Related papers: Towards 3-Dimensional Rewriting Theory
Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition left-FP3. Using this result, he constructed finitely presentable monoids with a…
In this paper we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorically as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For…
In this paper, we introduce monoidal rewriting systems (MRS), an abstraction of string rewriting in which reductions are defined over an arbitrary ambient monoid rather than a free monoid of words. This shift is partly motivated by logic:…
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by…
This paper investigates the class of finitely presented monoids defined by homogeneous (length-preserving) relations from a computational perspective. The properties of admitting a finite complete rewriting system, having finite derivation…
Rewriting methods have been developed for the study of coherence for algebraic objects. This consists in starting with a convergent presentation, and expliciting a family of generating confluences to obtain a coherent presentation -- one…
In this paper, we introduce a rewriting theory of linear monoidal categories. Those categories are a particular case of what we will define as linear (n, p)-categories. We will also define linear (n, p)-polygraphs, a linear adapation of…
Precategories generalize both the notions of strict $n$-category and sesquicategory: their definition is essentially the same as the one of strict $n$-categories, excepting that we do not require the various interchange laws to hold. Those…
This paper studies complete rewriting systems and biautomaticity for three interesting classes of finite-rank homogeneous monoids: Chinese monoids, hypoplactic monoids, and sylvester monoids. For Chinese monoids, we first give new…
A coherent presentation of an n-category is a presentation by generators, relations and relations among relations. Confluent and terminating rewriting systems generate coherent presentations, whose relations among relations are defined by…
In this paper we examine a number of term rewriting system for integer number representations, building further upon the datatype defining systems described in [2]. In particular, we look at automated methods for proving confluence and…
All current investigations to analyze the derivational complexity of term rewrite systems are based on a single termination method, possibly preceded by transformations. However, the exclusive use of direct criteria is problematic due to…
String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. Dixon, Duncan and Kissinger introduced string graphs, which are a combinatoric…
With a view on applications in computing, in particular concurrency theory and higher-dimensional rewriting, we develop notions of $n$-fold monoid and comonoid objects in $n$-fold monoidal categories and bicategories. We present a series of…
String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to…
We introduce homotopical methods based on rewriting on higher-dimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an…
In this paper we study rewriting systems for groups and monoids, focusing on situations where finite convergent systems may be difficult to find or do not exist. We consider systems which have no length increasing rules and are confluent…
A series of works has established rewriting as an essential tool in order to prove coherence properties of algebraic structures, such as MacLane's coherence theorem for monoidal categories, based on the observation that, under reasonable…
A partial monoid $P$ is a set with a partial multiplication $\times$ (and total identity $1_P$) which satisfies some associativity axiom. The partial monoid $P$ may be embedded in a free monoid $P^*$ and the product $\star$ is simulated by…
The concept of a system has proliferated through natural and social sciences. While myriad theories of systems exist, there is no mathematical general theory of systems. In this thesis, we take a first step towards formulating such a…