Deciding Identities in Nonassociative Algebras with Dynamic Programming

用动态规划确定非关联代数中的恒等式

• 批准号: 8905534
• 负责人: David Jacobs
• 金额: $ 3.02万
• 依托单位: Clemson University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-08-15 至 1992-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8905534&HistoricalAwards=false
• 关键词: Deciding; Identities; Nonassociative; Algebras; Dynamic

This investigation is concerned with algorithms and computational techniques to decide if a given nonassociative polynomial is an identity for a variety of nonassociative algebras. In general, this problem is computationally very difficult, and existing methods suffer from prohibitive time and memory requirements. The approach used in this investigation is to use a dynamic programming algorithm. The algorithm solves the problem by constructing a certain homomorphic image of the free algebra. The number of arithmetic operations required to construct the algebra is bound by a polynomial in the dimension of the resulting algebra (but for many varieties the algorithm might not be polynomial-time in the input size). Using this method, it appears that many questions, which previously appeared to be computationally impractical, may now be solved with existing computers. Preliminary tests show that this method can be used to decide in minutes if certain polynomials, up to degree 10, were identities in the variety of commutative, fourth-power-associative algebras. While the method looks promising, several theoretic issues need to be examined in order to make the method more practical. A goal is to implement the full version of the algorithm on several specific problems. One such problem is the search for Jordan s-identities. Another is determining whether there exist nonspecial Malcev algebras.

这项研究涉及算法和计算技术，以确定给定的非关联多项式是否是各种非关联代数的恒等式。 一般来说，这个问题在计算上非常困难，并且现有方法面临着令人望而却步的时间和内存需求。 本研究中使用的方法是使用动态规划算法。 该算法通过构造自由代数的某个同态图像来解决该问题。 构造代数所需的算术运算数量受所得代数维数中的多项式约束（但对于许多变体，该算法可能不是输入大小的多项式时间）。 使用这种方法，许多以前在计算上似乎不切实际的问题现在可以用现有的计算机来解决。 初步测试表明，该方法可用于在几分钟内确定某些多项式（最多 10 次）是否是各种可交换的四次方结合代数中的恒等式。 虽然该方法看起来很有前途，但需要研究几个理论问题才能使该方法更加实用。 目标是针对几个特定问题实现该算法的完整版本。 其中一个问题就是寻找乔丹的 S 身份。 另一个是确定是否存在非特殊马尔切夫代数。