47,141 research outputs found
Graphical Reasoning in Compact Closed Categories for Quantum Computation
Compact closed categories provide a foundational formalism for a variety of
important domains, including quantum computation. These categories have a
natural visualisation as a form of graphs. We present a formalism for
equational reasoning about such graphs and develop this into a generic proof
system with a fixed logical kernel for equational reasoning about compact
closed categories. Automating this reasoning process is motivated by the slow
and error prone nature of manual graph manipulation. A salient feature of our
system is that it provides a formal and declarative account of derived results
that can include `ellipses'-style notation. We illustrate the framework by
instantiating it for a graphical language of quantum computation and show how
this can be used to perform symbolic computation.Comment: 21 pages, 9 figures. This is the journal version of the paper
published at AIS
Algorithmic information and incompressibility of families of multidimensional networks
This article presents a theoretical investigation of string-based generalized
representations of families of finite networks in a multidimensional space.
First, we study the recursive labeling of networks with (finite) arbitrary node
dimensions (or aspects), such as time instants or layers. In particular, we
study these networks that are formalized in the form of multiaspect graphs. We
show that, unlike classical graphs, the algorithmic information of a
multidimensional network is not in general dominated by the algorithmic
information of the binary sequence that determines the presence or absence of
edges. This universal algorithmic approach sets limitations and conditions for
irreducible information content analysis in comparing networks with a large
number of dimensions, such as multilayer networks. Nevertheless, we show that
there are particular cases of infinite nesting families of finite
multidimensional networks with a unified recursive labeling such that each
member of these families is incompressible. From these results, we study
network topological properties and equivalences in irreducible information
content of multidimensional networks in comparison to their isomorphic
classical graph.Comment: Extended preprint version of the pape
Irrational Conformal Field Theory
This is a review of irrational conformal field theory, which includes
rational conformal field theory as a small subspace. Central topics of the
review include the Virasoro master equation, its solutions and the dynamics of
irrational conformal field theory. Discussion of the dynamics includes the
generalized Knizhnik-Zamolodchikov equations on the sphere, the corresponding
heat-like systems on the torus and the generic world- sheet action of
irrational conformal field theory.Comment: 195 pages, Latex, 12 figures, to appear in Physics Reports. Typos
corrected in Sections 13 and 14, and a footnote added in Section 1
Graphical Markov models, unifying results and their interpretation
Graphical Markov models combine conditional independence constraints with
graphical representations of stepwise data generating processes.The models
started to be formulated about 40 years ago and vigorous development is
ongoing. Longitudinal observational studies as well as intervention studies are
best modeled via a subclass called regression graph models and, especially
traceable regressions. Regression graphs include two types of undirected graph
and directed acyclic graphs in ordered sequences of joint responses. Response
components may correspond to discrete or continuous random variables and may
depend exclusively on variables which have been generated earlier. These
aspects are essential when causal hypothesis are the motivation for the
planning of empirical studies.
To turn the graphs into useful tools for tracing developmental pathways and
for predicting structure in alternative models, the generated distributions
have to mimic some properties of joint Gaussian distributions. Here, relevant
results concerning these aspects are spelled out and illustrated by examples.
With regression graph models, it becomes feasible, for the first time, to
derive structural effects of (1) ignoring some of the variables, of (2)
selecting subpopulations via fixed levels of some other variables or of (3)
changing the order in which the variables might get generated. Thus, the most
important future applications of these models will aim at the best possible
integration of knowledge from related studies.Comment: 34 Pages, 11 figures, 1 tabl
Billiard algebra, integrable line congruences, and double reflection nets
The billiard systems within quadrics, playing the role of discrete analogues
of geodesics on ellipsoids, are incorporated into the theory of integrable
quad-graphs. An initial observation is that the Six-pointed star theorem, as
the operational consistency for the billiard algebra, is equivalent to an
integrabilty condition of a line congruence. A new notion of the
double-reflection nets as a subclass of dual Darboux nets associated with
pencils of quadrics is introduced, basic properies and several examples are
presented. Corresponding Yang-Baxter maps, associated with pencils of quadrics
are defined and discussed.Comment: 18 pages, 8 figure
On external presentations of infinite graphs
The vertices of a finite state system are usually a subset of the natural
numbers. Most algorithms relative to these systems only use this fact to select
vertices.
For infinite state systems, however, the situation is different: in
particular, for such systems having a finite description, each state of the
system is a configuration of some machine. Then most algorithmic approaches
rely on the structure of these configurations. Such characterisations are said
internal. In order to apply algorithms detecting a structural property (like
identifying connected components) one may have first to transform the system in
order to fit the description needed for the algorithm. The problem of internal
characterisation is that it hides structural properties, and each solution
becomes ad hoc relatively to the form of the configurations.
On the contrary, external characterisations avoid explicit naming of the
vertices. Such characterisation are mostly defined via graph transformations.
In this paper we present two kind of external characterisations:
deterministic graph rewriting, which in turn characterise regular graphs,
deterministic context-free languages, and rational graphs. Inverse substitution
from a generator (like the complete binary tree) provides characterisation for
prefix-recognizable graphs, the Caucal Hierarchy and rational graphs. We
illustrate how these characterisation provide an efficient tool for the
representation of infinite state systems
Algebraic Aspects of Conditional Independence and Graphical Models
This chapter of the forthcoming Handbook of Graphical Models contains an
overview of basic theorems and techniques from algebraic geometry and how they
can be applied to the study of conditional independence and graphical models.
It also introduces binomial ideals and some ideas from real algebraic geometry.
When random variables are discrete or Gaussian, tools from computational
algebraic geometry can be used to understand implications between conditional
independence statements. This is accomplished by computing primary
decompositions of conditional independence ideals. As examples the chapter
presents in detail the graphical model of a four cycle and the intersection
axiom, a certain implication of conditional independence statements. Another
important problem in the area is to determine all constraints on a graphical
model, for example, equations determined by trek separation. The full set of
equality constraints can be determined by computing the model's vanishing
ideal. The chapter illustrates these techniques and ideas with examples from
the literature and provides references for further reading.Comment: 20 pages, 1 figur
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