Deciding Identities in Nonassociative Algebras with Dynamic Programming

这项研究涉及判定给定的非结合多项式是否为各种非结合代数的恒等式的算法和计算技术。一般来说，这个问题在计算上非常困难，现有的方法存在着时间和内存要求过高的问题。调查中使用的方法是使用动态规划算法。该算法通过构造一个自由代数的同态映象来解决这个问题。构造代数所需的算术运算数由结果代数的维度中的多项式限定(但对于许多变体，算法可能不是输入大小中的多项式时间)。使用这种方法，似乎许多以前在计算上看起来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，这种方法可以在几分钟内判定某些多项式，直到10次，是否是交换的四次结合代数的恒等式。虽然该方法看起来很有前途，但为了使该方法更实用，还需要研究几个理论问题。目标是在几个特定问题上实现该算法的完整版。其中一个问题是寻找乔丹·S的身份。二是判定是否存在非特殊的Malcev代数。

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.