相关论文: Binary trees and fibred categories
Tanglegrams are a special class of graphs appearing in applications concerning cospeciation and coevolution in biology and computer science. They are formed by identifying the leaves of two rooted binary trees. We give an explicit formula…
Bilinear maps and their classifying tensor products are well-known in the theory of linear algebra, and their generalization to algebras of commutative monads is a classical result of monad theory. Motivated by constructions needed in…
Artificial Intelligence (AI) has long pursued models, theories, and techniques to imbue machines with human-like general intelligence. Yet even the currently predominant data-driven approaches in AI seem to be lacking humans' unique ability…
We define and prove isomorphisms between three combinatorial classes involving labeled trees. We also give an alternative proof by means of generating functions.
This document is an introduction to two related formalisms to define Boolean functions: binary decision diagrams, and Boolean circuits. It presents these formalisms and several of their variants studied in the setting of knowledge…
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…
The main goal of this paper is to describe a data structure called binary join trees that are useful in computing multiple marginals efficiently using the Shenoy-Shafer architecture. We define binary join trees, describe their utility, and…
We study the synchronous and asynchronous automatic structures on the fundamental group of a graph of groups in which each edge group is finite. Up to a natural equivalence relation, the set of biautomatic structures on such a graph product…
This is a survey article on trees, with a modest number of proofs to give a flavor of the way these topologies can be efficiently handled. Trees are defined in set-theorist fashion as partially ordered sets in which the elements below each…
Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of…
The unprecedented pace of machine learning research has lead to incredible advances, but also poses hard challenges. At present, the field lacks strong theoretical underpinnings, and many important achievements stem from ad hoc design…
Lenses, optics and dependent lenses (or equivalently morphisms of containers, or equivalently natural transformations of polynomial functors) are all widely used in applied category theory as models of bidirectional processes. From the…
We develop layered monoidal theories -- a generalisation of monoidal theories combining formal descriptions of a system at different levels of abstraction. Via their representation as string diagrams, monoidal theories provide a graphical…
This paper introduces a series of methods for traversing binary decision trees using arithmetic operations. We present a suite of binary tree traversal algorithms that leverage novel representation matrices to flatten the full binary tree…
In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many…
A central limit theorem for binary tree is numerically examined. Two types of central limit theorem for higher-order branches are formulated. A topological structure of a binary tree is expressed by a binary sequence, and the…
The concept of n-categories and related subject is considered. An n-category is described as an n-graph with a composition. A new definition of operad is presented. Some illustrative examples are given.
The theory of finite automata concerns itself with words in a free monoid together with concatenation and without further structure. There are, however, important applications which use alphabets which are structured in some sense. We…
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…
Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…