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<!DOCTYPE html>
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content="A Multi-Modal Neural Geometric Solver with Textual Clauses Parsed from Diagram">
<meta name="keywords" content="Geometry problem solving, PGPSNet, Multi-Modal ">
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<title>PGPSNet: A Multi-Modal Neural Geometric Solver with Textual Clauses Parsed from Diagram</title>
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<h1 class="title is-1 publication-title">A Multi-Modal Neural Geometric Solver with Textual Clauses Parsed from Diagram</h1>
<div class="is-size-5 publication-authors">
<span class="author-block">
<a href="https://mingliangzhang2018.github.io/">Mingliang Zhang</a><sup>1,2</sup>,
</span>
<span class="author-block">
<a href="https://nlpr.ia.ac.cn/pal/People/YinFei.html">Fei Yin</a><sup>1,2</sup>,
</span>
<span class="author-block">
<a href="https://nlpr.ia.ac.cn/liucl/">Chenglin Liu</a><sup>1,2</sup>,
</span>
</div>
<div class="is-size-5 publication-authors">
<span class="author-block"><sup>1</sup>MAIS, Institute of Automation of Chinese Academy of Sciences,</span>
<span class="author-block"><sup>2</sup>School of Artifcial Intelligence, University of Chinese Academy of Sciences</span>
<br>
<span class="paper-block"><b style="color:#f41c1c">IJCAI 2023</b> </span>
</div>
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<a href="https://arxiv.org/pdf/2302.11097"
class="external-link button is-normal is-rounded is-dark">
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</span>
<span>Paper</span>
</a>
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<a href="https://arxiv.org/abs/2302.11097"
class="external-link button is-normal is-rounded is-dark">
<span class="icon">
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</span>
<span>arXiv</span>
</a>
</span>
<!-- Video Link. -->
<span class="link-block">
<a href="https://github.com/mingliangzhang2018/PGPS"
class="external-link button is-normal is-rounded is-dark">
<span class="icon">
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</span>
<span>code</span>
</a>
</span>
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<span class="link-block">
<a href="https://nlpr.ia.ac.cn/databases/CASIA-PGPS9K/"
class="external-link button is-normal is-rounded is-dark">
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<i class="far fa-images"></i>
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<span>dataset</span>
</a>
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<section class="hero is-light is-small">
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<!-- <h1 class="title is-1 mathvista"><img src="static/images/mathvista.png" style="width:1.0em;vertical-align: middle" alt="Logo"/>MathVista Dataset</h1> -->
<h1 class="title is-1 mathvista">
<!--<img src="static/images/mathvista.png" style="width:1em;vertical-align: middle" alt="Logo"/>-->
<span class="mathvista" style="vertical-align: middle">Introduction</span>
</h1>
</div>
</section>
<section class="section">
<div class="container" style="margin-bottom: 2vh;">
<!-- Abstract. -->
<div class="columns is-centered has-text-centered">
<div class="column is-four-fifths">
<div class="content has-text-justified">
<p>
<b>Geometry problem solving (GPS)</b> involves high-level mathematical reasoning, necessitating the ability to fuse <b>multi-modal</b> information and apply geometric knowledge. Recent neural solvers have shown great potential in this area but struggle with diagram presentation and modal fusion. To address these challenges, we introduce <b>PGPSNet</b>, a neural solver that effectively converts diagrams into basic textual clauses to describe their features. <b>PGPSNet</b> integrates multi-modal information through <b>structural and semantic pre-training</b>, <b>data augmentation</b>, and <b>self-limited decoding</b>. This endows it with rich knowledge of geometric theorems and representations, enhancing geometric understanding and reasoning.
</p>
<div id="results-carousel" class="carousel results-carousel">
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images/PGPSNet.png" alt="geometric reasoning" width="50%"/>
<p> Overview of <b>PGPSNet</b> solver. <b>PGPSNet</b> is a multi-modal
learning framework whose modal inputs contain not only the diagram and textual problem, but also the textual clauses parsed from
diagram. It generates the theorem-based interpretable solution program to solve geometry problem.
</p>
</div>
</div>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images/pipeline.png" alt="comparison" width="100%"/>
<p>Pipeline of structural and semantic pre-training. [M] denotes the mask token. <br>Class tags of [G], [N], [ARG], [P], [ANGID] represent
tokens of general, variable, argument, point and angle ID, respectively. <br>Section tags of [S], [C], [T] refer to tokens of structure, condition and
target, respectively.
</p>
</div>
</div>
</div>
<br><br>
<p>
To support GPS research, we have constructed a large-scale, finely-annotated dataset named <b>PGPS9K</b>, which includes detailed diagram annotations and interpretable solution programs. PGPS9K dataset has fve properties:
<ul class="custom-list">
<li><b>Theorem-based:</b> Solving problems in PGPS9K need to apply geometric theorem knowledge to carry out algebraic calculation and get numerical results finally.</li>
<li><b>Diagram-dependent:</b> Above 90% of problems must be solved using the diagrams because necessary conditions such as variable content and geometric structure are displayed via visual form instead of text.</li>
<li><b>Abstract:</b> The diagram is integrated with basic geometric primitives (point, line, circle) and non-geometric primitives (text, symbol). No complex semantic scenarios are involved in textual problem except abstract geometric conditions.</li>
<li><b>Fine-grained:</b> Problems with the same diagram vary in conditions or targets. Slight distinctions in textual problems usually lead to completely different solutions to problems.</li>
<li><b>Condition-redundancy:</b> Lots of conditions in semantic clauses or textual problem are not needed in problem solving at hand.</li>
</ul>
</div>
</div>
</div>
<!--/ Abstract. -->
</div>
</section>
<!-- DATASET SECTION -->
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<!-- <h1 class="title is-1 mathvista"><img src="static/images/mathvista.png" style="width:1.0em;vertical-align: middle" alt="Logo"/>MathVista Dataset</h1> -->
<h1 class="title is-1 mathvista">
<!--<img src="static/images/mathvista.png" style="width:1em;vertical-align: middle" alt="Logo"/>-->
<span class="mathvista" style="vertical-align: middle">PGPS9K Dataset</span>
</h1>
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</section>
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<div class="columns is-centered has-text-centered">
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<h2 class="title is-3">Introduction</h2>
<div class="content has-text-justified">
<p> -->
<section class="section">
<div class="container">
<div class="columns is-centered has-text-centered">
<!-- <div class="column is-full-width has-text-centered"> -->
<div class="column is-four-fifths">
<h2 class="title is-3">Overview</h2>
<div class="content has-text-justified">
<p>
The <b>Plane Geometry Problem Solving Dataset (PGPS9K)</b> was constructed by the State Key Laboratory of Multimodal Artificial Intelligence Systems (MAIS), <a href="http://www.ia.cas.cn/">Institute of Automation of Chinese Academy of Sciences</a>(CASIA). The samples in <b>PGPS9K</b> are labeled with both fine-grained diagram annotation and interpretable solution program, where the diagram annotation is converted into structural clauses and semantic clauses to effectively describe multi-level information in geometry diagram.
</p>
<p>
<span class="mathvista">PGPS9K</span> is composed of 9,022 geometry problems paired with non-duplicate 4,000 geometry diagrams, where 2,891 problems paired with 1,738 diagrams are selected from <a href="https://lupantech.github.io/inter-gps/">Geometry3K dataset</a>, the rest of problems are collected from five popular textbooks across grades 6-12 on <a href="https://www.mheducation.com/">mathematics curriculum websites</a>. Our <b>PGPS9K</b> is divided into 30 problem types as exhibited in the following figure, covering almost all problem types of plane geometry problem in corresponding grades.
</p>
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<img src="static/images/distributions.png" alt="distribution" width="80%"/>
<p> Distribution of problem types of PGPS9K dataset.</p>
</div>
</div>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images/Comparison.png" alt="comparison" width="100%"/>
<p>Comparison with existing GPS datasets. <br>Type, OP and PL represent problem type, operator number and program length, respectively.
</p>
</div>
</div>
</div>
<!-- <div class="content has-text-centered">
<img src="static/images/source_dataset.png" alt="geometric reasoning" width="60%"/>
<p> Summary of the 31 different source datasets in <img src="static/images/mathvista.png" style="width:1.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">MathVista</span>.
</p>
</div>
<div class="content has-text-centered">
<img src="static/images/our_new_3_datasets.png" alt="geometric reasoning" width="70%"/>
<p> Examples of our newly annotated datasets: IQTest, FunctionQA, and PaperQA.</p>
</div> -->
</div>
</div>
</div>
<div class="columns is-centered m-6">
<div class="column is-full has-text-centered content">
<h2 class="title is-3">Examples</h2>
<p>
The annotations of <b>PGPS9K</b> include diagram annotation and solution program, where the diagram annotation is to extract structural and semantic information in diagram and the solution program defines the solution steps of problem. Diagram annotation, textual clauses and solution program are shown as follows.
</p>
<div id="results-carousel" class="carousel results-carousel">
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images/examples/example.png" alt="example" width="80%"/>
<p>Example presentation of PGPS9K dataset.</p>
</div>
</div>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images/examples/templates.png" alt="example" width="100%"/>
<p> Templates of textual clauses. The symbols of &, *, $, % denote point, line, variable and angle ID, respectively</p>
</div>
</div>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images/examples/solution program.png" alt="example" width="45%"/>
<p>Annotation of solution program and its interpretability</p>
</div>
</div>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images/examples/oprator.png" alt="example" width="100%"/>
<p>Program sets defined in our solution program, consisting of 34 operators and 55 operands, <br>where operands involve 11 problem
variables, 7 process variables, 26 augments and 11 constants</p>
</div>
</div>
</div>
<br><br>
</div>
</div>
<!-- <div class="columns is-centered m-6">
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<h2 class="title is-3">Statistics</h2>
<p>Notable statistics of <img src="static/images/mathvista.png" style="width:1.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">MathVista</span></p>
<div id="results-carousel" class="carousel results-carousel">
<div class="box m-5">
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<img src="static/images/statistics/context-dist.png" alt="context" class="stats-image">
<p>Distribution of visual context types within <img src="static/images/mathvista.png" style="width:1.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">MathVista</span></p>
</div>
</div>
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<div class="content has-text-centered">
<img src="static/images/statistics/category.png" alt="category" class="stats-image"/>
<p>Category distribution of problems within <img src="static/images/mathvista.png" style="width:1.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">MathVista</span></p>
</div>
</div>
<div class="box m-5">
<div class="content has-text-centered">
<img src="static/images/statistics/grade-levels.png" alt="grade" class="stats-image"/>
<p>Distribution of questions across different grade levels within <img src="static/images/mathvista.png" style="width:1.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">MathVista</span></p>
</div>
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<p>Distribution of the number of words per question in <img src="static/images/mathvista.png" style="width:1.0em;vertical-align: middle" alt="Logo"/>
<span class="mathvista">MathVista</span>.
Questions with a length greater than 60 are categorized as 61 for visualization simplicity
</p>
</div>
</div>
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<h2 class="title is-3">Numerical answer accuracies of state-of-the-art GPS solvers.</h2>
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<table class="js-sort-table" id="results">
<thead>
<tr>
<th class="js-sort-number" style="vertical-align: middle; color: gray;">Method</th>
<th colspan="3" style="color: gray;">Geometry3K</th>
<th colspan="3" style="color: gray;">PGPS9K</th>
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<tr>
<th></th>
<th class="js-sort-number" style="color: gray;">Completion</th>
<th class="js-sort-number" style="color: gray;">Choice</th>
<th class="js-sort-number" style="color: gray;">Top-3</th>
<th class="js-sort-number" style="color: gray;"><b>Completion<b></th>
<th class="js-sort-number" style="color: gray;">Choice</th>
<th class="js-sort-number" style="color: gray;">Top-3</th>
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<tr>
<td>Human Expert [Lu et al., 2021]</td>
<td>-</td>
<td>90.9</td>
<td>-</td>
<td>-</td>
<td>-</td>
<td>-</td>
</tr>
<tr>
<td>Baseline (Neural Solver) [Lu et al., 2021]</td>
<td>-</td>
<td>35.9</td>
<td>-</td>
<td>-</td>
<td>-</td>
<td>-</td>
</tr>
<tr>
<td>InterGPS (Predict)* [Lu et al., 2021]</td>
<td>44.6</td>
<td>56.9</td>
<td>-</td>
<td>-</td>
<td>-</td>
<td>-</td>
</tr>
<tr>
<td>InterGPS (Diagram GT)* [Lu et al., 2021]</td>
<td>64.2</td>
<td>71.7</td>
<td>-</td>
<td>-</td>
<td>-</td>
<td>-</td>
</tr>
<tr>
<td>InterGPS (All GT)* [Lu et al., 2021]</td>
<td><b class="best-score-text">69.0</b></td>
<td>75.9</td>
<td>-</td>
<td>-</td>
<td>-</td>
<td>-</td>
</tr>
<tr>
<td>NGS# [Chen et al., 2021]</td>
<td>35.3</td>
<td>58.8</td>
<td>62.0</td>
<td>34.1</td>
<td>46.1</td>
<td>60.9</td>
</tr>
<tr>
<td>Geoformer# [Chen et al., 2022]</td>
<td>36.8</td>
<td>59.3</td>
<td>62.5</td>
<td>35.6</td>
<td>47.3</td>
<td>62.3</td>
</tr>
<tr>
<td>PGPSNet</td>
<td>65.0</td>
<td><b class="best-score-text">77.9</b></td>
<td><b class="best-score-text">80.7</b></td>
<td><b class="best-score-text">62.7</b></td>
<td><b class="best-score-text">70.4</b></td>
<td><b class="best-score-text">79.5</b></td>
</tr>
</tbody>
</table>
* denotes results re-produced with the authors' code. <br>
# denotes methods re-implemented by us.
<h2 class="title is-3">Ablation studies on Geometry3K.</h2>
<table class="js-sort-table" id="results">
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<tr>
<th rowspan="2" class="js-sort-number" style="vertical-align: middle; color: gray;">Self-limited Decoder</th>
<th rowspan="2" class="js-sort-number" style="vertical-align: middle; color: gray;">Data Aug</th>
<th rowspan="2" class="js-sort-number" style="vertical-align: middle; color: gray;">Structural Clauses</th>
<th rowspan="2" class="js-sort-number" style="vertical-align: middle; color: gray;">Pre-trained LM</th>
<th colspan="3" class="js-sort-number" style="vertical-align: middle; color: gray;">Ans acc</th>
<th colspan="3" class="js-sort-number" style="vertical-align: middle; color: gray;">Prog acc</th>
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<tr>
<th class="js-sort-number" style="vertical-align: middle; color: gray;">Completion</th>
<th class="js-sort-number" style="vertical-align: middle; color: gray;">Choice</th>
<th class="js-sort-number" style="vertical-align: middle; color: gray;">Top-3</th>
<th class="js-sort-number" style="vertical-align: middle; color: gray;">Completion</th>
<th class="js-sort-number" style="vertical-align: middle; color: gray;">Choice</th>
<th class="js-sort-number" style="vertical-align: middle; color: gray;">Top-3</th>
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</thead>
<tbody>
<tr>
<td>✔</td>
<td>✘</td>
<td>✔</td>
<td>✘</td>
<td>32.5</td>
<td>52.2</td>
<td>57.6</td>
<td>27.2</td>
<td>47.3</td>
<td>53.1</td>
</tr>
<tr>
<td>✘</td>
<td>✔</td>
<td>✔</td>
<td>✘</td>
<td>28.2</td>
<td>48.3</td>
<td>50.7</td>
<td>25.4</td>
<td>42.7</td>
<td>45.6</td>
</tr>
<tr>
<td>✔</td>
<td>✔</td>
<td>✔</td>
<td>✘</td>
<td>36.6</td>
<td>59.5</td>
<td>62.4</td>
<td>33.9</td>
<td>52.8</td>
<td>58.6</td>
</tr>
<tr>
<td>✔</td>
<td>✔</td>
<td>✘</td>
<td>✘</td>
<td>38.4</td>
<td>61.7</td>
<td>64.8</td>
<td>34.8</td>
<td>54.2</td>
<td>59.2</td>
</tr>
<tr>
<td>✔</td>
<td>✔</td>
<td>✔</td>
<td>✔</td>
<td>48.1</td>
<td>67.5</td>
<td>71.4</td>
<td>45.4</td>
<td>62.0</td>
<td>68.1</td>
</tr>
<tr>
<td>✔</td>
<td>✔</td>
<td>✔</td>
<td>✔</td>
<td><b class="best-score-text">65.0</b></td>
<td><b class="best-score-text">77.9</b></td>
<td><b class="best-score-text">80.7</b></td>
<td><b class="best-score-text">62.8</b></td>
<td><b class="best-score-text">72.4</b></td>
<td><b class="best-score-text">78.2</b></td>
</tr>
</tbody>
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<p>🚨 For more details, please refer to <a href="https://arxiv.org/pdf/2302.11097">this link</a>
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<pre><code>@inproceedings{Zhang2023PGPS,
title = {A Multi-Modal Neural Geometric Solver with Textual Clauses Parsed from Diagram},
author = {Zhang, Ming-Liang and Yin, Fei and Liu, Cheng-Lin},
booktitle = {IJCAI},
year = {2023},
}
</code></pre>
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