c) Problem description and MPF formalization. The red part indicates the unfaithhful formalization and blue part indicates the correct answer.
Figure 12: Failed case of AutoGPS.
(2) Incorrect matching of geometric elements and text labels (e.g., GPT-4o in Figure 15).
(3) Incorrect understanding of geometric relationships (e.g., Intern VL3 in Figure 15).
In most cases, MILMs simultaneously suffered from multiple issues, occasionally arriving correct conclusions while demonstrating incorrect outcomes across the majority of scenarios.
Notably, relatively few errors were found in the algebraic transformation process, which was surpris- ing. These results provide us with some insights about how to further improve the geometric reasoning capability of MLLMs: (1) Enhancing the understanding of fine- fine-grained geometric primitives and relationships, (2) Improving the theorem application capability.
Solving Efficiency Analysis. During the experiments, we set up a 1800-second timeout for the DSR to ensure high performance. In fact, most problems were solved in a much shorter time, as shown in Figure 13. For example, on the Geometry3K dataset with GPT-4o for the MPF module, the average solving time was only 57.59 seconds, with 99% of problems solved within 360 seconds (Figure 13a). demonstrates the high efficiency of our DSR, which stems from high-quality formalization provided by the MPF and the efficient reasoning algorithm of the DSR.
Ablation Study Geometry Validation. Geometry validation not only ensures the correctness the generated formalization but also completes the implicit geometric relationships that are not explicitly stated in the diagram. We conducted an ablation study to evaluate the performance our DSR with and without geometry validation, as shown in Table 5. Without geometry validation, the DSR was unable to fully capture the geometric relationships, even with the ground-truth formalization, reaching an accuracy ceiling around 56%. There are two reasons for this performance barrier: (1) Inconsistent formalization makes the problem unsolvable. (2) Many geometric relationships are implicit but important for problem solving. Since the Geometry Validation phase detects inconsistencies and supplements implicit geometric relationships, it provides the symbolic representation with error-free free and complete geometric relationships, thereby significantly improving the performance of DSR.
Ablation Study of Reasoning Strategies. There are two reasoning strategies in our DSR: Deductive Reasoning (DR) and Algebraic Reasoning (AR). evaluate their effectiveness, we conducted an ablation study by removing one of the two reasoning strategies and comparing the performance of our DSR using the GKOmetry3K ground-trui truth formalization, as shown Table 6. When either reasoning strategy was removed, the DSR failed to solve most of the problems, where the accuracy dropped to 5.5% for both cases. Only when both reasoning strategies were employed did the DSR achieve a significant accuracy of 94.5%. DR's role is to derive new geometric relationships from existing ones through the application of geometric theorems, while AR's s role is to solve algebraic equations and obtain values for unknown variables. Since geometry problems often require both
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Mean (57.595)
Mean
Percentage of Problemss (86)
50
85th percentil (90.94s)
percentie (131.654)
Probi ems (%g
85th percentile (141.68s)
percentile (212.05s)
95th percentie (241.70s)
30
95th percentile (267.555)
99th percentie (354.10g
percentile (618.735)
25
23
600
800
1000
1200
1400
1800
200
400
600
800
1000
Solving Time (5)
Salving Time (s)
(a) Geometry3K with GPT-4o's formalization.
(b) PGPS9K with GPT-4o's formalization.
40
Mean (53.90s)
Mean (75.923)
Percentage of Problems (%)
85th percentle (07.06s)
of Problems (%)
85th percentile (135,795)
percentll [130. 05x]
percentile (219.318)
25th percentlle 1224.435 43c)
95th pencentile (271.665)
percentlee 39s)
30
percentile (674.575)
20
15
10
600
1200
Solving Time
Solving Time (9)
(c) Geometry3K with InternLL3's formalization.
(d) PGPS9K with Intern VL.3's formalization.
Figure 13: Solving time distribution of DSR.
theorem applications and algebraic transformations, the two reasoning strategies are complementary and indispensable, collaboratively enhancing the overall reasoning capability of the DSR.
Ablation Study of Feedback and Refinement. To rigorously evaluate the utility of our Feedback and Refinement mechanism, we conducted an ablation study comparing with two standard multi-pass strategies: @ (a naive approach) and Major@ (Self-Consistence) the Refine@5 5 strategy demonstrates clear superiority in both efficacy and efficiency. Efficacy is evidenced by Refine @5 significantly outperforming Major@5 SSeff-Consistency) by substantial margin +4.9% to +7.0% across all datasets (Geo3K, PGPS9K, and GeoQA). This finding strongly suggests that the directed refinement based explicit feedback is vastly more effective for error correction than simply aggregating an undirected consensus from independent samples. Regarding Efficiency, Refine @5 achieves this superior performance with an average of only 1.36-1.51 total forward passes, dramatically lower than the 5 full, independent passes required by both Pass @ and M5 This confirms that mechanism not only yields higher accuracy but also does in computationally efficient manner.
As shown in Table 7,
Table 5: Completion accuracy of DSR with and without geometry validation Geometry3K dataset.
Formalization
Geometry Validation
MPF (GPTT-4)
Ground Truth
55.5
$5.4
75.4
94.5
Table 6: Completion accuracy of different reasoning strategies Geometry3K dataset.
W/O DR
With DR
W/O AR
With AR
5.5
94.5
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Table 7: Completion accuracy of different forward strategies.
Strategy
Geo3K
PGPS9K
GeoQA
Refino@5
71.2%
71.9% 68.2%
73.3%
Pass
70.0%
Major@5
66.3%
67.2%
64.9%
C LIMITATIONS AND FUTURE DIRECTIONS
Limited Geometric Formal Representation. The formal language used in our work is expressive enough to represent most geometric configurations in the datasets. However, it is not sufficient to represent some corner cases, such three tangential circles. General-purpose formal languages such as Lean (De Moura et al., 2015; Moura & Ullrich, 2021) and Coq (Bertot & Castéran, 2013) still require large amount of groundwork to describe the geometry problems present. Formalization with those languages requires deep expertise in both geometry and formal languages, which is high barrier for most researchers. This challenge complicates the annotation of large- scale geometry datasets. Additionally, due to the limited existing training data, LLMS are not able to understand such complex formal languages well, resulting in low performance (Wu et al., 2022; Li et al., 2024a). In this paper, we do not pursue complete solution geometry representation, as it is separate and extremely challenging research topic that demands substantial investment from the mathematical formalization community.
Large-Scale Annotated Geometry Datasets for AI Community. The creation of large-scale geometry datasets remains challenging due to the labor-intensive nature of manual collection and annotation(Huang al., 2024). Current benchmarks like Geometry3K (Lu et al., 2021) and PGPS9K (Zhang et al., 2023) not only suffer from limited scale but also lack detailed step-by-ster solutions. Our AutoGPS addresses these limitations by enabling the automated generation of high-quality formalizations and rigorous reasoning steps, which can be utilized to create large-scale datasets with comprehensive solution documentation. This will empower the community develop more robust neural models with enhanced reasoning fidelity and reduced hallucination risks.
Trade-off between Proof Conciseness and Computational Efficiency. The proposed methodol- ogy employs deductive reasoning- based algorithm that operates by exhaustively searching and applying all potential axioms to progressively expand the reasoning hypergraph, aiming to enable backtracking identify the shortest proof path. The imperative to find the shortest proof requires exhaustive exploration of various proof approaches to find the optimal path, which fundamentally limits algorithmic efficiency through combinatorial explosion. Should computational efficiency be prioritized over proof conciseness. heuristic search strategies would be a superior alternative. though the expense of guaranteeing solution optimality. This fundamental tension reveals an inherent trade-off between the dual goals of minimal proof determination and highly efficient problem solving, requiring us find a balance between the two objectives in future work.
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D
SOLUTION EXAMPLES
D.1 STEPWISE REASONING PROCESS FOR NOISE DATA
M
6
5
Problem Text
N Q O P. Find length of Q P.
Q
N
35
3
5
X
O
P
Solution by AutoGPS
Step
Known facts
start
PQ, 6 MN, 5
NO, MP, ZOMP
NQ 5 = MQ, N on MO
Step 2:
Line Segment Split: Q MP
MP MQ PQ
Step
Same Angle: NMP, Q on MP
= / /MMP = /NMQ
Step
Corresponding Angle Theorem:
OQ
LMNQ LOOP, MPO LMQN
Step
Line Segment Split: N on MO
= MO MN NO
Step
Same Angle: / O.PP, N MO
/ NMP /OMP
Step
Substitution MP MQ + PQ, a PQ, 5 MQ = 5 MP
Step8:
Substitution
MN, MO MN NO, Transtivity of Equivalence
NO
MO
Step
LNMP LOMP, LNMP LNMQ
LNMQ LOMP
Step 10
Solve Linear Equation System (3 5
MO 9.6 MO
Step
Angle-Angle Similarity Theorem LNMQ
LOMP, LNNQ MOP, <MPO MQQ
4 MNQ A MOP
MN
Step Similar Definition
MNQQ ~ MOP LNMQ = LOMP, ratio
MQ
sim_ratio
LNN MOP
MP
MN
6
Step Substitution
MN, 9.6
MO, ratio
sim_ ratio
MO
9.6
MQ MP
Step 14:
Substitution sim_ratio
5 +
MP,
MQ
sim_ratio
(5 x)
6
Step
Transtivity Equivalence
sim_ratio
sim_ratio
(5 + 2) 9.6
9.6
x)
Step 16;
Solve Linear Equation System
x. PQ
= 3 PQ
9.6
(5 + x)
Figure 14: Reasoning process of AutoGPS for noise problem.
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D.2 REASONING PROCESSES COMPARISON
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2000, 814
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Step 11:
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360
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Figure 15: Comparison of reasoning processes (1). All three MLLMS generated correct answer but provided logically incorrect reasoning process. Symbolic-based method, InterGPS, provided the theorem applied but lacked detailed reasoning process. In constract, AutoGPS generated logically coherent stepwise solution with correct answer.
23

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Published as a conference paper at ICLR 2026
A
B
3
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5
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Figure 16: Comparison of reasoning processes (2). MLLMs captured the correct geometric infor- mation but generated wrong reasoning processes, leading to incorrect answers. Symbolic-based method, InterGPS, failed to solve this problem. AutoGPS generated a concise stepwise solution with correct answer.
24

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Published as a conference paper at ICLR 2026
InteraV
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12+
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balkne XX2 XVW M00 6x
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Figure 17: Comparison of reasoning processes (3). MILMs failed to capture correct geometric relationships and generated wrong reasoning processes. Symbolic-based method, InterGPS, provided correct answer but lacked detailed reasoning process. leading to the wrong answers. AutoGPS generated human-readable reasoning process with correct answer.
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Published as a conference paper at ICLR 2026
E HYPERGRAPH EXAMPLES
Stop
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conclusion
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premise
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premice
prémise
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premise
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conclusion
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21
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45
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Step 9 Evaluste
condusionn
0 ZUK
Figure 18: Minimal reasoning hypergraph example (1). The red nodes indicate literals. The blue boxes indicate derivation hyperedges connecting the premise nodes to the conclusion nodes. The hyperedges are topologically sorted indicate the order of reasoning.
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Forduain (undevon
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Circle O has a radius of inches, Radius B is perpendicular chord C which is inches long Find O.X.
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Figure 19: Minimal reasoning hypergraph example (2). The red nodes indicate literals. The blue boxes indicate derivation hyperedges connecting the premise nodes to the conclusion nodes. The hyperedges are topologically sorted to indicate the order of reasoning.
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Published as a conference paper at ICLR 2026
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Figure 20: Minimal reasoning hypergraph example (3). The red nodes indicate literals. The blue boxes indicate derivation hyperedges connecting the premise nodes to the conclusion nodes. The hyperedges are topologically sorted indicate the order of reasoning.
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and Published as a conference paper at ICLR 2026
F FORMAL LANGUAGE DEFINITIONS
We have appropriately simplified the formal language used in InterGPS (Lu et al., 2021), and described its syntax via context-free grammar (CFG). This mathematical definition facilitates efficient parsing of the formal language. Let the special character set be SPECIAL_CHAR = {_ (........ *//.,. 8,'). The grammar defined as a quadruple G (N.) where:
N flogic_ form, args, arg, id, expr)
Σ = UPPER_LETTER U ALPHA NUM U SPECIAL_CHAE
S logic_form
P:
logic_ form + id args id expr
args + arg (, arg)*
arg logic_ formfrrmmm id expr
id + UPPER LETTER (ALPHA NALPHA_NUMU NUM (1)"
expr [-] ANUB (ALPHA_ NUM SUNCIUUUUUS CHAR)
UPPER_LETTER = ( B, 1.......]] 2
ALPHA_NUP {a,..., 2) J (A,..,,ZUU ANUB = ALPHA__NNEA LAPPAANNUML NUMU (_A)
detailed illustrative example of the formal language provided below:
Table Predicate and literal definitions for the formal language (1).
Literals
Explanation
Line(A.B)
A line segment with endpoints A and B The angle with point as vertex Angle ABC with B as the vertex Triangle with vertices A, B, and C Quadrilateral with vertices A. B, C, and D Parallclogram with vertices A, C, and D Square with vertices A, B, C, and
Angle(A)
Angle(A.B,C)
Triangle(A,B.C)
Quadrlateera(AA.B...D))
Parallelogram(AABB...D))
Square(A,B,C.DD)
Rcctanglo(A.B.C.))
Rectangle with vertices A, B, C, and D Rhombus with vertices A, B, C. and D Trapezoid with vertices A, B, C, and D Kite with vertices A, B, and Polygon with vertices A, B, C, etc.
Chombus(A.B,C.D)
Trapezoid(A.B.C.D)
Kite(A.B.C.D)
Polygon(A.B.C....
Pentagon(A.BBC.DD.E)
Pentagon with vertices A, B, C, D, and E Hexagon with vertices A, B, C. D. E, and Heptagon with vertices A, C, D. Octagon with vertices A, B, C, D, F, G. Circle with center
Hex.gom(A..BC...E..F)
Heppaagam(A.B...DDEFFGG
ActagootAAB.C...DEE.E....
Circle(A)
Circle(O.
Circle with center O and radius Minor with A and B endpoints circle Are that passes through points A. and Sector of a circle with center O and points A and B on the circumference
Arc(A.B)
Arc(A.B.C)
Sector(O.A,B)
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Table Predicate and literal definitions for the formal language (2).
Literals
Explanation
Equiaaeal(PolyeentAAA....DD
Polygon ABCD is equilateral
Regular(Polygom(AAB....DD
Polygon ABCD) is regular
SraaO(())ape(...)
Area the Shape
PerimeterOO(Shappel
Perimeter of the Shape
RadiusOf(Cinlle(O))
Radius of the circle O
DiameterOf((Cicleo(O)
Diameter of the circle O Circumference of the circle Measure of the angle ABC Measure of the are AB Length of the line segment AB Point A lies
Cirumffrenn:oO(Ciicll()))
CeasureeO(AAnglc(AA. B, C))
MeasureOf(AAn((A B))
LengthOf(Line((A, B))
PoinLLisoonLime((LLLnmeCBBB1.))
segment BC
Point A lies on the circle with center O and radius
Puaalel(LLiocAABBBLLnnCCC.))))))
Line AB is parallel to Line CD
Line AB is perpendicular to Line CD
Line AB bisects angle XAY
Triangle ABC is congruent triangle DEF Triangle ABC is similar to triangle DEF
Tangent(LineAAADD.CicccddC..)))
Line AB is tangent to circle O with radius
Secam(LLin(AAB,,.iccce((0.)
Line AB is secant circle O with radius First shape is cireumscribed the second shape First shape inscribed the second shape
InnericclinnShpppc....Shaa.....
IsMidpoinnOf((CLinn(AAABB)
Point C the midpoint of line AB
Point O is the centroid of triangle ABC
SlincenterOOOOOTTTinggAAABBBBBB
Point O is the incenter of triangle ABC
Line OA is radius of circle with radius
AB a diameter circle with radius Line AB a midsegment triangle DEF Line AB is a chord of circle O with radius r Line AB is the perpendicular bisector of line CD Line EF is the median trapezoid ABCD
CeetraadccaaassttttooolilieeBBLl IsMedianOfi
IsMedianOf[Limm
Line EF is median of triangle ABC
SinOf(var)
Sine of var (var can be variable, measure of angle or arc)
CosOf(var)
Cosine of var (var can be variable, measure of angle are) Tangent of var (var be variable, measure of angle
Tanrf(var)
arc)
CotOf(var)
Cotangent of var (var can be variable, measure of angle or arc)
HalfOf(var)
Half of (var can be variable, length, measure, area, etc.) Square root of var (var can be variable, length, measure, area, etc.)
Sqr(O((vr))
RatioOf(yarl,var))
Ratio of varl to var2 (can be variables, lengths, measures, areas, etc.)
Addition of varl, var2, and possibly variables Multiplication varl, var2, possibly more variables Subtraction of var2 from varl Division of varl by var2 varl raised the power of var2 varl equals var2 b is equivalent Equals(a. b)) Find the value the variable
Mul(varl,
Sub(varl, var2)
Div(varl, var2)
Now(varl,vard)
Equals(varl, var2)
Find(var)
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Published as a conference paper at ICLR 2026
G PROMPT TEMPLATE
In our experiments, we involve the use of multimodal large language models to accomplish the following tasks:
Choice Task. Solving multiple-choice questions.
Completion Task. Solving fill-in-the-blank questions.
Formalization Task. Direct formalization of problems without pre-formalization.
Alignment Task. Aligning the results with pre-formalization specifications.
The corresponding prompt templates for each task presented below. The image input is omitted for brevity. The variables are colored in red to indicate the need for replacement with the corresponding content.
Prompt Template for Choice Task
This is geometry problem. The problem text given as
(prohlem text)
There are several choices:
Choices are: A. [content A), B. [content B), C. [content C), D. (content D)
**
Describe the figures and label information in the geometry diagram.
Solve the problem step by step and give your final choice in the form of "choice". If your choice is A, give "A" last.
Prompt Template for Formalization Task
Given the geometry problem with problem text "" [problem text]"", we use logic forms describe information of this problem. The logic forms are defined as follows: "'plaintxxt
HH Prodicate Definitions iipredicatel__efiitttia
**
Identify the geometric figures in the diagram and list the known value information in the diagram. Formalize the problem with the given logic forms. Give your final logic forms in single plaintext
code block.
**Note:**
A line named with endpoints and B, then it is expressed Line(A, B) rather than Line(t).
circle with center with radius of is expressed Circle(O. radius. o) and Equals(radius 5).
expressed as Equuls@[.engthhO((inneee An angle ABC with measure 30 degrees is expressed as (quals(MeasurrOO((AA) (nngle,, B, C)). 30).
line segment with length
B)), 10).
An are AB with measure 60 degrees is expressed as (quals(MMaaurreOO((Arrc B)), 60).
point A lies segment BC is expressed PointLieSOnLineAA. Line(B, C)).
point lies on circle with center
radius expressed PoinLLesOOnCirleCA. Circle(O. r)).
the goal find area a shaded region, use the arithmetic operation expression with other regular figures to represent the shaded region. For example, Sub(AreaOF(Cirrl(()) AreaOf((riangle(),
E, F))).
Each problem should have goal in the form of Find(...). for example, LenddLeegttOO((Lin((X, Y))).
Please formalize problem faithfully. not add any extra information or deduction.
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Prompt Template for Completion Task
This is geometry problem. The problem text is given (problem text)
**
Describe the figures and label information in the geometry diagram. Solve the problem step by step and give the final answer in the form of "answer" and round it 3rd decimals, the answer is 5, give "5.000" last.
Prompt Template for Alignment Task
Given the geometry problem with problem text "1 (prohlem text) "* logic forms describe the information of this problem. logic forms are defined as follows: "'plaintxxt
### Predicate Definitions [lpredicatel_definnttio.
We have previously parsed the diagram and text. The diagram logic forms are: "plaintext
Madiagaan_ooiic_ormmul
Midiagram_logic_-orre
And the text logic forms are: "pllinnext
Mttoxt_loglc lttec__
**
Replace point identifier and replace Shape(S) with specific geometric figures.
Check if the problem is correctly converted to logic forms. Combine the final diagram logic forms and text logic forms in the format of plain text code block.
**Note:**
A circle with center with a radius of 5 is expressed as Circle(O, radius) and EqualssRadusOOO(Crreeeee 5).
A circle with center O with diameter of 10 is expressed Circle(O, radius) and 10).
EapualsDDaneteerOCCCcceeeee
line segment with length
expressed as (puull(LengthOOR[....
B)), 10).
An angle ABC with measure 30 degrees is expressed as Equals(MeasureOO((AngggAAAA B, C)). 30).
An are AB with measure 60 degrees is expressed as Equals(MeaauureO((AAr
B)), 60).
If the goal is find the of shaded region, the arithmetic operation expression with other regular figures represent the shaded region. For example, Sub(AreaOF((inellCC)) (ranOR(Traanle((), E, F)))
32


Reviews:
[{"review_id": "ZCCTpmDW3M", "reviewer": "Reviewer_bJ89", "summary": "This paper propose AutoGPS, a neural symbolic plang geometry problem solver. AutoGPS consists of a MPF to parse the plane geometry diagram into symbolic language, and further use a DSR to conduct deductive symbolic reasoning based on a hyper graph to solve the problem, and finally obtain the answer by extracting the solution path from the graph. Experiments on Geometry3K and PGPS9K show the proposed method is effective on solving PGPs.", "questions": "I would like to see what is the performance of the diagram parsing in AutoGPS by using the feedback and refinement process, comparing to the original PGDP work, which already achieved 99% accuracy on the PGDP5K dataset.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1.\tThe design of the framework is reasonable. AutoGPS first parse the diagram into symbolic languages, further construct a hyper graph of the given problem and search the solution path on the graph to finally get the solution steps, ensure the interpretability.\n2.\tComparing to the mentioned previous works in this paper, AutoGPS improved the problem solving accuracy than previous baselines, which show the effectiveness of the proposed method.\n3.\tIn the human evaluation, AutoGPS output stepwise solutions with ideal accuracy, outperformend other pure MLLMs.", "weaknesses": "1. As I know, using hyper graph in solving plane geometry probelms (PGPs) was already explored by FGeo-HyperGNet [1], this paper not give appropriate discussion on this previous work, what is the difference of the hyper graph between AutoGPS and HyperGNet. Meanwhile, HyperGNet achieved 91.99% on Geometry3K which is higher than AutoGPS with 81.6.\n\n2. To this end, the contribution of this paper is limited, as the diagram parsing mainly relies on PGDP and the hypergraph has already been used in other work. Despite the author proposing a deductive symbolic reasoning framework to assemble these modules and give stepwise solutions that are easy for humans to read, I think this paper does not satisfy the bar of the ICLR community.\n\n3. The description of symbolic solvers is not appropriate. Indeed, symbolic solvers do not directly output the results without solution steps, like Inter-GPS, which was proposed along with the Geometry3K dataset. Inter-GPS uses a theorem predictor (also has a search algorithm) to predict the theorems at each step, and based on these theorems, to conduct symbolic reasoning on the existing problem conditions to get the final target answer. And the reliable interpretability is one of the advantages of symbolic solvers. The author said symbolic solvers are hard for humans to read, it is easy to tackle by using an LLM to translate the symbolic solution steps into human language, which will be used to understand.\n\n4. Limited experiment dataset, benchmark. This paper only conducted experiments on the Geometry3K and PGPS9K, while actually PGPS9K was expanded from Geometry3K. It is necessary to conduct experiments on popular benchmarks in the PGP solving area, such as Math-Vista Geo, Math-Verse. The current experiments are not convincing. \n\n5. Limited generalization ability of AutoGPS. As mentioned in the above weakness, I am wondering if AutoGPS is hard to solve other domains of PGPs, such as GeoQA, which is also a foundation dataset for the research of solving PGPs, as the diagram parsing module in the AutoGPS is leveraged from PGDP, which is a delicate model designed for parsing diagrams in the style of Geometry3K. If AutoGPS is not able to solve problems in GeoQA and other benchmarks, the scope of this work is not sufficient for ICLR.\n\nRef:\n\n[1] FGeo-HyperGNet: Geometric Problem Solving Integrating FormalGeo Symbolic System and Hypergraph Neural Network, in IJCAI 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QIXbludNf4", "reviewer": "Reviewer_U8MK", "summary": "This paper addresses the challenges of multimodal understanding and rigorous reasoning in automated geometry problem-solving. It proposes a neuro-symbolic framwork called AutoGPS which is composed by the Multimodal Problem Formalizer (MPF) and the Deductive Symbolic Reasoner (DSR) modules. MPF is used to multimodal understanding, translating the diagram and natural text to rigorous formal languages, while DSR is used to perform deductive reasoning to infer the final answer on a meticulously constructed supergraph. Experimental results demonstrate the effectiveness of the proposed AutoGPS framework.", "questions": "Besides the weakness above, I wonder what the average time cost of AutoGPS is for solving a geometry problem compared to other baselines.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The design of the overall neuro-symbolic architecture is reasonable, and the MPF and DSR modules are well defined.\n2. The introduction of the multimodal alignment stage in MPF is effective in filling the missing semantic information that is not captured by the pixel-level diagram parser.\n3. The experimental results are convincing and demonstrate the effectiveness of the AutoGPS framework.\n4. The writing and structure of the paper are clear and easy to follow.", "weaknesses": "1. My main concern lies in the originality of this paper. First, in MPF, the text parser $M_t$ should properly cite InterGPS (Line 214). Second, in Line 274, the authors state that \"solving algebraic relations remains out of its scope\". However, in my understanding, AlphaGeometry adopts a DD + AR symbolic reasoning engine, where AR stands for algebraic reasoning. Therefore, the DSR module appears highly similar to the DD + AR reasoning engine of AlphaGeometry, as well as to the hypergraph expansion component. It would be helpful if the authors could clarify the unique innovations of AutoGPS.\n2. Since MPF heavily relies on the diagram parser (PGDPNet), which is trained solely on PGDP and Geometry3K-style data, it may struggle to seamlessly generalize to other styles of geometric diagrams. This limitation further constrains the scalability of AutoGPS.\n3. The compared methods are limited. The authors should include comparisons with more recent neuro-symbolic approaches, such as FGeo-HyperGNet [1].\n\n[1] FGeo-HyperGNet: Geometric Problem Solving Integrating FormalGeo Symbolic System and Hypergraph Neural Network", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nhtiupjm0e", "reviewer": "Reviewer_pNN4", "summary": "This paper presents AutoGPS, a framework for automated geometry problem solving that integrates multimodal formalization with deductive reasoning. Unlike traditional text-only or vision-only systems, AutoGPS unifies textual problem statements and geometric diagrams into a consistent symbolic representation, which is then solved through formal logic-based reasoning. The system includes:\n1. A multimodal formalization module that parses text and visual diagrams into formal symbolic predicates.\n2. A deductive reasoning engine that performs geometric theorem proving through symbolic inference.\n3. A data generation and verification pipeline for training and evaluating multimodal geometry reasoning systems.\n\nExperiments demonstrate AutoGPS’s superiority over both LLM-based and neuro-symbolic baselines on geometry reasoning benchmarks, including synthetic and real-world datasets. The approach significantly improves consistency between textual and visual modalities and exhibits better generalization to novel problem types.", "questions": "As discussed above.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Innovative integration of modalities:\nThe combination of natural language, diagram understanding, and formal reasoning is both technically challenging and conceptually elegant. The multimodal formalization module is well-motivated and executed with a clear architecture.\n\n2. Clear reasoning pipeline:\nThe formal-to-symbolic transition is described thoroughly, including explicit steps for entity detection, relation extraction, and logical grounding. The modular design supports interpretability and verifiability.\n\n3. Strong empirical performance:\nExperimental results demonstrate substantial improvements over both pure neural models (e.g., GPT-4V, Flamingo) and previous neuro-symbolic systems. The inclusion of accuracy, formal consistency, and visual grounding metrics provides a holistic evaluation.", "weaknesses": "1. Limited scalability and automation.\nThe formalization process still relies partly on rule-based heuristics for entity alignment and relation mapping. It is unclear how the system scales to more complex, noisy, or real-world diagrams with ambiguous geometry.\n\n2. Dataset construction bias.\nThe training and evaluation datasets appear to be semi-synthetic or curated from well-structured geometry problems. There is limited discussion on how AutoGPS performs on imperfect or non-standard problem statements often found in educational or competition settings.\n\n3. Insufficient discussion of failure cases.\nThe paper would benefit from analyzing situations where the symbolic representation fails (e.g., misalignment between text and diagram) and discussing how these errors propagate through reasoning.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FTA6kK6qTb", "reviewer": "Reviewer_JksY", "summary": "This paper introduces a novel neuro-symbolic framework for GPS problems, featuring two key components: the Multimodal Problem Formalizer (MPF) and the Deductive Symbolic Reasoner (DSR). These components work collaboratively to address the challenges inherent in GPS problems. Experimental results demonstrate that the proposed method yields very promising outcomes, indicating significant potential in this area of research.", "questions": "Please refer to the `Weakness` section.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "1. This paper is technically sound. The authors introduce two novel modules, the Multimodal Problem Formalizer (MPF) and the Deductive Symbolic Reasoner (DSR), both of which require substantial human effort to develop. These components have the potential to greatly benefit the GPS research community.\n2. The experiments presented in this paper are both promising and comprehensive. The proposed method can be integrated with various base MLLMs, significantly improving their performance, especially in completion tasks.\n3. Overall, the paper is well-written and easy to follow.", "weaknesses": "1. The authors state that MPF and DSR collaborate to formalize the original GPS problem. However, it is unclear whether this process may lead to incorrect problem formulations. The paper would benefit from a discussion and analysis of both successful and failed cases, including a reported success rate for the formalization process. Such information would be valuable for subsequent research to better understand the strengths and limitations of the proposed approach.\n2. The paper does not report the API or inference costs associated with each method. As neuro-symbolic approaches often introduce additional inference and interaction steps when addressing GPS problems, it is important to assess whether these extra costs are justified by the observed performance improvements. Additionally, it would be worthwhile to investigate whether simpler techniques, such as self-consistency, could achieve similar performance gains within comparable evaluation budgets.\n3. The discussion of related work could be expanded. In particular, the paper should cite and analyze earlier neuro-symbolic methods for mathematical reasoning problems, such as [1], [2], [3], and [4]. This would help position the proposed approach within the broader context of existing research.\n\n**Reference** \n\n[1] Zenan Li, Zhi Zhou, Yuan Yao, Xian Zhang, Yu-Feng Li, Chun Cao, Fan Yang, Xiaoxing Ma. **Neuro-Symbolic Data Generation for Math Reasoning.** NeurIPS 2024.\n\n[2] Ning Shang, Yifei Liu, Yi Zhu, Li Lyna Zhang, Weijiang Xu, Xinyu Guan, Buze Zhang, Bingcheng Dong, Xudong Zhou, Bowen Zhang, Ying Xin, Ziming Miao, Scarlett Li, Fan Yang, Mao Yang. **rStar2-Agent: Agentic Reasoning Technical Report.** Arxiv 2025.\n\n[3] Zenan Li, Zhaoyu Li, Wen Tang, Xian Zhang, Yuan Yao, Xujie Si, Fan Yang, Kaiyu Yang, Xiaoxing Ma. **Proving Olympiad Inequalities by Synergizing LLMs and Symbolic Reasoning.** ICLR 2025\n\n[4] Weiming Wu, Zi-kang Wang, Jin Ye, Zhi Zhou, Yu-Feng Li, Lan-Zhe Guo. **NeSyGeo: A Neuro-Symbolic Framework for Multimodal Geometric Reasoning Data Generation.** Arxiv 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)