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.
翻译：如果 $\ mathcal{ P} = left\ langle A \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
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