117 research outputs found

    Category Theory for Autonomous Robots: The Marathon 2 Use Case

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    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

    Mathematical Foundations for a Compositional Account of the Bayesian Brain

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    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

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    Rank-based linkage is a new tool for summarizing a collection SS 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 SS. Rank-based linkage is applied to the KK-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 KK-nearest neighbor graph on SS. In SK2|S| K^2 steps it builds an edge-weighted linkage graph (S,L,σ)(S, \mathcal{L}, \sigma) where σ({x,y})\sigma(\{x, y\}) is called the in-sway between objects xx and yy. Take Lt\mathcal{L}_t to be the links whose in-sway is at least tt, and partition SS into components of the graph (S,Lt)(S, \mathcal{L}_t), for varying tt. 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

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    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

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    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

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    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

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    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

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    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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