If $\mathcal{P} = \left\langle A \, | \,R \right\rangle$ is a monoid presentation, then the relation words in $\mathcal{P}$ are just the set of words on the left or right hand side of any pair in $R$. A word $w\in A ^*$ is said to be a piece of $\mathcal{P}$ if $w$ is a factor of at least two distinct relation words, or $w$ occurs more than once as a factor of a single relation word (possibly overlapping). A finitely presented monoid is a small overlap monoid if no relation word can be written as a product of fewer than $4$ pieces. In this paper, we present a quadratic time algorithm for computing normal forms of words in small overlap monoids where the coefficients are sufficiently small to allow for practical computation. Additionally, we show that the uniform word problem for small overlap monoids can be solved in linear time.

中文翻译：

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小重叠类曲面中范式的显式算法

如果$ \ mathcal {P} = \ left \ langle A \，| | \，R \ right \ rangle $是一个等边线表示，因此$ \ mathcal {P} $中的关系词只是$ R $中任何对的左手或右手的一组单词。如果$ w $是至少两个不同关系词的因数，或者$ w $作为一个因数不止一次出现，则将A ^ * $中的单词$ w \视为$ \ mathcal {P} $的一部分单个关系词（可能重叠）。如果没有任何关系词可以写成少于$ 4 $的乘积，则有限表示的monoid是一个小的重叠的monoid。在本文中，我们提出了一种二次时间算法，用于计算小重叠单面体中的普通形式的单词，其中系数足够小，可以进行实际计算。此外，我们表明，小重叠单面体的统一词问题可以在线性时间内解决。