Higher-dimensional rewriting is founded on a duality of rewrite systems and cell complexes, connecting computational mathematics to higher categories and homotopy theory: the two sides of a rewrite rule are two halves of the boundary of an (n+1)-cell, which are diagrams of n-cells. We study higher-dimensional diagram rewriting as a mechanism of computation, focussing on the matching problem for rewritable subdiagrams within the combinatorial framework of diagrammatic sets. We provide an algorithm for subdiagram matching in arbitrary dimensions, based on new results on layerings of diagrams, and derive upper bounds on its time complexity. We show that these superpolynomial bounds can be improved to polynomial bounds under certain acyclicity conditions, and that these conditions hold in general for diagrams up to dimension 3. We discuss the challenges that arise in dimension 4.

翻译：高维重写建立在重写系统与胞腔复形的对偶性之上，将计算数学与高范畴论及同伦理论联系起来：重写规则的两侧是(n+1)-胞腔边界的两个半部，它们由n-胞腔的图解构成。我们研究作为计算机制的高维图解重写，重点关注图形式集组合框架中可重写子图的匹配问题。我们基于图解分层的新结果，提出任意维数下子图匹配的算法，并推导其时间复杂度上界。我们证明在某些无环条件下，这些超多项式上界可改进为多项式上界，且这些条件对不超过3维的图解普遍成立。最后讨论在4维情形中出现的挑战。