Related papers: When Only Topology Matters
We introduce string diagrams as a formal mathematical, graphical language to represent, compose, program and reason about games. The language is well established in quantum physics, quantum computing and quantum linguistic with the…
Graph transformation is concerned with the manipulation of graphs by means of rules. Graph grammars have been traditionally studied using techniques from category theory. In previous works, we introduced Matrix Graph Grammars (MGGs) as a…
A method of {\it topological grammars} is proposed for multidimensional data approximation. For data with complex topology we define a {\it principal cubic complex} of low dimension and given complexity that gives the best approximation for…
This paper exposes the language of geometric contexts and elementary schemes, which is a functorial formalism to study categories of geometric objects such as schemes, topological manifolds, differential manifolds, analytic manifolds, etc.…
The language of graph theory, or network science, has proven to be an exceptional tool for addressing myriad problems in neuroscience. Yet, the use of networks is predicated on a critical simplifying assumption: that the quintessential unit…
In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much…
Several old and recent classes of picture grammars, that variously extend context-free string grammars in two dimensions, are based on rules that rewrite arrays of pixels. Such grammars can be unified and extended using a tiling based…
Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations…
We introduce context-free languages of morphisms in monoidal categories, extending recent work on the categorification of context-free languages, and regular languages of string diagrams. Context-free languages of string diagrams include…
String diagrams are pictorial representations for morphisms of symmetric monoidal categories. They constitute an intuitive and expressive graphical syntax, which has found application in a very diverse range of fields including concurrency…
Graphical (Linear) Algebra is a family of diagrammatic languages allowing to reason about different kinds of subsets of vector spaces compositionally. It has been used to model various application domains, from signal-flow graphs to Petri…
This is a motivating tutorial introduction to a semantic analysis of programming languages using a graphical language as the representation of terms, and graph rewriting as a representation of reduction rules. We show how the graphical…
As one of the mainstream models of artificial intelligence, world models allow agents to learn the representation of the environment for efficient prediction and planning. However, classical world models based on flat tensors face several…
We investigate three formalisms to specify graph languages, i.e. sets of graphs, based on type graphs. First, we are interested in (pure) type graphs, where the corresponding language consists of all graphs that can be mapped…
We provide a pipeline for generating syntax diagrams (also called railroad diagrams) from context free grammars. Syntax diagrams are a graphical representation of a context free language, which we formalize abstractly as a set of mutually…
Recently researchers working in the LFG framework have proposed algorithms for taking advantage of the implicit context-free components of a unification grammar [Maxwell 96]. This paper clarifies the mathematical foundations of these…
We introduce regular languages of morphisms in free monoidal categories, with their associated grammars and automata. These subsume the classical theory of regular languages of words and trees, but also open up a much wider class of…
We show that a compact rigid balanced braided monoidal category with enough compact projective objects gives rise to a system of mapping class group representations compatible with the gluing along marked intervals. A motivation to consider…
Orthogonal drawings, i.e., embeddings of graphs into grids, are a classic topic in Graph Drawing. Often the goal is to find a drawing that minimizes the number of bends on the edges. A key ingredient for bend minimization algorithms is the…
Our research integrates graph data with Large Language Models (LLMs), which, despite their advancements in various fields using large text corpora, face limitations in encoding entire graphs due to context size constraints. This paper…