Solving Algebraic Problems with Geometry Diagrams (APGDs) poses a significant challenge in artificial intelligence due to the complex and diverse geometric relations among geometric objects. Problems typically involve both textual descriptions and geometry diagrams, requiring a joint understanding of these modalities. Although considerable progress has been made in solving math word problems, research on solving APGDs still cannot discover implicit geometry knowledge for solving APGDs, which limits their ability to effectively solve problems. In this study, a systematic and modular three-phase scheme is proposed to design an algorithm for solving APGDs that involve textual and diagrammatic information. The three-phase scheme begins with the application of the state-transformer paradigm, modeling the problem-solving process and effectively representing the intermediate states and transformations during the process. Next, a generalized APGD-solving approach is introduced to effectively extract geometric knowledge from the problem’s textual descriptions and diagrams. Finally, a specific algorithm is designed focusing on diagram understanding, which utilizes the vectorized syntax-semantics model to extract basic geometric relations from the diagram. A method for generating derived relations, which are essential for solving APGDs, is also introduced. Experiments on real-world datasets, including geometry calculation problems and shaded area problems, demonstrate that the proposed diagram understanding method significantly improves problem-solving accuracy compared to methods relying solely on simple diagram parsing.

The first step in solving geometry problems is to understand problems, and automatic understanding of geometry problems by computers has always been a challenge due to the massive advanced knowledge implied in the text and diagram. This paper proposes a method for geometry problem understanding based on vectorized Syntax-Semantics (S ² ) model. The proposed method divides the understanding of geometry problems into three parts. Firstly, we modified and optimized vectorized S ² model for understanding explicit arithmetic word problems, and applied it to the text understanding of geometry problems to extract basic geometric relations. Then, based on the idea that a diagram is an extension of problem text, we designed vectorized S ² model of diagram understanding according to the same framework as that of text understanding. All geometry diagrams are transformed into vectors for understanding in a uniform way. Finally, we designed a derived relations generation model based on the diagramet theory to extract derived geometric relations from the basic relations. Experimental results show that the proposed method is effective in understanding geometry problems with diagrams.

Solving arithmetic word problems that entail deep implicit relations is still a challenging problem. However, significant progress has been made in solving Arithmetic Word Problems (AWP) over the past six decades. This paper proposes to discover deep implicit relations by qualia inference to solve Arithmetic Word Problems entailing Deep Implicit Relations (DIR-AWP), such as entailing commonsense or subject-domain knowledge involved in the problem-solving process. This paper proposes to take three steps to solve DIR-AWPs, in which the first three steps are used to conduct the qualia inference process. The first step uses the prepared set of qualia-quantity models to identify qualia scenes from the explicit relations extracted by the Syntax-Semantic (S²) method from the given problem. The second step adds missing entities and deep implicit relations in order using the identified qualia scenes and the qualia-quantity models, respectively. The third step distills the relations for solving the given problem by pruning the spare branches of the qualia dependency graph of all the acquired relations. The research contributes to the field by presenting a comprehensive approach combining explicit and implicit knowledge to enhance reasoning abilities. The experimental results on Math23K demonstrate hat the proposed algorithm is superior to the baseline algorithms in solving AWPs requiring deep implicit relations.

In recent years, there has been a significant increase in the design of neural network models for solving math word problems (MWPs). These neural solvers have been designed with various architectures and evaluated on diverse datasets, posing challenges in fair and effective performance evaluation. This paper presents a comparative study of representative neural solvers, aiming to elucidate their technical features and performance variations in solving different types of MWPs. Firstly, an in-depth technical analysis is conducted from the initial deep neural solver DNS to the state-of-the-art GPT-4. To enhance the technical analysis, a unified framework is introduced, which comprises highly reusable modules decoupled from existing MWP solvers. Subsequently, a testbed is established to conveniently reproduce existing solvers and develop new solvers by combing these reusable modules, and finely regrouped datasets are provided to facilitate the comparative evaluation of the designed solvers. Then, comprehensive testing is conducted and detailed results for eight representative MWP solvers on five finely regrouped datasets are reported. The comparative analysis yields several key findings: (1) Pre-trained language model-based solvers demonstrate significant accuracy advantages across nearly all datasets, although they suffer from limitations in math equation calculation. (2) Models integrated with tree decoders exhibit strong performance in generating complex math equations. (3) Identifying and appropriately representing implicit knowledge hidden in problem texts is crucial for improving the accuracy of math equation generation. Finally, the paper also discusses the major technical challenges and potential research directions in this field. The insights gained from this analysis offer valuable guidance for future research, model development, and performance optimization in the field of math word problem solving.