Constructions, inductive types and strong normalization

This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and type-checking, based on the equality-as-judgement presentation. We present a set-theoretic notion of model, CC-structures, and use this to give a new strong normalization proof based on a modification of the realizability interpretation. An extension of the core calculus by inductive types is investigated and we show, using the example of infinite trees, how the realizability semantics and the strong normalization argument can be extended to non-algebraic inductive types. We emphasize that our interpretation is sound for large eliminations, e.g. allows the definition of sets by recursion. Finally we apply the extended calculus to a non-trivial problem: the formalization of the strong normalization argument for Girard's System F. This formal proof has been developed and checked using the..

A computable expression of closure to efficient causation

International audienceIn this paper, we propose a mathematical expression of closure to efficient causation in terms of lambda-calculus; we argue that this opens up the perspective of developing principled computer simulations of systems closed to efficient causation in an appropriate programming language. An important implication of our formulation is that, by exhibiting an expression in lambda-calculus, which is a paradigmatic formalism for computability and programming, we show that there are no conceptual or principled problems in realizing a computer simulation or model of closure to efficient causation. We conclude with a brief discussion of the question whether closure to efficient causation captures all relevant properties of living systems. We suggest that it might not be the case, and that more complex definitions could indeed create crucial some obstacles to computability

Testing a Random Number Generator: formal properties and automotive application

L'elaborato analizza un metodo di validazione dei generatori di numeri casuali (RNG), utilizzati per garantire la sicurezza dei moderni sistemi automotive. Il primo capitolo fornisce una panoramica della struttura di comunicazione dei moderni autoveicoli attraverso l'utilizzo di centraline (ECU): vengono riportati i principali punti di accesso ad un automobile, assieme a possibili tipologie di hacking; viene poi descritto l'utilizzo dei numeri casuali in crittografia, con particolare riferimento a quella utilizzata nei veicoli. Il secondo capitolo riporta le basi di probabilità necessarie all'approccio dei test statistici utilizzati per la validazione e riporta i principali approcci teorici al problema della casualità. Nei due capitoli centrali, viene proposta una descrizione dei metodi probabilistici ed entropici per l'analisi di dati reali utilizzati nei test. Vengono poi descritti e studiati i 15 test statistici proposti dal National Institute of Standards and Technology (NIST). Dopo i primi test, basati su proprietà molto semplici delle sequenze casuali, vengono proposti test più sofisticati, basati sull'uso della trasformata di Fourier (per testare eventuali comportamenti periodici), dell'entropia (strettamente connessi con la comprimibilità della sequenza), o sui random path. Due ulteriori test, permettono di valutare il buon funzionamento del generatore, e non solo delle singole sequenze generate. Infine, il quinto capitolo è dedicato all'implementazione dei test al fine di testare il TRNG delle centraline

Beyond Logic. Proceedings of the Conference held in Cerisy-la-Salle, 22-27 May 2017

The project "Beyond Logic" is devoted to what hypothetical reasoning is all about when we go beyond the realm of "pure" logic into the world where logic is applied. As such extralogical areas we have chosen philosophy of science as an application within philosophy, informatics as an application within the formal sciences, and law as an application within the field of social interaction. The aim of the conference was to allow philosophers, logicians and computer scientists to present their work in connection with these three areas. The conference took place 22-27 May, 2017 in Cerisy-la-Salle at the Centre Culturel International de Cerisy. The proceedings collect abstracts, slides and papers of the presentations given, as well as a contribution from a speaker who was unable to attend

Towards a formally verified functional quantum programming language

This thesis looks at the development of a framework for a functional quantum programming language. The framework is first developed in Haskell, looking at how a monadic structure can be used to explicitly deal with the side-effects inherent in the measurement of quantum systems, and goes on to look at how a dependently-typed reimplementation in Agda gives us the basis for a formally verified quantum programming language. The two implementations are not in themselves fully developed quantum programming languages, as they are embedded in their respective parent languages, but are a major step towards the development of a full formally verified, functional quantum programming language. Dubbed the “Quantum IO Monad”, this framework is designed following a structural approach as given by a categorical model of quantum computation