Long reals

The familiar continuum R of real numbers is obtained by a well-known procedure which, starting with the set of natural numbers N=\omega, produces in a canonical fashion the field of rationals Q and, then, the field R as the completion of Q under Cauchy sequences (or, equivalently, using Dedekind cuts). In this article, we replace \omega by any infinite suitably closed ordinal \kappa in the above construction and, using the natural (Hessenberg) ordinal operations, we obtain the corresponding field \kappa-R, which we call the field of the \kappa-reals. Subsequently, we study the properties of the various fields \kappa-R and develop their general theory, mainly from the set-theoretic perspective. For example, we investigate their connection with standard themes such as forcing and descriptive set theory

A Computable Economist’s Perspective on Computational Complexity

A computable economist.s view of the world of computational complexity theory is described. This means the model of computation underpinning theories of computational complexity plays a central role. The emergence of computational complexity theories from diverse traditions is emphasised. The unifications that emerged in the modern era was codified by means of the notions of efficiency of computations, non-deterministic computations, completeness, reducibility and verifiability - all three of the latter concepts had their origins on what may be called "Post's Program of Research for Higher Recursion Theory". Approximations, computations and constructions are also emphasised. The recent real model of computation as a basis for studying computational complexity in the domain of the reals is also presented and discussed, albeit critically. A brief sceptical section on algorithmic complexity theory is included in an appendix.

A Computable Economist’s Perspective on Computational Complexity

A computable economist's view of the world of computational complexity theory is described. This means the model of computation underpinning theories of computational complexity plays a central role. The emergence of computational complexity theories from diverse traditions is emphasised. The unifications that emerged in the modern era was codified by means of the notions of efficiency of computations, non-deterministic computations, completeness, reducibility and verifiability - all three of the latter concepts had their origins on what may be called 'Post's Program of Research for Higher Recursion Theory'. Approximations, computations and constructions are also emphasised. The recent real model of computation as a basis for studying computational complexity in the domain of the reals is also presented and discussed, albeit critically. A brief sceptical section on algorithmic complexity theory is included in an appendix

A Primer on the Tools and Concepts of Computable Economics

Computability theory came into being as a result of Hilbert's attempts to meet Brouwer's challenges, from an intuitionistc and constructive standpoint, to formalism as a foundation for mathematical practice. Viewed this way, constructive mathematics should be one vision of computability theory. However, there are fundamental differences between computability theory and constructive mathematics: the Church-Turing thesis is a disciplining criterion in the former and not in the latter; and classical logic - particularly, the law of the excluded middle - is not accepted in the latter but freely invoked in the former, especially in proving universal negative propositions. In Computable Economic an eclectic approach is adopted where the main criterion is numerical content for economic entities. In this sense both the computable and the constructive traditions are freely and indiscriminately invoked and utilised in the formalization of economic entities. Some of the mathematical methods and concepts of computable economics are surveyed in a pedagogical mode. The context is that of a digital economy embedded in an information society

Distributed Differential Privacy and Applications

Recent growth in the size and scope of databases has resulted in more research into making productive use of this data. Unfortunately, a significant stumbling block which remains is protecting the privacy of the individuals that populate these datasets. As people spend more time connected to the Internet, and conduct more of their daily lives online, privacy becomes a more important consideration, just as the data becomes more useful for researchers, companies, and individuals. As a result, plenty of important information remains locked down and unavailable to honest researchers today, due to fears that data leakages will harm individuals. Recent research in differential privacy opens a promising pathway to guarantee individual privacy while simultaneously making use of the data to answer useful queries. Differential privacy is a theory that provides provable information theoretic guarantees on what any answer may reveal about any single individual in the database. This approach has resulted in a flurry of recent research, presenting novel algorithms that can compute a rich class of computations in this setting. In this dissertation, we focus on some real world challenges that arise when trying to provide differential privacy guarantees in the real world. We design and build runtimes that achieve the mathematical differential privacy guarantee in the face of three real world challenges: securing the runtimes against adversaries, enabling readers to verify that the answers are accurate, and dealing with data distributed across multiple domains

CALF: Categorical Automata Learning Framework

Automata learning is a popular technique used to automatically construct an automaton model from queries, and much research has gone into devising specific adaptations of such algorithms for different types of automata. This thesis presents a unifying approach to many existing algorithms using category theory, which eases correctness proofs and guides the design of new automata learning algorithms. We provide a categorical automata learning framework---CALF---that at its core includes an abstract version of the popular L* algorithm. Using this abstract algorithm we derive several concrete ones. We instantiate the framework to a large class of Set functors, by which we recover for the first time a tree automata learning algorithm from an abstract framework, which moreover is the first to cover also algebras of quotiented polynomial functors. We further develop a general algorithm to learn weighted automata over a semiring. On the one hand, we identify a class of semirings, principal ideal domains, for which this algorithm terminates and for which no learning algorithm previously existed; on the other hand, we show that it does not terminate over the natural numbers. Finally, we develop an algorithm to learn automata with side-effects determined by a monad and provide several optimisations, as well as an implementation with experimental evaluation. This allows us to improve existing algorithms and opens the door to learning a wide range of automata