Live Graph Databases Using DCR Graphs

Nowadays, it is of uttermost importance for companies that want to be relevant on the market to produce more while making fewer mistakes. Good management practices recommend the replication of critical business operations, like hiring a new employee and the set up he has to go through to have the company tools available, or the steps and decisions required when producing some daily report. The possibility of creating and refining these processes through business process systems to better suit the daily activity of an enterprise has a direct impact on the overall productivity, organization, and cost-reduction. The commonly used process systems make use of notations that are like state machines, having a somewhat imperative style depicting a narrow path where every decision in a process is sequential – providing the user no chance to offer input on how the process carries out – and struggle to take data into account. The proposal of several declarative languages and notations meant to solve this problem, easily incorporating data alongside the specified workflow, and providing actual control to the end-user on how the processes are accomplished by stating what can/needs to be done rather than how to do it in a step-by-step fashion. With this dissertation we present ReDa, a novel declarative, dynamic, and reactive data-centric process language, and the mapping from its specification to a running system (the operational semantics) implemented using the mechanisms of a graph-database, namely neo4j. We also present and evaluate a prototype of a business process system able to emulate the process via a reactive application, addressing the challenges of having a system that interacts with a dynamic process, and the solutions adopted.Atualmente, para que uma empresa possa ser relevante no mercado é bastante importante que a sua produção aumente e que a sua taxa de erros diminua. Regras de boa prática no que toca ao controlo de processos de uma empresa recomendam que as suas tarefas mais críticas sejam efetuadas da mesma forma independentemente de quem as executa, tal como a contratação de um novo empregado e todos os passos que ele precisa de executar para que reúna as condições necessárias para trabalhar, ou quais os pontos-chave obrigatórios a seguir quando se submetem relatórios. A possibilidade de criar e ajustar estes processos ao dia a dia de uma empresa tem um impacto direto na sua produtividade, organização e redução de custos. Os sistemas de processos mais utilizados adotam notações semelhantes a máquinas de estado, onde definem as suas atividades de uma forma sequencial e têm dificuldade em incorporar dados no processo. A proposta de várias linguagens de processos declarativas tem como objetivo solucionar este problema, permitindo a definição do processo e dos seus dados de forma simultânea e flexível, pois ao invés de se definir uma sequência de execução é possível estabelecer o que pode/tem de ser feito. Com esta dissertação apresentamos a ReDa, uma nova linguagem declarativa, dinâmica e reativa centrada em dados, e um mapeamento desta especificação para um sistema de execução que utiliza os mecanismos de uma base de dados de grafos, nomeadamente o neo4j. Apresentamos e avaliamos também um protótipo de sistema de gestão de processos capaz de emular processos ReDa através de uma aplicação reativa, abordando os desafios de desenvolver um sistema que interaja com um processo dinâmico e as soluções adotadas

From constructive field theory to fractional stochastic calculus. (II) Constructive proof of convergence for the L\'evy area of fractional Brownian motion with Hurst index α∈(1/8,1/4)\alpha\in(1/8,1/4)

{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha<1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a difficult task because of the low H\"older regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard" tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates, and call for an extension of Gaussian tools such as for instance the Malliavin calculus. After a first introductory paper \cite{MagUnt1}, this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as L\'evy area

Cyclic Directed Probabilistic Graphical Model: A Proposal Based on Structured Outcomes

In the process of building (structural learning) a probabilistic graphical model from a set of observed data, the directional, cyclic dependencies between the random variables of the model are often found. Existing graphical models such as Bayesian and Markov networks can reflect such dependencies. However, this requires complicating those models, such as adding additional variables or dividing the model graph into separate subgraphs. Herein, we describe a probabilistic graphical model - probabilistic relation network - that allows the direct capture of directional cyclic dependencies during structural learning. This model is based on the simple idea that each sample of the observed data can be represented by an arbitrary graph (structured outcome), which reflects the structure of the dependencies of the variables included in the sample. Each of the outcomes contains only a part of the graphical model structure; however, a complete graph of the probabilistic model is obtained by combining different outcomes. Such a graph, unlike Bayesian and Markov networks, can be directed and can have cycles. We explored the full joint distribution and conditional distribution and conditional independence properties of variables in the proposed model. We defined the algorithms for constructing of the model from the dataset and for calculating the conditional and full joint distributions. We also performed a numerical comparison with Bayesian and Markov networks. This model does not violate the probability axioms, and it supports learning from observed data. Notably, it supports probabilistic inference, making it a prospective tool in data analysis and in expert and design-making applications.Comment: 41 pages, 11 figures, arXiv:2206.06089v

Brick polytopes, lattice quotients, and Hopf algebras

This paper is motivated by the interplay between the Tamari lattice, J.-L. Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf algebra on binary trees. We show that these constructions extend in the world of acyclic $k$-triangulations, which were already considered as the vertices of V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural surjection from the permutations to the acyclic $k$-triangulations. We show that the fibers of this surjection are the classes of the congruence $\equiv^k$ on $\mathfrak{S}_n$ defined as the transitive closure of the rewriting rule $U ac V_1 b_1 \cdots V_k b_k W \equiv^k U ca V_1 b_1 \cdots V_k b_k W$ for letters $a < b_1, \dots, b_k < c$ and words $U, V_1, \dots, V_k, W$ on $[n]$. We then show that the increasing flip order on $k$-triangulations is the lattice quotient of the weak order by this congruence. Moreover, we use this surjection to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on permutations, indexed by acyclic $k$-triangulations, and to describe the product and coproduct in this algebra and its dual in term of combinatorial operations on acyclic $k$-triangulations. Finally, we extend our results in three directions, describing a Cambrian, a tuple, and a Schr\"oder version of these constructions.Comment: 59 pages, 32 figure