117 research outputs found
Category Theory for Autonomous Robots: The Marathon 2 Use Case
Model-based systems engineering (MBSE) is a methodology that exploits system
representation during the entire system life-cycle. The use of formal models
has gained momentum in robotics engineering over the past few years. Models
play a crucial role in robot design; they serve as the basis for achieving
holistic properties, such as functional reliability or adaptive resilience, and
facilitate the automated production of modules. We propose the use of formal
conceptualizations beyond the engineering phase, providing accurate models that
can be leveraged at runtime. This paper explores the use of Category Theory, a
mathematical framework for describing abstractions, as a formal language to
produce such robot models. To showcase its practical application, we present a
concrete example based on the Marathon 2 experiment. Here, we illustrate the
potential of formalizing systems -- including their recovery mechanisms --
which allows engineers to design more trustworthy autonomous robots. This, in
turn, enhances their dependability and performance
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Proceedings of the 33rd Annual Workshop of the Psychology of Programming Interest Group
This is the Proceedings of the 33rd Annual Workshop of the Psychology of Programming Interest Group (PPIG). This was the first PPIG to be held physically since 2019, following the two online-only PPIGs in 2020 and 2021, both during the Covid pandemic. It was also the first PPIG conference to be designed specifically for hybrid attendance. Reflecting the theme, it was hosted by Music Computing Lab at the Open University in Milton Keynes
Mathematical Foundations for a Compositional Account of the Bayesian Brain
This dissertation reports some first steps towards a compositional account of
active inference and the Bayesian brain. Specifically, we use the tools of
contemporary applied category theory to supply functorial semantics for
approximate inference. To do so, we define on the `syntactic' side the new
notion of Bayesian lens and show that Bayesian updating composes according to
the compositional lens pattern. Using Bayesian lenses, and inspired by
compositional game theory, we define fibrations of statistical games and
classify various problems of statistical inference as corresponding sections:
the chain rule of the relative entropy is formalized as a strict section, while
maximum likelihood estimation and the free energy give lax sections. In the
process, we introduce a new notion of `copy-composition'.
On the `semantic' side, we present a new formalization of general open
dynamical systems (particularly: deterministic, stochastic, and random; and
discrete- and continuous-time) as certain coalgebras of polynomial functors,
which we show collect into monoidal opindexed categories (or, alternatively,
into algebras for multicategories of generalized polynomial functors). We use
these opindexed categories to define monoidal bicategories of cilia: dynamical
systems which control lenses, and which supply the target for our functorial
semantics. Accordingly, we construct functors which explain the bidirectional
compositional structure of predictive coding neural circuits under the free
energy principle, thereby giving a formal mathematical underpinning to the
bidirectionality observed in the cortex. Along the way, we explain how to
compose rate-coded neural circuits using an algebra for a multicategory of
linear circuit diagrams, showing subsequently that this is subsumed by lenses
and polynomial functors.Comment: DPhil thesis; as submitted. Main change from v1: improved treatment
of statistical games. A number of errors also fixed, and some presentation
improved. Comments most welcom
Rank-based linkage I: triplet comparisons and oriented simplicial complexes
Rank-based linkage is a new tool for summarizing a collection of objects
according to their relationships. These objects are not mapped to vectors, and
``similarity'' between objects need be neither numerical nor symmetrical. All
an object needs to do is rank nearby objects by similarity to itself, using a
Comparator which is transitive, but need not be consistent with any metric on
the whole set. Call this a ranking system on . Rank-based linkage is applied
to the -nearest neighbor digraph derived from a ranking system. Computations
occur on a 2-dimensional abstract oriented simplicial complex whose faces are
among the points, edges, and triangles of the line graph of the undirected
-nearest neighbor graph on . In steps it builds an
edge-weighted linkage graph where
is called the in-sway between objects and . Take to be
the links whose in-sway is at least , and partition into components of
the graph , for varying . Rank-based linkage is a
functor from a category of out-ordered digraphs to a category of partitioned
sets, with the practical consequence that augmenting the set of objects in a
rank-respectful way gives a fresh clustering which does not ``rip apart`` the
previous one. The same holds for single linkage clustering in the metric space
context, but not for typical optimization-based methods. Open combinatorial
problems are presented in the last section.Comment: 37 pages, 12 figure
Dynamic Tracing: a graphical language for rewriting protocols
The category Set* of sets and partial functions is well-known to be traced
monoidal, meaning that a partial function S+U -/-> T+U can be coherently
transformed into a partial function S -/-> T. This transformation is generally
described in terms of an implicit procedure that must be run. We make this
procedure explicit by enriching the traced category in Cat#, the symmetric
monoidal category of categories and cofunctors: each hom-category has such
procedures as objects, and advancement through the procedures as arrows. We
also generalize to traced Kleisli categories beyond Set*, providing a
conjectural trace operator for the Kleisli category of any polynomial monad of
the form t+1. The main motivation for this work is to give a formal and
graphical syntax for performing sophisticated computations powered by graph
rewriting, which is itself a graphical language for data transformation
A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks
Regulatory networks depict promoting or inhibiting interactions between
molecules in a biochemical system. We introduce a category-theoretic formalism
for regulatory networks, using signed graphs to model the networks and signed
functors to describe occurrences of one network in another, especially
occurrences of network motifs. With this foundation, we establish functorial
mappings between regulatory networks and other mathematical models in
biochemistry. We construct a functor from reaction networks, modeled as Petri
nets with signed links, to regulatory networks, enabling us to precisely define
when a reaction network could be a physical mechanism underlying a regulatory
network. Turning to quantitative models, we associate a regulatory network with
a Lotka-Volterra system of differential equations, defining a functor from the
category of signed graphs to a category of parameterized dynamical systems. We
extend this result from closed to open systems, demonstrating that
Lotka-Volterra dynamics respects not only inclusions and collapsings of
regulatory networks, but also the process of building up complex regulatory
networks by gluing together simpler pieces. Formally, we use the theory of
structured cospans to produce a lax double functor from the double category of
open signed graphs to that of open parameterized dynamical systems. Throughout
the paper, we ground the categorical formalism in examples inspired by systems
biology.Comment: 33 pages. Added several examples, plus minor revision
Topología del problema de migración de datos
Estudia la topología del problema de migración de datos, realizando un mapeo de una estructura de datos a otra estructura, utilizando herramientas matemáticas. Este trabajo pretende desarrollar una nueva metodología para que a través de los grafos y sus propiedades se pueda estudiar el problema de migración. En general, no siempre es posible encontrar una migración de datos. En este trabajo presentamos y usamos un invariante numérico (llamado complejidad para el problema de migración) como herramienta de apoyo para evaluar la topología del problema de migración. Además, este invariante numérico permite conocer casi-migraciones óptimas de datos. El aporte de una nueva metodología con herramientas matemáticas no convencionales, debe hacer posible la ampliación de los resultados en esta nueva corriente. De donde se desprende que, este trabajo solo se limitará a estudiar un contexto definido del problema que es el estudio del homomorfismo de grafos usando la complejidad para realizar casi-migraciones de una estructura de datos a otra estructura
The free energy principle induces neuromorphic development
We show how any finite physical system with morphological, i.e. three-dimensional embedding or shape, degrees of freedom and locally limited free energy will, under the constraints of the free energy principle, evolve over time towards a neuromorphic morphology that supports hierarchical computations in which each ‘level’ of the hierarchy enacts a coarse-graining of its inputs, and dually, a fine-graining of its outputs. Such hierarchies occur throughout biology, from the architectures of intracellular signal transduction pathways to the large-scale organization of perception and action cycles in the mammalian brain. The close formal connections between cone-cocone diagrams (CCCD) as models of quantum reference frames on the one hand, and between CCCDs and topological quantum field theories on the other, allow the representation of such computations in the fully-general quantum-computational framework of topological quantum neural networks
Synergy between Quantum Computers and Databases
Academia, industry, and societies are showing increasing interest in the possibilities of quantum computing. The research in the intersection of quantum computing and databases is still in its initial steps. This work represents several crucial data management and query processing problems that will benefit from quantum computing. We outline how quantum computing will tackle these challenges and what kind of outcomes and speed-ups we expect. We discuss the position of quantum computing in data management and raise awareness of possible security threats in encryption. We aim to be realistic and point out technical difficulties that currently restrict implementations.Peer reviewe
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