利用计数函数对自由群和自由幺半群的有效计算
Efficient computations with counting functions on free groups and free monoids
摘要：
给出了判定非交换自由群或幺半群上两个给定的计数函数之间是否有界距离以及判定非交换自由群上两个给定的计数拟同构是否上同调的有效算法.我们工作在多带图灵机模型与非常数时间算术运算。在整数系数的情况下，我们构造了一个线性空间和时间复杂度的算法（假设在幺半群的情况下，秩至少为$3$）。在有理系数的情况下，我们证明了时间复杂度是$O(N\log N)$，其中$N$表示输入的大小，即它是一样快的有理数（使用哈维-范德胡芬算法整数乘法）。这些算法是基于我们以前的工作，其特征有界计数函数。
英文摘要：
We present efficient algorithms to decide whether two given counting functions on non-abelian free groups or monoids are at bounded distance from each other and to decide whether two given counting quasimorphisms on non-abelian free groups are cohomologous. We work in the multi-tape Turing machine model with non-constant time arithmetic operations. In the case of integer coefficients we construct an algorithm of linear space and time complexity (assuming that the rank is at least $3$ in the monoid case). In the case of rational coefficients we prove that the time complexity is $O(N\log N)$, where $N$ denotes the size of the input, i.e. it is as fast as addition of rational numbers (implemented using the Harvey--van der Hoeven algorithm for integer multiplication). These algorithms are based on our previous work which characterizes bounded counting functions.
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