Related papers: Universal Constructions for (Co)Relations: categor…
There are many category-theoretic notions of algebraic theory, including Lawvere theories, monads, PROPs and operads. The first central notion of this thesis is a common generalisation of these, which we call a proto-theory. In order to…
Our starting point is a particular `canvas' aimed to `draw' theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this paper we consider the conceptual foundations for this canvas, and how these can…
Tasks that model the relation between pairs of tokens in a string are a vital part of understanding natural language. Such tasks, in general, require exhaustive pair-wise comparisons of tokens, thus having a quadratic runtime complexity in…
String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and…
Diagrammatically speaking, grammatical calculi such as pregroups provide wires between words in order to elucidate their interactions, and this enables one to verify grammatical correctness of phrases and sentences. In this paper we also…
In the graphical calculus of planar string diagrams, equality is generated by exchange moves, which swap the heights of adjacent vertices. We show that left- and right-handed exchanges each give strongly normalizing rewrite strategies for…
Coherence theorems for covariant structures carried by a category have traditionally relied on the underlying term rewriting system of the structure being terminating and confluent. While this holds in a variety of cases, it is not a…
Many theories of semantic interpretation use lambda-term manipulation to compositionally compute the meaning of a sentence. These theories are usually implemented in a language such as Prolog that can simulate lambda-term operations with…
Constellations are partial algebras that are one-sided generalisations of categories. It has previously been shown that the category of inductive constellations is isomorphic to the category of left restriction semigroups. Here we consider…
Bimonoidal categories (also known as rig categories) are categories with two monoidal structures, one of which distributes over the other. We formally define sheet diagrams, a graphical calculus for bimonoidal categories that was informally…
Architecture styles characterise families of architectures sharing common characteristics. We have recently proposed configuration logics for architecture style specification. In this paper, we study a graphical notation to enhance…
We relate the computational complexity of finite strings to universal representations of their underlying symmetries. First, Boolean functions are classified using the universal covering topologies of the circuits which enumerate them. A…
Tree-width is an invaluable tool for computational problems on graphs. But often one would like to compute on other kinds of objects (e.g. decorated graphs or even algebraic structures) where there is no known tree-width analogue. Here we…
Graph compositions generalize both integer compositions and partitions of a finite set. We develop formulas, generating functions and recurrence relations for composition counting functions for several families of graphs.
The analysis of games played on graph-like structures is of increasing importance due to the prevalence of social networks, both virtual and physical, in our daily life. As well as being relevant in computer science, mathematical analysis…
We develop a comprehensive string diagrammatic treatment of electrical circuits. Building on previous, limited case studies, we introduce controlled sources and meters as elements, and the impedance calculus, a powerful toolbox for…
We generalize the notion of linear chord diagrams to the case of matched sets of size $k$, which we call $k$-chord diagrams. We provide formal generating functions and recurrence relations enumerating these $k$-chord diagrams by the number…
Parametric entities appear in many contexts, be it in optimisation, control, modelling of random quantities, or uncertainty quantification. These are all fields where reduced order models (ROMs) have a place to alleviate the computational…
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…
Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm…