利用问题公式化、等式约束和真值表不变性在Maple中实现CAD
An implementation of CAD in Maple utilising problem formulation, equational constraints and truth-table invariance
摘要：
柱形代数分解（CAD）是研究半代数集的重要工具，在代数几何及其他领域有着广泛的应用。我们最近报道了在Maple中实现CAD的新实现，该实现了Collins的原始算法和McCallum对投影的后续改进。我们的实现与Maple的内置CAD命令相反，基于一个完全独立的理论。虽然最初是作为比较算法的调查工具开发的，但我们发现并报告说，我们的代码提供了目前在任何其他现有实现中不可用的功能。一个特别重要的功能是生成顺序不变CAD的能力。这使我们能够扩展的实现，以产生CAD不变的方程约束（ECCAD）或真值表的公式序列（TTICAD）。这个新功能包含在我们代码的第二个版本中，以及考虑问题公式化的命令，这可能是CAD易处理性的主要因素。在报告中，我们描述了新的功能和它所引发的一些理论发现。我们描述了如何使用等式约束产生的CAD能够利用不仅改进的投影，而且在提升阶段的改进。我们还提出了一个扩展到原来的TTICAD算法，增加了TTICAD的适用性和相对于其他算法的好处。代码和演示该软件包完整功能的Maple工作表/ pdf介绍可在网上免费获得。
英文摘要：
Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of semi-algebraic sets, with applications within algebraic geometry and beyond. We recently reported on a new implementation of CAD in Maple which implemented the original algorithm of Collins and the subsequent improvement to projection by McCallum. Our implementation was in contrast to Maple's in-built CAD command, based on a quite separate theory. Although initially developed as an investigative tool to compare the algorithms, we found and reported that our code offered functionality not currently available in any other existing implementations. One particularly important piece of functionality is the ability to produce order-invariant CADs. This has allowed us to extend the implementation to produce CADs invariant with respect to either equational constraints (ECCADs) or the truth-tables of sequences of formulae (TTICADs). This new functionality is contained in the second release of our code, along with commands to consider problem formulation which can be a major factor in the tractability of a CAD. In the report we describe the new functionality and some theoretical discoveries it prompted. We describe how the CADs produced using equational constraints are able to take advantage of not just improved projection but also improvements in the lifting phase. We also present an extension to the original TTICAD algorithm which increases both the applicability of TTICAD and its relative benefit over other algorithms. The code and an introductory Maple worksheet / pdf demonstrating the full functionality of the package are freely available online.
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