Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Definability equals recognizability for graphs of bounded treewidth

We prove a conjecture of Courcelle, which states that a graph property is definable in MSO with modular counting predicates on graphs of constant treewidth if, and only if it is recognizable in the following sense: constant-width tree decompositions of graphs satisfying the property can be recognized by tree automata. While the forward implication is a classic fact known as Courcelle's theorem, the converse direction remained openComment: 21 pages, an extended abstract will appear in the proceedings of LICS 201

Causal Inference from Statistical Data

The so-called kernel-based tests of independence are developed for automatic causal discovery between random variables from purely observational statistical data, i.e., without intervention. Beyond the independence relations, the complexity of conditional distriubtions is used as an additional inference principle of determining the causal ordering between variables. Experiments with simulated and real-world data show that the proposed methods surpass the state-of-the-art approaches

Graph parameters and the speed of hereditary properties

In this thesis we study the speed of hereditary properties of graphs and how this defines some of the structure of the properties. We start by characterizing several graph parameters by means of minimal hereditary classes. We then give a global characterization of properties of low speed, before looking at properties with higher speeds starting at the Bell number. We then introduce a new parameter, clique-width, and show that there are an infinite amount of minimal hereditary properties with unbounded clique-width. We then look at the factorial layer in more detail and focus on P7-free bipartite graphs. Finally we discuss word-representable graphs

Generalizing graph decompositions

The Latin aphorism ‘divide et impera’ conveys a simple, but central idea in mathematics and computer science: ‘split your problem recursively into smaller parts, attack the parts, and conquer the whole’. There is a vast literature on how to do this on graphs. But often we need to compute on other structures (decorated graphs or perhaps algebraic objects such as groups) for which we do not have a wealth of decomposition methods. This thesis attacks this problem head on: we propose new decomposition methods in a variety of settings. In the setting of directed graphs, we introduce a new tree-width analogue called directed branch-width. We show that parameterizing by directed branch-width allows us to obtain linear-time algorithms for problems such as directed Hamilton Path and Max-Cut which are intractable by any other known directed analogue of tree-width. In fact, the algorithmic success of our new measure is more far-reaching: by proving algorithmic meta-theorems parameterized by directed branch-width, we deduce linear-time algorithms for all problems expressable in a variant of monadic second-order logic. Moving on from directed graphs, we then provide a meta-answer to the broader question of obtaining tree-width analogues for objects other than simple graphs. We do so introducing the theory of spined categories and triangualtion functors which constitutes a vast category-theoretic abstraction of a definition of tree-width due to Halin. Our theory acts as a black box for the definition and discovery of tree-width-like parameters in new settings: given a spined category as input, it yields an appropriate tree-width analogue as output. Finally we study temporal graphs: these are graphs whose edges appear and disappear over time. Many problems on temporal graphs are intractable even when their topology is severely restricted (such as being a tree or even a star); thus, to be able to conquer, we need decompositions that take temporal information into account. We take these considerations to heart and define a purely temporal width measure called interval-membership-width which allows us to employ dynamic programming (i.e. divide and conquer) techniques on temporal graphs whose times are sufficiently well-structured, regardless of the underlying topology