GAPS: Geometry-Aware Problem Solver

TL;DR

GAPS introduces a geometry-aware problem solver that jointly handles geometry CAL and PRV problems via a unified solution-program generator, guided by a problem-type classifier and a joint diagram-text encoder. It advances geometry reasoning by separately generating operators and operands from domain-specific languages, and by enhancing geometry element recognition through a geometry elements enhancement technique, plus a Hierarchical Beam Search to explore operator-operand combinations. Empirical results on UniGeo and PGPS9K show substantial gains, most notably a 5.3% CAL improvement and a 41.1% PRV improvement over Geoformer, with PRV accuracy reaching 97.5%. The work demonstrates that a single generator with a discriminative problem-type module, along with targeted enhancements and data augmentation, can robustly solve diverse geometry problems and guide future multi-type geometry reasoning systems.

Abstract

Geometry problem solving presents a formidable challenge within the NLP community. Existing approaches often rely on models designed for solving math word problems, neglecting the unique characteristics of geometry math problems. Additionally, the current research predominantly focuses on geometry calculation problems, while overlooking other essential aspects like proving. In this study, we address these limitations by proposing the Geometry-Aware Problem Solver (GAPS) model. GAPS is specifically designed to generate solution programs for geometry math problems of various types with the help of its unique problem-type classifier. To achieve this, GAPS treats the solution program as a composition of operators and operands, segregating their generation processes. Furthermore, we introduce the geometry elements enhancement method, which enhances the ability of GAPS to recognize geometry elements accurately. By leveraging these improvements, GAPS showcases remarkable performance in resolving geometry math problems. Our experiments conducted on the UniGeo dataset demonstrate the superiority of GAPS over the state-of-the-art model, Geoformer. Specifically, GAPS achieves an accuracy improvement of more than 5.3% for calculation tasks and an impressive 41.1% for proving tasks. Notably, GAPS achieves an impressive accuracy of 97.5% on proving problems, representing a significant advancement in solving geometry proving tasks.

Figures (7)

• Figure 1: Two geometry math problems and related solution programs from the UniGeo dataset. Particularly, the problems in the blue rectangle box belongs to the calculation problem, whereas the problems in the orange rectangle box belongs to the proving problem. The operand "C_x" refers to the x-th constant, the operand "N_x" refers to the x-th numbers in the problem text, "V_x" refers to the results of the previous sub-program that at the x-th index of the total program, and the operand "E_x" refers to the x-th geometry element from the problem text. The operator "R_x" refers to the x-th pre-defined theorem.
• Figure 2: First, the diagram is divided into patches, and along with the problem text, both are transformed into vectors. These vectors are then concatenated and passed to the joint diagram and problem encoder, producing the joint representations denoted as $H$. Subsequently, the geometry-specific program generator starts to generate the solution program, each solution program is represented by several sub-program $r^{i}_{sub}$. Each sub-program $r^{i}_{sub}$ contains one operator $op_{i}$ and corresponding operands $\{oe^{i}\}$, where the query vector $q^{op}$ is used for generating the operator $op^{i}$, and the query vector $q^{oe}$ is used for generating operands in the sub-program. After each sub-program is generated, the corresponding cache token is updated by replacing its vector by the $q^{oe}_{t}$, which is the query vector used for generating the last operand in the sub-program. Specifically, $P_{type}$ is used to produce the mask to multiply with the probabilities of operators and operands, thus influencing GAPS to select $op$ and $oe$ from the domain-specific language (DSL) of the corresponding problem type. This mechanism enables GAPS to adapt and solve different kinds of geometry math problems effectively.
• Figure 3: An example of normalized representation of the solution program.
• Figure 4: Overall results comparison in top-10 accuracy between GAPS and baselines on all sub-tasks by the radar chart.
• Figure 5: (a) Numbers of occurring indices of correct predictions in top-10 predictions candidates. We discard 2th, 3th, and 7th indices because numbers are zero. (b) Distributions of number of operands following each operator in ground truth solution programs from CAL problems and PRV problems. (c) The number of incorrect predictions due to wrong operators or wrong operands generated by GAPS in all sub-tasks. The "Angle", "Length", and "Other" belong to CAL problems, and the others belong to PRV problems.