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REVIEW 3 major objections 5 minor 1 cited by

Enhancing the Geometric Problem-Solving Ability of Multimodal LLMs via Symbolic-Neural Integration

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Symbolically generated geometry data and inference-time verification make multimodal LLMs solve geometry problems more accurately and with fewer hallucinations.

desk verdict Solid symbolic-data-generation engineering, but the headline gains on Geometry3K and PGPS9K may be inflated by train/test overlap that the paper never addresses. read the letter →

arxiv 2504.12773 v1 pith:IYL44EXB submitted 2025-04-17 cs.CL cs.AI

Yicheng Pan , Zhenrong Zhang , Pengfei Hu , Jiefeng Ma , Jun Du , Jianshu Zhang , Quan Liu , Jianqing Gao
show 1 more author
Feng Ma
This is my paper · ORCID
classification cs.CLcs.AI
keywords geometryproblemsolvingmultimodallargelanguagemodelssymbolicreasoningchain-of-thoughtdatasynthesisinference-timeverificationGeoGenLogic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's central claim is that the scarcity of correct step-by-step geometry solutions is the main bottleneck for multimodal large language models on geometry problems, and that a formal symbolic reasoner can remove that bottleneck. GeoGen samples geometric premises, renders diagrams, runs theorem-based forward search to find derivable conclusions, and rewrites the resulting formal proofs into natural-language question-answer pairs. From this pipeline the authors build GeoExpand (about 45k samples over existing benchmark images) and GeoSynth (about 62k samples from freshly synthesized diagrams). They then train GeoLogic, an LLM that translates natural-language reasoning steps into formal statements a symbolic system can verify, and use it to run step-level tree search over a model's candidate reasoning steps at inference. The paper reports consistent accuracy gains across several multimodal LLMs and geometry benchmarks, arguing that symbolic supervision during training and symbolic checking during inference together make geometry reasoning more reliable.

What carries the argument

The load-bearing object is the formal dependency graph produced by GeoGen's Target Finder. Starting from a set of geometry literals, forward search applies every theorem whose premises are satisfied and records each derived literal with its premises; a chosen target is then traced backward through this graph, the subgraph is topologically sorted, and linearized into condition-theorem-conclusion triples. These triples are the common currency of the whole method: they are rewritten into natural-language chain-of-thought solutions for training, they are the supervision that trains GeoLogic to translate natural language back into formal language, and they are the unit that the symbolic system verifies during inference. The Plotter renders the diagrams from the same sampled predicates, so every training image is aligned with the formal knowledge used to solve it.

What would settle it

Inspect the list of 4,849 source images used for GeoExpand: if any official Geometry3K or PGPS9K test image is among them, rerun the GeoGen-SFT experiments with test images excluded and compare the accuracy tables. The presence of even one test image in the expansion sources would directly inflate the reported benchmark gains.

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Extended reading notes

Core claim

On its own terms, this paper establishes that a symbolic geometry engine can serve twice in a multimodal LLM pipeline: as a generator of formally correct reasoning data and as a verifier of the model's own reasoning. GeoGen produces 45,526 question-answer pairs from 4,849 existing images and 62,868 pairs from 129,230 synthesized diagrams, each with multi-step chain-of-thought solutions. Training on this data plus an existing instruction-tuning set improves completion and choice accuracy on the GeoQA, Geometry3K, PGPS9K, and MathVista-mini-GPS benchmarks for several 2B-7B multimodal models; the best 7B model reaches 63.9% completion and 74.0% choice accuracy on MathVista-mini-GPS. GeoLogic, a 3B translator model, converts each predicted natural-language step into a formal condition-theorem-conclusion triple, and the symbolic system checks it; step-level tree search over valid candidates improves accuracy up to a point, with the paper attributing the later decline to the verifier checking necessary but not sufficient conditions.

Load-bearing premise

The load-bearing premise is that the GeoExpand training examples generated from the 4,849 Geometry3K and PGPS9K images are disjoint from the test splits of those same datasets; the paper reports accuracy on those test sets but never states that test images were excluded.

Editorial extensions

If this is right

  • Adding GeoExpand and GeoSynth to the existing instruction-tuning data improves both completion and choice accuracy across several open multimodal LLMs from 2B to 7B parameters.
  • The GeoGen-SFT-7B model reaches 63.9% completion and 74.0% choice accuracy on the MathVista-mini-GPS geometry subset, and 78.0% choice accuracy on GeoQA, the best among the multimodal LLM solvers compared.
  • GeoLogic translates natural-language steps to formal triples with 94.16% exact-match accuracy on the authors' test set, which is what makes inference-time symbolic verification feasible.
  • Step-level tree search with symbolic verification improves accuracy as search width grows from 1 to 4, then declines at larger widths, consistent with the fast verification mode checking necessary conditions only.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct extension would be to apply the same generate-and-verify loop to other multimodal reasoning domains that have formal checkers, such as arithmetic word problems, programming, or physics, by swapping the geometry theorem set for the domain's rules.
  • Because the fast verifier checks only necessary visual conditions, building a sufficiency-checking verifier, or scoring candidate steps by how much they contribute to the final answer, could plausibly reverse the observed accuracy decline at larger search widths.
  • The 94.16% translator accuracy implies roughly one in seventeen formal steps is mistranslated, so improving GeoLogic's translation or adding self-consistency across candidate translations may raise the ceiling of verification more than simply widening the search.
  • If the training-time gains survive a clean train/test split, the pipeline's value may be larger as a data generator than as a verifier, since the supervised fine-tuning gains are consistent across models while the inference-time gains are modest.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper presents GeoGen, a pipeline that automatically generates step-by-step geometry solutions by combining random diagram synthesis, symbolic forward search with traceback, and LLM-based translation into natural language. The pipeline is used to create two training datasets: GeoExpand, with 45,526 QA pairs derived from 4,849 images from Geometry3K and PGPS9K, and GeoSynth, with 62,868 pairs from 129,230 synthesized diagrams. The authors also train GeoLogic, an LLM that translates natural language reasoning steps into formal predicates, and integrate it with a symbolic verifier in a step-level tree search during inference. Fine-tuning Qwen2.5-VL, InternVL2.5, and LLaVA-v1.6 on a mixture of Geo170K, GeoExpand, and GeoSynth yields consistent accuracy gains on GeoQA, Geometry3K, PGPS9K, and MathVista-mini-GPS. The authors report state-of-the-art results among MLLM-based geometry solvers, including 78.0% choice accuracy on GeoQA with GeoGen-SFT-7B.

Significance. If the reported gains are not an artifact of training/test overlap, the paper makes a useful contribution: it demonstrates a fully symbolic route to generating large volumes of chain-of-thought supervision that is guaranteed correct at the formal level, and it shows one way to use formal verification as a post-hoc check on MLLM outputs. Strengths include open-sourced code, experiments across multiple model families, manual verification of 100 translated solutions, and a GeoLogic translation accuracy of 94.16% on a custom test set. The main risk is the lack of any statement about excluding the benchmark test splits from the GeoExpand expansion; this must be resolved before the empirical claims can be accepted.

major comments (3)
  1. [Section 3.2 and Section 4.1.3] The manuscript does not establish that GeoExpand is disjoint from the evaluation sets. GeoExpand is built from 4,849 original images from Geometry3K and PGPS9K (Section 3.2), and the evaluation in Section 4.1.3 uses the standard test splits of those datasets (601 and 1,000 images). No sentence states that test-split images were excluded from the expansion, and no image-level deduplication is reported. If test images are among the GeoExpand sources, the model is trained on the exact diagrams it is later asked to solve, which alone can inflate the Completion and Choice accuracies in Tables 1 and 2 and the MathVista results, since MathVista-mini-GPS also draws from Geometry3K and GeoQA+. The authors should state the split policy, verify non-overlap at the image level, and re-run the experiments with any overlapping test images removed from the training expansion (or report results on a held-out set of images never used in GeoExpand).
  2. [Section 4.1.3] The Completion metric is defined through an LLM-based automatic verifier (Qwen2.5-7B) that judges whether the final three sentences of the response contain the correct numerical answer, but no validation of this verifier is reported. If the verifier is systematically lenient, or if it is affected by the same fine-tuning data, the completion numbers in Tables 1 and 2 may not be comparable across models or to the published results marked with asterisks. The paper should report exact-match or rule-based numerical extraction as a second metric and give verifier agreement on a human-annotated sample.
  3. [Section 4.5 and Figure 4] The evidence for the inference-time search is weak and partly inconsistent. The text states that search width 1 and width 16 are comparable, but Figure 4 shows sampling widths only from 20 to 24 and exhibits small, non-monotonic differences. Moreover, the fast verification mode checks only necessary conditions, which the authors themselves note can admit many invalid steps, and the random selection among valid steps may approximate random sampling. The paper should provide the full accuracy-versus-width curve, including the stated widths 1 and 16, report confidence intervals or significance tests, and clarify whether the observed differences support the claim that verification improves accuracy.
minor comments (5)
  1. [Section 1] There are incomplete and misspelled phrases, for example 'we find that the in MLLMs with symbolic systems' and 'the symblic system'; these typographical errors should be corrected.
  2. [Sections 3.1.2 and 4.3] Typographical errors: 'constructoin' should be 'construction' in Section 3.1.2, and 'evaluation ion GeoTest' should be 'evaluation on GeoTest' in Section 4.3.
  3. [Section 3.2] The sentence 'as shown in Figure 1' refers to a distribution of reasoning steps, but Figure 1 is the framework diagram; no figure showing the step-length distribution is provided.
  4. [Figure 4] The x-axis is labelled 'Sampling Width' with ticks 20-24, which conflicts with the text's discussion of widths 1 and 16; the label, the values, or the text should be harmonized.
  5. [Section 4.1.3] The combined evaluation set is called GeoTest, but Table 2 also reports MathVista-mini-GPS without clarifying whether it is part of GeoTest; the relationship between the two should be stated explicitly.

Circularity Check

1 steps flagged · score 6.0 of 10

GeoExpand is generated from Geometry3K and PGPS9K images while Tables 1–2 score the test sets of those same datasets; without an explicit split-exclusion statement, the reported gains are potentially trained-on-test predictions.

  1. fitted input called prediction [Section 3.2 (Dataset Construction via GeoGen) and Section 4.1.3 (Evaluation Metric)]
    "As a result, we generate over 45,526 QA pairs from 4,849 original images in the Geometry3K and PGPS9K datasets. ... We evaluate model performance on the test sets of GeoQA, Geometry3K, and PGPS9K, which contain 754, 601, and 1,000 samples respectively."

    The SFT corpus GeoExpand is defined by expanding images from Geometry3K and PGPS9K, while Tables 1-2 report accuracy on test subsets of those same datasets. The paper never states that the 601 Geometry3K and 1,000 PGPS9K test images were excluded from the 4,849 source images. If any overlap exists, the model has been trained on the exact diagram it is later asked to solve; those Completion/Choice 'predictions' are re-presentations of training input, not independent generalization. The claimed consistent improvement on these two benchmarks is therefore forced by training/evaluation source identity, not solely by symbolic-neural integration.

full rationale

The only substantive circularity concern is benchmark-level rather than derivation-level. GeoGen's symbolic reasoning engine, target-finder, and GeoLogic translator form an independent pipeline; the formal system's theorems (FormalGeo, expanded to 210 theorems) are not defined in terms of the MLLM outputs, and GeoLogic's translation accuracy is measured against the generator's formal triples. No load-bearing argument reduces to a self-citation. However, the paper's central empirical claim rests on evaluating on Geometry3K and PGPS9K test sets while constructing GeoExpand from the same datasets, and the text contains no sentence declaring that test-split images were withheld. MathVista-mini-GPS is only partially independent, since it also draws from Geometry3K and GeoQA+, and Geo170K (derived from GeoQA+) is included in training. The GeoQA cross-model gains provide some independent grounding because GeoExpand is not built from GeoQA, but the Geometry3K/PGPS9K improvements in Tables 1 and 2 cannot be cleanly interpreted without a split-overlap statement. This is a partial circularity of the fitted-input-called-prediction form, not a self-citation chain or definitional identity; score 6 reflects the partial nature of the circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the correctness of FormalGeo's symbolic engine, the validity of randomly generated diagrams, faithful LLM translation at scale, and, critically, on an unstated guarantee that GeoExpand training images are disjoint from the Geometry3K and PGPS9K test images. The latter is unverified and could change the interpretation of the benchmark results. No new physical or theoretical entities are introduced; GeoLogic is a trained model rather than an invented entity.

free parameters (4)
  • Per-theorem-sequence cap in GeoSynth = 400
    Hand-chosen cap on Q&A pairs per unique theorem sequence to promote diversity (Section 3.2). This affects dataset composition and downstream accuracy.
  • Overly complex node filtering threshold in Target Finder = unspecified
    Nodes deeper in the dependency graph are filtered as too complex; the threshold is not quantified (Section 3.1.2). It shapes the difficulty distribution of generated questions.
  • Search width K for GeoLogic tree search = varied 20 to 24 in Figure 4
    Number of candidate steps sampled per iteration (Algorithm 1). Results in Figure 4 are sensitive to this value, and no principled choice is given.
  • Predicate count limit per synthesized diagram = unspecified
    To keep images clean, the number of predicates per diagram is limited; the exact cap and sampling distribution are not reported (Section 3.2).
assumptions (6)
  • domain assumption FormalGeo's 88 predicates and its theorem set correctly encode plane geometry and produce sound conclusions.
    GeoGen's 'guaranteed correctness' rests on the formal system; Sections 3.1.2 and 4.1.2.
  • domain assumption Randomly sampled predicate combinations always yield consistent, non-degenerate diagrams.
    Plotter assigns coordinates under constraints; no rejection or degeneracy handling is described in Section 3.1.1.
  • domain assumption Qwen2.5-32B-Instruct's translation of formal reasoning paths into natural language preserves logical content.
    Only 100 samples were manually checked (Section 4.1.2), so this is an assumption for the remaining 45k plus 62k samples.
  • domain assumption GeoExpand source images from Geometry3K and PGPS9K do not include the test images used in Tables 1 and 2.
    Section 3.2 creates GeoExpand from those datasets; Section 4.1.3 evaluates on their test sets; no exclusion is stated.
  • domain assumption GeoLogic's formal translations are accurate enough that symbolic verification reflects whether MLLM steps are correct.
    Reported 94.16% exact-match accuracy on a custom test set (Section 4.5); the remaining 5.84% translation errors propagate into verification.
  • domain assumption The Qwen2.5-7B answer verifier correctly judges completion answers from the final three sentences.
    Completion accuracy in Tables 1 and 2 relies on this LLM checker (Section 4.1.3).

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Cite this review

Pith. "Pith review of Enhancing the Geometric Problem-Solving Ability of Multimodal LLMs via Symbolic-Neural Integration." pith.science (2026). https://pith.science/paper/IYL44EXB

@misc{pith2026250412773,
  author       = {Pith},
  title        = {Pith review of: Enhancing the Geometric Problem-Solving Ability of Multimodal LLMs via Symbolic-Neural Integration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IYL44EXB}},
  note         = {Machine review of arXiv:2504.12773}
}
read the original abstract

Recent advances in Multimodal Large Language Models (MLLMs) have achieved remarkable progress in general domains and demonstrated promise in multimodal mathematical reasoning. However, applying MLLMs to geometry problem solving (GPS) remains challenging due to lack of accurate step-by-step solution data and severe hallucinations during reasoning. In this paper, we propose GeoGen, a pipeline that can automatically generates step-wise reasoning paths for geometry diagrams. By leveraging the precise symbolic reasoning, \textbf{GeoGen} produces large-scale, high-quality question-answer pairs. To further enhance the logical reasoning ability of MLLMs, we train \textbf{GeoLogic}, a Large Language Model (LLM) using synthetic data generated by GeoGen. Serving as a bridge between natural language and symbolic systems, GeoLogic enables symbolic tools to help verifying MLLM outputs, making the reasoning process more rigorous and alleviating hallucinations. Experimental results show that our approach consistently improves the performance of MLLMs, achieving remarkable results on benchmarks for geometric reasoning tasks. This improvement stems from our integration of the strengths of LLMs and symbolic systems, which enables a more reliable and interpretable approach for the GPS task. Codes are available at https://github.com/ycpNotFound/GeoGen.

Figures

Figures reproduced from arXiv: 2504.12773 by the authors.

Figure 1
Figure 1. Framework of our proposed integration methods of symbolic system and MLLMs. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 1
Figure 1. During the training phase, we conduct extensive SFT [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Framework of our proposed GeoGen pipeline. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figures from the paper (3 more)
Figure 3
Figure 3. Figure 3: We conduct ablation study to examine the impact [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]
Figure 4
Figure 4. Figure 4: Accuracy trends as we vary the search width during [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: A typical case with model predictions improving as [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Towards Geometry Problem Solving in the Large Model Era: A Survey

    cs.CV 2025-06 conditional novelty 4.0 of 10

    A survey that organizes geometry problem-solving research into benchmark construction, parsing, and reasoning, and proposes a unified parse-then-reason paradigm for the large-model era.

Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.