Formalizing Constructive Analysis: A comparison of minimal systems and a study of uniqueness principles.

Αυτή η διατριβή εξετάζει ορισμένες πλευρές της τυποποίησης και της αξιωματικοποίησης της κατασκευαστικής ανάλυσης. Η έρευνα στους κλάδους της κατασκευαστικής ανάλυσης που αντιστοιχούν στις διάφορες εκδοχές κατασκευαστικότητας διεξάγεται σε μια πλειάδα τυπικών ή όχι συστημάτων, των οποίων οι σχέσεις είναι ασαφείς. Αυτό το πρόβλημα αποβαίνει κρίσιμο για την ανάπτυξη της σχετικά νέας περιοχής των κατασκευαστικών ανάστροφων μαθηματικών. Η εργασία αυτή συμβάλλει σε μια πιο καθαρή εικόνα. Το Μέρος 1 περιέχει μία ακριβή σύγκριση των δύο ευρύτερα χρησιμοποιούμενων συστημάτων που τυποποιούν τον κοινό πυρήνα της κατασκευαστικής, της ενορατικής, της αναδρομικής και της κλασικής ανάλυσης, των Μ και EL, των Kleene και Troelstra, αντιστοίχως. Αποδεικνύεται ότι το EL είναι ασθενέστερο από το M και ότι η διαφορά τους αποτυπώνεται από μια αρχή η οποία εγγυάται την ύπαρξη χαρακτηριστικής συνάρτησης για κάθε αποκρίσιμο κατηγόρημα φυσικών αριθμών. Με παρόμοια επιχειρήματα προκύπτουν συγκρίσεις για τα περισσότερα από τα χρησιμοποιούμενα ελαχιστικά συστήματα. Στην κατασκευαστική ανάλυση χρησιμοποιούνται διάφορες αρχές επιλογής, συνέχειας και άλλες. Στο Μέρος 2, μελετώνται σχέσεις μεταξύ πολλών από αυτές, στις εκδοχές τους με μία συνθήκη μοναδικότητας, ένα χαρακτηριστικό από το οποίο απορρέουν ενδιαφέρουσες ιδιότητες, καθώς και σχέσεις μεταξύ αυτών των αρχών και μη κατασκευαστικών λογικών αρχών, στο πνεύμα των ανάστροφων μαθηματικών.This dissertation investigates certain aspects of the formalization and axiomatization of constructive analysis. The research in the branches of constructive analysis corresponding to the various forms of constructivism is carried out in a multitude of formal or informal systems, whose relations are unclear. This problem becomes quite crucial for the development of the relatively new field of constructive reverse mathematics. This work contributes to a clearer picture. Part 1 contains a precise comparison of the two most widely used systems which formalize the common core of constructive, intuitionistic, recursive and classical analysis, namely Kleene's M and Troelstra's EL. It is shown that EL is weaker than M and that their difference is captured by a function existence principle asserting that every decidable predicate of natural numbers has a characteristic function. Applying similar arguments, comparisons of most of the used minimal systems are obtained. In constructive analysis, various forms of choice principles, continuity principles and many others are used. Part 2 studies relations between many of them, in their versions having a uniqueness condition, a feature from which interesting properties follow, as well as relations between these principles and non-constructive logical principles, in the spirit of reverse mathematics

A theoretical constructivisation of mathematical economics

This thesis deals with some problems in mathematical economics, looked at constructively; that is, with intuitionistic logic. In particular, we look at the connection between approximate Pareto optima and approximate equilibria. We then examine the classically vacuous, but constructively nontrivial, problem of locating the exact point where a line segment crosses the boundary of a convex subset of RN. We also prove the pointwise continuity of an associated boundary crossing mapping. Turning to a rather different aspect of the theory, we discuss Ekeland's Theorem giving approximate minima of certain functions, as well as some fundamental notions in related areas of optimisation. The thesis ends with a discussion of some problems associated with the possible constructivisation of McKenzie's proof of the existence of competitive equilibria

Contributions to the privacy provisioning for federated identity management platforms

Identity information, personal data and user’s profiles are key assets for organizations and companies by becoming the use of identity management (IdM) infrastructures a prerequisite for most companies, since IdM systems allow them to perform their business transactions by sharing information and customizing services for several purposes in more efficient and effective ways. Due to the importance of the identity management paradigm, a lot of work has been done so far resulting in a set of standards and specifications. According to them, under the umbrella of the IdM paradigm a person’s digital identity can be shared, linked and reused across different domains by allowing users simple session management, etc. In this way, users’ information is widely collected and distributed to offer new added value services and to enhance availability. Whereas these new services have a positive impact on users’ life, they also bring privacy problems. To manage users’ personal data, while protecting their privacy, IdM systems are the ideal target where to deploy privacy solutions, since they handle users’ attribute exchange. Nevertheless, current IdM models and specifications do not sufficiently address comprehensive privacy mechanisms or guidelines, which enable users to better control over the use, divulging and revocation of their online identities. These are essential aspects, specially in sensitive environments where incorrect and unsecured management of user’s data may lead to attacks, privacy breaches, identity misuse or frauds. Nowadays there are several approaches to IdM that have benefits and shortcomings, from the privacy perspective. In this thesis, the main goal is contributing to the privacy provisioning for federated identity management platforms. And for this purpose, we propose a generic architecture that extends current federation IdM systems. We have mainly focused our contributions on health care environments, given their particularly sensitive nature. The two main pillars of the proposed architecture, are the introduction of a selective privacy-enhanced user profile management model and flexibility in revocation consent by incorporating an event-based hybrid IdM approach, which enables to replace time constraints and explicit revocation by activating and deactivating authorization rights according to events. The combination of both models enables to deal with both online and offline scenarios, as well as to empower the user role, by letting her to bring together identity information from different sources. Regarding user’s consent revocation, we propose an implicit revocation consent mechanism based on events, that empowers a new concept, the sleepyhead credentials, which is issued only once and would be used any time. Moreover, we integrate this concept in IdM systems supporting a delegation protocol and we contribute with the definition of mathematical model to determine event arrivals to the IdM system and how they are managed to the corresponding entities, as well as its integration with the most widely deployed specification, i.e., Security Assertion Markup Language (SAML). In regard to user profile management, we define a privacy-awareness user profile management model to provide efficient selective information disclosure. With this contribution a service provider would be able to accesses the specific personal information without being able to inspect any other details and keeping user control of her data by controlling who can access. The structure that we consider for the user profile storage is based on extensions of Merkle trees allowing for hash combining that would minimize the need of individual verification of elements along a path. An algorithm for sorting the tree as we envision frequently accessed attributes to be closer to the root (minimizing the access’ time) is also provided. Formal validation of the above mentioned ideas has been carried out through simulations and the development of prototypes. Besides, dissemination activities were performed in projects, journals and conferences.Programa Oficial de Doctorado en Ingeniería TelemáticaPresidente: María Celeste Campo Vázquez.- Secretario: María Francisca Hinarejos Campos.- Vocal: Óscar Esparza Martí

Interpretation of quantum mechanics : a case-study in the sociology of science

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