有限不确定性代数结构的秩计算
Computing rank of finite algebraic structures with limited nondeterminism
摘要：
一个有限代数结构的秩是在一个二元运算的闭包下表示每一个其他元素所需的最小元素数。在组的情况下，以前计算秩的最佳算法使用多对数空间。我们减少了最好的上限的复杂性计算秩群和拟群。本文证明了这些代数结构的秩问题可以通过高度限制的计算模型来验证，只给出非常短的正确性证书。具体来说，我们证明了问题的决定是否有限quasigroup的秩，给定的凯莱表，小于一个指定的数字是可判定的电路的深度$O(\log \log n)$增加了$O(\log^2 n)$不确定性位。此外，如果拟群是一个群，那么这个问题也可以由图灵机使用$O(\log n)$空间和$O(\log^2 n)$位的非确定性来判定，该图灵机具有多次读取非确定性位的能力。最后，我们对其他代数结构和其他类型的秩的相关问题也给出了类似的结果。这些新的上限是显著的改进，特别是对于组。一般来说，有限不确定性的镜头提供了一个简单的方法来改善许多简单的算法，像这里提出的，我们怀疑它将是特别有用的其他代数算法。
英文摘要：
The rank of a finite algebraic structure with a single binary operation is the minimum number of elements needed to express every other element under the closure of the operation. In the case of groups, the previous best algorithm for computing rank used polylogarithmic space. We reduce the best upper bounds on the complexity of computing rank for groups and for quasigroups. This paper proves that the rank problem for these algebraic structures can be verified by highly restricted models of computation given only very short certificates of correctness. Specifically, we prove that the problem of deciding whether the rank of a finite quasigroup, given as a Cayley table, is smaller than a specified number is decidable by a circuit of depth $O(\log \log n)$ augmented with $O(\log^2 n)$ nondeterministic bits. Furthermore, if the quasigroup is a group, then the problem is also decidable by a Turing machine using $O(\log n)$ space and $O(\log^2 n)$ bits of nondeterminism with the ability to read the nondeterministic bits multiple times. Finally, we provide similar results for related problems on other algebraic structures and other kinds of rank. These new upper bounds are significant improvements, especially for groups. In general, the lens of limited nondeterminism provides an easy way to improve many simple algorithms, like the ones presented here, and we suspect it will be especially useful for other algebraic algorithms.
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