论文标题
自动建造奥林匹克几何学问题的图
Automatically Building Diagrams for Olympiad Geometry Problems
论文作者
论文摘要
我们提出了一种自动构建奥林匹克级几何问题问题的方法，并在新的开源软件工具《几何模型构建器》（GMB）中实现我们的方法。我们方法的核心是一种新的特定领域的语言，即几何模型构建语言（GMBL），用于指定几何问题以及对构建图的其他元数据。 GMBL程序指定（1）如何参数化几何对象（或几何对象集）并初始化这些参数化的数量，（2）哪个数量直接从其他数量计算，以及（3）其他约束以累积到（不同的）损失功能中。 GMBL程序诱导了（通常）可处理的数值优化问题，其解决方案对应于原始问题陈述的图，并且我们可以使用梯度下降可靠地解决。自2000年以来，在国际数学奥林匹克运动会出现的39个几何问题中，可以在我们的逻辑中表达36个几何问题，我们的系统平均可以为94％的图表生成图表。据我们所知，我们的方法是第一个在自动化几何图构造中生成这种复杂问题的模型的方法。
We present a method for automatically building diagrams for olympiad-level geometry problems and implement our approach in a new open-source software tool, the Geometry Model Builder (GMB). Central to our method is a new domain-specific language, the Geometry Model-Building Language (GMBL), for specifying geometry problems along with additional metadata useful for building diagrams. A GMBL program specifies (1) how to parameterize geometric objects (or sets of geometric objects) and initialize these parameterized quantities, (2) which quantities to compute directly from other quantities, and (3) additional constraints to accumulate into a (differentiable) loss function. A GMBL program induces a (usually) tractable numerical optimization problem whose solutions correspond to diagrams of the original problem statement, and that we can solve reliably using gradient descent. Of the 39 geometry problems since 2000 appearing in the International Mathematical Olympiad, 36 can be expressed in our logic and our system can produce diagrams for 94% of them on average. To the best of our knowledge, our method is the first in automated geometry diagram construction to generate models for such complex problems.
