Geometric Reasoning with Logic and Algebra
Geometric Reasoning with Logic and Algebra
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DOI：
10.1016/0004-3702(88)90049-5
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发表时间：
1988-12
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影响因子：
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通讯作者：
D. Arnon
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文献类型：
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作者：
D. Arnon
几何推理所涉及的（几何）对象通常可以用真实的数的一阶理论的语言公式来定义。在几何推理中出现的某些问题可以被转换成以下形式之一：查询问题-某个对象集合是否具有某个（一阶）属性;约束问题-给定一个定义对象的量化公式，通过一个无量化器的公式找到一个等价的定义;显示问题-描述一个对象，例如确定它的维度或指定它的拓扑。在过去的十五年中，这些问题的精确解的可行算法已经被发现，实现，并用于解决非平凡问题。我们给出的例子属于这些方法的范围内的问题，提供了一个教程介绍目前使用的主要算法，并描述了使用这些算法的样本问题的解决方案。
Geometric reasoning is concerned with (geometric) objects that often are definable by formulae of the language of the first-order theory of the real numbers. Certain problems that arise in geometric reasoning can be cast into one of the following forms:queryproblem—does a certain collection of objects possess a certain (first-order) property;constraintproblem—given a quantified formula defining an object, find an equivalent definition by a quantifier-free formula;displayproblem—describe an object, e.g. determine its dimension or specify its topology. In the last fifteen years, feasible algorithms for the exact solution of these problems have been discovered, implemented, and used to solve nontrivial problems. We give examples of problems that fall within the scope of these methods, provide a tutorial introduction to the principal algorithms currently in use, and describe the solution of the sample problems using those algorithms.