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E-GPS: Explainable Geometry Problem Solving via Top-Down Solver and Bottom-Up Generator
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Proposes E-GPS, a symbolic geometry problem solver that searches top-down from the problem target to produce explainable, short solution steps, and uses a bottom-up generator to augment training data for a transformer theorem predictor.
Problem
Existing geometry problem solvers lack explainable reasoning and solving steps, and small, incompletely annotated datasets hinder learning of geometric theorem knowledge.
Approach
Solve geometry problems top-down by decomposing the target into child targets via theorem premises, so every condition serves the goal; generate augmented well-annotated problems bottom-up by forward-applying theorems to train a better theorem predictor.
Key results
- E-GPS reaches 64.7% accuracy on Geometry3K with 3.49-4.19 average theorem steps versus Inter-GPS 57.5% with 7.10 steps (KB1).
- With KB2, E-GPS* achieves 67.9% accuracy at 1.63-2.28 steps versus GeoDRL 68.4% at 2.34 steps.
- On aug-Geo3K one-step theorem prediction, E-GPS reaches R@5 of 88.9% versus Inter-GPS 59.5%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use E-GPS instead of Inter-GPS with KB1 | Geometry problem solving · Geometry3K | average steps: 7.10 -> 3.49∼4.19 | Helped |
| use E-GPS with KB2 instead of GeoDRL* | Geometry problem solving · Geometry3K | accuracy: 68.4 -> 67.9 | Hurt |
| use E-GPS with KB2 instead of GeoDRL* | Geometry problem solving · Geometry3K | average steps: 2.34 -> 1.63∼2.28 | Helped |
| use E-GPS with ground-truth parsing instead of Inter-GPS with ground-truth parsing | Geometry problem solving · Geometry3K, KB1 | accuracy: 78.3 -> 79.8 | Helped |
| use E-GPS with ground-truth parsing instead of GeoDRL* with ground-truth parsing | Geometry problem solving · Geometry3K, KB2 | accuracy: 89.4 -> 89.8 | Helped |
| train the theorem predictor with BU-PG-augmented data instead of Inter-GPS's training setup | E-GPS theorem predictor · aug-Geo3K, one-step theorem prediction | R@5: 59.5 -> 88.9 | Helped |
| train the theorem predictor with BU-PG-augmented data instead of Inter-GPS's training setup | E-GPS theorem predictor · aug-Geo3K, multi-step theorem prediction | R@5: 31.7 -> 80.3 | Helped |
| use one-step theorem prediction instead of theorem traversal (ablation) | E-GPS · Geometry3K, KB1 | accuracy: 77.0 -> 79.8 | Helped |
| use one-step theorem prediction instead of theorem traversal (ablation) | E-GPS · Geometry3K, KB1 | application search: 22.3 -> 11.5 | Helped |
| use one-step theorem prediction instead of random theorem selection (ablation) | E-GPS · Geometry3K, KB1 | accuracy: 78.1 -> 79.8 | Helped |
| use one-step theorem prediction instead of multi-step theorem prediction (ablation) | E-GPS · Geometry3K, KB1 | accuracy: 79.6 -> 79.8 | No clear effect |
| use one-step theorem prediction instead of multi-step theorem prediction (ablation) | E-GPS · Geometry3K, KB1 | application search: 12.0 -> 11.5 | Helped |
Surprising findings
- E-GPS's accuracy on Geometry3K (64.7%) is below Inter-GPS-style comparisons only slightly and falls short of GeoDRL with KB2 at 68.4% vs 67.9%, so the method trades a small accuracy gap for fewer steps.
Limitations
- Does not guarantee the shortest solution; sometimes selects longer theorem chains (e.g., Pythagoras and Sine instead of Cosine).
- Performance depends on diagram and text parsing quality; fails on ambiguous shaded regions or shape ambiguity.
- Relies on a predefined theorem knowledge base, which limits coverage.
Experimental setup
Datasets: Geometry3K aug-Geo3K
Benchmarks: Geometry3K aug-Geo3K
Metrics: accuracy average steps R@1 R@3 R@5
Baselines: FiLM FiLM-BERT FiLM-BART Inter-GPS GeoDRL
Builds on: Inter-GPS GeoDRL PGDPNet
Closest papers by meaning
- Hilbert-Geo: Solving Solid Geometric Problems by Neural-Symbolic Reasoning (CVPR 2026 · similarity 0.86)
- GeoX: Geometric Problem Solving Through Unified Formalized Vision-Language Pre-training (ICLR 2025 · similarity 0.85)
- Geoint-R1: Formalizing Multimodal Geometric Reasoning with Dynamic Auxiliary Constructions (CVPR 2026 · similarity 0.84)
- GeoBench: Rethinking Multimodal Geometric Problem-Solving via Hierarchical Evaluation (ICLR 2026 · similarity 0.84)
- GNS: Solving Plane Geometry Problems by Neural-Symbolic Reasoning with Multi-Modal LLMs (AAAI 2025 · similarity 0.83)
- GeoDial: A Multimodal Dialog Tutoring Dataset for Geometry Problem-Solving with Visual Tutor Turns (NeurIPS 2026 · similarity 0.82)
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