Many temporal safety properties of higher-order programs go beyond simple event sequencing and require an automaton register (or “accumulator”) to express, such as input-dependency, event summation, resource usage, ensuring equal event magnitude, computation cost, etc. Some steps have been made towards verifying more basic temporal event sequences via reductions to fair termination [Murase et al. 2016] or some input-dependent properties through deductive proof systems [Nanjo et al. 2018]. However, there are currently no automated techniques to verify the more general class of register-automaton safety properties of higher-order programs.

We introduce an abstract interpretation-based analysis to compute dependent, register-automata effects of recursive, higher-order programs. We capture properties of a program’s effects in terms of automata that summarizes the history of observed effects using an accumulator register. The key novelty is a new abstract domain for context-dependent effects, capable of abstracting relations between the program environment, the automaton control state, and the accumulator value. The upshot is a dataflow type and effect system that computes context-sensitive effect summaries. We demonstrate our work via a prototype implementation that computes dependent effect summaries (and validates assertions) for OCaml-like recursive higher order programs. As a basis of comparison, we describe reductions to assertion checking for effect-free programs, and demonstrate that our approach outperforms prior tools Drift, MoCHi and RCaml/PCSat.

Polynomial models are ubiquitous in computer science, arising in the study of automata and formal languages, optimisation, game theory, control theory, and numerous other areas. In this thesis, we consider models described by polynomial systems of equations and difference equations, where the system evolves through a set of discrete time steps with polynomial updates at every step. We explore three aspects of “zero problems” for polynomial models: zero testing for algebraic expressions given by polynomials, determining the existence of zeros for polynomial systems and determining the existence of zeros for sequences satisfying recurrences with polynomial coefficients.

In the first part, we study identity testing for algebraic expressions involving radicals. That is, given a k-variate polynomial represented by an algebraic circuit and k real radicals, we examine the complexity of determining whether the polynomial vanishes on the radical input. We improve on the existing PSPACE bound, placing the problem in coNP assuming the Generalised Riemann Hypothesis (GRH). We further consider a restricted version of the problem, where the inputs are square roots of odd primes, showing that it can be decided in randomised polynomial time assuming GRH.

We next consider systems of polynomial equations, and study the complexity of determining whether a system of polynomials with polynomial coefficients has a solution. We present a number-theoretic approach to the problem, generalising techniques used for identity testing, showing the problem belongs to the complexity class AM assuming GRH. We discuss how the problem relates to determining the dimension of a complex variety, which is also known to belong to AM assuming GRH.

In the final part of this thesis, we turn our attention to sequences satisfying recurrences with polynomial coefficients. We study the question of whether zero is a member of a polynomially recursive sequence arising as a sum of two hypergeometric sequences. More specifically, we consider the problem for sequences where the polynomial coefficients split over the field of rationals Q. We show its relation to the values of the Gamma function evaluated at rational points, which allows to establish decidability of the problem under the assumption of the Rohrlich-Lang conjecture. We propose a different approach to the problem based on studying the prime divisors of the sequence, allowing us to establish unconditional decidability of the problem.

There are many ways to teach proof competence. Students try to use “perfect” proofs from textbooks or experts to create their own proofs in exercises, and they often fail. In particular, many students struggle with properly understanding the structure of proofs. We believe that—for the education of Computer Science students who are used to programming—it is natural to use proof assistants to learn to develop and write proofs. In addition, the immediate feedback from proof assistants helps students to check their proofs for correctness and completeness.

However, proof assistants are not made for beginners. Moreover, learning formal and mathematical language at the same time is often overwhelming. We develop the web interface ProofBuddy based on the proof assistant Isabelle as a tool that (also) targets for beginners. We chose Isabelle, because we assume that the proximity of its Isar language to natural/mathematical language supports the learning process. Pedagogically, we experiment with a setting in which students constantly switch between pen-and-paper-proofs and their formal counterparts. ProofBuddy is designed to support this back-and-forth approach.

In the talk, I report on the architecture and the user interface of ProofBuddy, give a short introduction to Isar within ProofBuddy and present our pedagogical approach.

Blind Signatures are a useful primitive for privacy preserving applications such as electronic payments, e-voting, anonymous credentials, and more.

However, existing practical blind signature schemes based on standard assumptions require either pairings or lattices. We present the first practical construction of a round-optimal blind signature in the random oracle model based on standard assumptions without resorting to pairings or lattices. In particular, our construction is secure under the strong RSA assumption and DDH (in pairing-free groups). For our construction, we provide a NIZK-friendly signature based on strong RSA, and efficiently instantiate a variant of Fischlin's generic framework (CRYPTO'06). Our Blind Signature scheme has signatures of size 4.28 KB and communication cost 10.98 KB. On the way, we develop techniques that might be of independent interest. In particular, we provide efficient *relaxed* range-proofs for large ranges with subversion zero-knowledge and compact commitments to elements of arbitrary groups.

Joint work with Michael Reichle and Ky Nguyen.

The Ahlbrandt-Ziegler result in model theory asserts that a countably categorical theory is characterised, up to bi-interpretability, by the topological automorphism group of its (unique) countable model, thus permitting a study of these theories by group-theoretic properties.

Removing the countable categoricity assumption necessitates using topological groupoids rather than groups. We will describe a groupoidal generalisation of the Ahlbrandt-Ziegler result using topos theory.

Our result extends the results in the topos-theoretic literature of Awodey, Butz, Caramello, Forssell and Moerdijk, while being orthogonal to the parallel generalisation of Ben Yaacov.

We will start a new series of non-permanent weekly seminars with an introduction section, during which everyone can briefly share their name, background, and research topic (if desired). This is a casual and friendly setting, so don't be shy!

After the session, we will have some refreshment and play some games, everyone is welcome.

Session types are a typed approach to ensure the communication correctness of message-passing programs. They have been extensively studied from a theoretical standpoint, and have been also incorporated into diverse programming languages and models.

Perhaps surprisingly, there is no single, canonical session type system, but many. It is fair to say that expressiveness has been key to the success and impact of session types: enhancing the expressivity of typed processes and/or the properties enforced by typing is often a strong motivation for developing new typed frameworks.

In this rich and diverse context, rigorously contrasting different variants of session types from an expressiveness perspective is a fascinating and non-trivial challenge. In this talk I introduce session types briefly and overview some recent works related to their expressiveness; they cover various angles, including “propositions-as-sessions”, i.e., the Curry-Howard correspondence between session types and linear logic.

Büchi's seminal 1962 paper established the decidability of the monadic second-order (MSO) theory of the structure (Nat; <), and in so doing brought to light the profound connections between mathematical logic and automata theory. This talk presents our recent results on the decidability of expansions (Nat; <, P1, …, Pd) for various unary predicates P1, …, Pd. We focus in particular on predicates of arithmetic origin, such as Pow-k (powers of k) and N-k (k-th powers). The following is a representative sample of the results yielded by our techniques, which combine automata theory, algebra, number theory, dynamical systems, and combinatorics:

- The MSO theory of (Nat; <, Pow-2, Pow-3) is decidable.

- The MSO theory of (Nat; <, Pow-2, Pow-3, Pow-6) is decidable.

- The MSO theory of (Nat; <, Pow-2, Pow-3, Pow-5) is decidable subject to Schanuel's conjecture.

- The MSO theory of (Nat; <, Squares, Pow-4) is decidable.

- The MSO theory of (Nat; <; Squares, Pow-2) is Turing-equivalent to the MSO theory of (Nat; <, S), where S is the unary predicate corresponding to the binary expansion of s = sqrt(2). As s is widely believed to be a weakly normal number, the corresponding MSO theory is in turn expected to be decidable.

The talk will give an overview of our techniques in general, and the automata theory and algebra that enable the above results in particular.

Dans ce premier exposé basé sur l’article « Univalence in simplicial sets » (https://arxiv.org/abs/1203.2553) je détaillerai la notion de « fibration universelle » qui en est l’ingrédient essentiel. J’examinerai en parallèle ce que devient une telle construction lorsqu’on remplace les ensembles simpliciaux par le cadre beaucoup plus simple des ensembles tout court.