Translating a Math Word Problem to an Expression Tree

By Lei Wang, Yan Wang, Deng Cai, Dongxiang Zhang, Xiaojiang Liu. 2018

179 citations recorded by the source corpus.

Sequence-to-sequence (SEQ2SEQ) models have been successfully applied to automatic math word problem solving. Despite its simplicity, a drawback still remains: a math word problem can be correctly solved by more than one equations. This non-deterministic transduction harms the performance of maximum likelihood estimation. In this paper, by considering the uniqueness of expression tree, we propose an equation normalization method to normalize the duplicated equations. Moreover, we analyze the performance of three popular SEQ2SEQ models on the math word problem solving. We find that each model has its own specialty in solving problems, consequently an ensemble model is then proposed to combine their advantages. Experiments on dataset Math23K show that the ensemble model with equation normalization significantly outperforms the previous state-of-the-art methods.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Natural Language Processing

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Citations in the corpus

MWPToolkit: An Open-Source Framework for Deep Learning-Based Math Word Problem Solvers (2021, methodology): “Furthermore, we implement and compare 17 MWP solving models, including Seq2Seq, Seq2tree, Graph2Tree, and commonly-used non-pretrained models and pretrained models, on 4 widely-used single equation generation benchmark datasets and 2 multiple equations generation benchmarks.”
Generate & Rank: A Multi-task Framework for Math Word Problems (2021, neutral): “DNS (Wang et al., 2017) is the ﬁrst to apply vanilla RNN-based models to MWP. Math-EN (Wang et al., 2018b) introduces equation normalization and compares three Seq2Seq models on MWP solving.”
DREAM: A Challenge Data Set and Models for Dialogue-Based Reading Comprehension (2019, background): “• Co-Matching This state-of-the-art multiple-choice reading comprehension model explicitly treats question and answer option as two sequences and jointly matches them against a given document (Wang et al., 2018b).”
A Goal-Driven Tree-Structured Neural Model for Math Word Problems (2019, methodology): “Researchers have proposed several approaches [Ling et al. , 2017; Wang et al. , 2017; Huang et al. , 2018; Wang et al. , 2018a] to solving MWPs based on the Seq2Seq model [Sutskever et al. , 2014].”
Modeling Intra-Relation in Math Word Problems with Different Functional Multi-Head Attentions (2019, methodology): “, we normalize the equations to postfix expressions following the rules in Wang et al. (2018a), ensuring that every problem is corresponding to a unique equation.”
Template-Based Math Word Problem Solvers with Recursive Neural Networks (2019, methodology): “(Wang et al. 2018b) modeled the math expression as a tree structure and made the ﬁrst attempt to apply deep reinforcement learning for iterative tree construction.”
Graph-to-Tree Learning for Solving Math Word Problems (2020, methodology): “Some researchers have proposed seq2seq models to improve solution expression generation using implicit (Wang et al., 2018; Chiang and Chen, 2019) and explicit (Wang et al.”
A Knowledge-Aware Sequence-to-Tree Network for Math Word Problem Solving (2020, methodology): “Wang et al. (2018b) incorporated reinforcement learning into the model to construct a math expression step by step.”
Investigating Math Word Problems using Pretrained Multilingual Language Models (2021, methodology): “How to use machine learning and NLP to solve Math Word Problems (MWPs) has attracted much attention in recent years (Hosseini et al., 2014; Kushman et al., 2014; Roy et al., 2015; Ling et al., 2017; Wang et al., 2017, 2018; Amini et al., 2019).”
GeoQA: A Geometric Question Answering Benchmark Towards Multimodal Numerical Reasoning (2021, background): “…problems, including traditional machine learning approaches (Kushman et al., 2014; Zhou et al., 2015; Huang et al., 2016) and network-based models (Wang et al., 2017, 2018; Xie and Sun, 2019), while solving geometric problems has been rarely investigated (Seo et al., 2014, 2015a; Sachan et al.,…”
Neural-Symbolic Solver for Math Word Problems with Auxiliary Tasks (2021, background): “…methods (Shi et al., 2015; Koncelkedziorski et al., 2015; Huang et al., 2017; Liang et al., 2018a), to deep learning meth-ods (Ling et al., 2017; Wang et al., 2017, 2018b; Huang et al., 2018; Wang et al., 2018a; Xie and Sun, 2019; Wang et al., 2019; Zhang et al., 2020a,b; Qin et al., 2020; Shen…”
Learning by Fixing: Solving Math Word Problems with Weak Supervision (2020, methodology): “Researchers have recently focused on solving MWPs using neural-symbolic models (Ling et al. 2017; Wang, Liu, and Shi 2017; Huang et al. 2018; Wang et al. 2018; Xie and Sun 2019).”
Ape210K: A Large-Scale and Template-Rich Dataset of Math Word Problems (2020, methodology): “Seq2SeqET (Wang et al., 2018) uses the same method as DNS, but changes the output sequence to an expression tree.”
Large Language Models are Better Reasoners with Self-Verification (2022, methodology): “Early work attempted to solve complex reasoning tasks using Seq2Seq models (Wang et al., 2018; Li et al., 2019).”
HMS: A Hierarchical Solver with Dependency-Enhanced Understanding for Math Word Problem (2021, methodology): “Better results can be achieved with better modeling and representations of complex problem structures (Wang et al. 2018a).”
Tree-structured Decoding for Solving Math Word Problems (2019, methodology): “…et al., 2018) 47.2% Classiﬁcation Self-Attention (Robaidek et al., 2018) 56.8% Bi-LSTM (Robaidek et al., 2018) 57.9% Generation DNS (Wang et al., 2017) 58.1% BiLSTM+Sufﬁx+EN (Wang et al., 2018a) 66.7% Semantically-Aligned † (Chiang and Chen, 2018) 65.8% T-RNN (Wang et al., 2019) 66.9% Ours 69.0%”
Inter-GPS: Interpretable Geometry Problem Solving with Formal Language and Symbolic Reasoning (2021, background): “, 2019) and expression trees (Wang et al., 2018; Qin et al., 2020; Hong et al., 2021).”
A Bottom-Up DAG Structure Extraction Model for Math Word Problems (2021, methodology): “For Seq2Seq models, as the model will result in different result for a+b and b+a, some methods apply “equation normalization” which explicitly adjusts the order of operands in the equation (Wang et al. 2018) in the training data to get a consistent order.”
Solving Math Word Problems with Teacher Supervision (2021, methodology): “On the other hand, the decoder in Seq2Seq is popularly constructed as tree-based math expression generators [Wang et al. , 2018; Liu et al. , 2019].”
A Survey of Deep Learning for Mathematical Reasoning (2022, unknown): “For instance, deep learning-based math word problem solvers have adopted a sequence-to-sequence framework with attention mechanisms to generate mathematical expressions as intermediate steps (Wang et al., 2018a; Chiang and Chen, 2019).”