Depuis plusieurs années, je participe au groupe de travail Homenet de l'IETF, l'organisme qui définit les normes qui régissent l'Internet. Dans cet exposé, sur l'exemple de Homenet et de mes aventures à l'IETF, j'essaierai de donner une idée de ce qu'est le processus de normalisation, pourquoi il est important pour l'Informatique et pour l'Internet.

Cet exposé est une version étendue et légèrement censurée de l'exposé que j'ai donné aux Journées PPS le 12 octobre 2017.

In higher-order model-checking (HOMC), functional programs are represented as higher-order recursion schemes (HORS), a kind of grammar with parameters which can be functions. These grammars generate infinite trees, over which we want to check formulas of monadic second-order logic (MSO). This problem is decidable, as proved first by Ong in 2006, and then by many others including Kobayashi and Ong (2009) whose approach use intersection types. It turns out that linear logic is a powerful and enlightening tool to reason about HOMC, as shown by Grellois and Melliès in the last years.

In general, the complexity of the HOMC problem is n-EXPTIME, where n is the order of the HORS of interest. In this talk, we explain that we can refine this complexity measure: motivated by linear logic, we introduce *linear* HORS and a linear-nonlinear model of automaton checking fragments of MSO. We then show that HOMC is in fact n-EXPTIME complete for n the *linear* order of the linear HORS generating the tree of interest. We believe that this explains why HOMC, in spite of its huge theoretical complexity, has been successfully used in practice by Kobayashi's team.

This linear framework allows to reprove in a much simpler way three existing results extending the usual HOMC problem, and notably to deal with call-by-value programs in a smooth way (the usual HOMC approach considering call-by-name).

This is joint work with Pierre Clairambault and Andrzej Murawski.

In this talk, we introduce a new game semantics framework for concurrency based on event structures, extending the work of Rideau and Winskel. In this framework, we can extend the notions of innocence and well-bracketing to the concurrent (and non-deterministic) case, generalizing the so-called “Abramsky cube”.

This talk focuses on the deterministic case. I will first introduce the concurrent strategies and their composition, in the existing linear setting. I will then present our extension to nonlinearity using copy indices and symmetry to represent uniformity. I will then present our notions of concurrent innocence & well-bracketing, to finish on our result of intensional full abstraction for PCF. Time permitting, I will discuss extensions of this result to non-angelic nondeterminism and probabilities.

Les algèbres dendriformes sont des structures algébriques introduites par Loday. Elle offrent un moyen de découper un produit associatif en deux produits non nécessairement associatifs. L'étude de ces algèbres, réalisée avec le point de vue fourni par la théorie des opérades, fait apparaître la combinatoire des arbres binaires et des mélanges d'arbres. Nous définissons ici une opérade à un paramètre entier généralisant l'opérade diassociative. Par dualité de Koszul, nous obtenons des généralisations de l'opérade dendriforme. Les algèbres sur ces opérades permettent de découper un produit associatif en plusieurs parties, avec certaines relations de compatibilité. Les propriétés combinatoires et algébriques de ces structures sont passées en revue.

There are two possible computational interpretations of second-order arithmetic: Girard's system F or Spector's bar recursion and its variants. While the logic is the same, the programs obtained from these two interpretations have a fundamentally different computational behavior and their relationship is not well understood. We make a step towards a comparison by defining the first translation of system F into a simply-typed total language with a variant of bar recursion. This translation relies on a realizability interpretation of second-order arithmetic. Due to Gödel's incompleteness theorem there is no proof of termination of system F within second-order arithmetic. However, for each individual term of system F there is a proof in second-order arithmetic that it terminates, with its realizability interpretation providing a bound on the number of reduction steps to reach a normal form. Using this bound, we compute the normal form through primitive recursion. Moreover, since the normalization proof of system F proceeds by induction on typing derivations, the translation is compositional. The flexibility of our method opens the possibility of getting a more direct translation that will provide an alternative approach to the study of polymorphism, namely through bar recursion.

Les tableaux boisés sont des objets combinatoires intervenant notamment dans l'étude de certains modèles de mécanique statistique. Cet exposé porte sur les arbres non ambigus, qui sont un cas particulier de tableaux boisés reliés aux permutations ayant leurs excédences (w(i)>i) en début de mot. Nous avons obtenu, lors d'un travail commun avec J.-C. Aval, A. Boussicault, F. Hivert et P. Laborde-Zubieta, des résultats énumératifs et bijectifs sur ces objets, que je présenterai ici après avoir introduit le cadre de cette étude.

Gradual typing is an approach for designing programming languages that integrate static and dynamic type checking. Gradual typing gives the programmer fine-grained control over which regions of a program are statically checked and which regions are dynamically checked. Over the last decade, there has been renewed interest in such an integration partly due to the rise in popularity of dynamic languages. But as small programs grow into large programs, so does the need for early error detection and modularity, which is provided by static type checking. Gradual typing provides a practical migration path for dynamically typed programs to become more statically typed. This talk will give a glimpse at the state of the art in gradual typing. It will describe a) the major challenges in the design and implementation of gradually typed languages, b) the research that has addressed many of of these challenges, and c) the open problems that need to be solved. The research in gradual typing spans both theoretical and practical questions, from mechanized metatheory to efficient compilation.

I will present a new proof assistant, Globular, which allows the formalization of compositional proofs in higher category theory. The interface is geometrical, with proofs depicted as string diagrams, and allowing direct click-and-drag manipulation of proofs in a simple and intuitive way, up to the level of 4-categories. I will show how to build a simple proof from scratch, and present some sophisticated formalized proofs from topology and algebra. I will also give some details of the theoretical basis, which gives a new and simple way to understand higher category theory. This talk will be accessible, and will not require any previous knowledge.

Web site: https://ncatlab.org/nlab/show/Globular Paper: https://arxiv.org/abs/1610.06908

Testing theory focuses on contextual equivalences that were proposed in the 80s as an alternative to bisimulation equivalence. During the last decade testing equivalences proved useful in constructing semantic models of session types and in laying the foundations of web-service technologies. As result, testing theory is more useful and richer than ever. In this seminar I will recall the chief ideas behind testing theory, and argue that we lack a general methodology to reason on testing equivalences. I will also outline the evidence that a meta-theory may exist, and some open questions.

The main purpose of this talk is to introduce a new automata model, hybrid set-vector automata, that naturally embed deterministic finite state automata and finite automata weighted over a field.

We take a category-theoretic approach, which provides a neat understanding of minimisation. It is well known that category theory offers a unifying view of some automata theory results. For example, minimisation of deterministic automata (over finite words) and Shützenberger’s automata weighted over fields, arise from the same categorical reasons.

In the first part of the talk, I will discuss about how to model and minimise automata in categories. Traditionally, automata are seen either as algebras for a functor plus a final map, possibly in a monoidal category, or as coalgebras for a functor plus an initial map. We propose yet another view of automata as functors from an input category to an output category.

The new hybrid set-vector automata can be modelled by taking the output category to be a free-colimit completion of the category of finite-dimensional vector spaces under a certain class of colimits.

This is joint work with Thomas Colocombet.

A very natural notion of equivalence of programs is the observational equivalence: two programs are equivalent if they have the same behaviour in any environment. More specifically in the lambda-calculus two terms M and N are equivalent if for every context C, the terms C[M] and C[N] are either both solvable or both unsolvable. Respecting this equivalence is an important property of some models of the lambda-calculus, called the full abstraction. It is well known that the model of infinitely extensional Böhm trees is fully abstract. The question we are interested in is what becomes of these notions when we introduce probabilistic choice in the calculus.

The solvability has a natural extension, which is the convergence probability, hence two terms are observationally equivalent if they have the same convergence probability under any context. There is also a simple way to define probabilistic Bohm trees, as we can associate to any term a subprobability distribution over head normal forms. But it is not obvious that these generalizations of the deterministic notions are still equivalent. In this talk we will show how to prove that probabilistic Böhm trees actually form a model, and how to get a separability result to ensure this model is fully abstract.

In functional programs, also called higher-order programs, functions may take functions themselves as arguments. As a result, their model-checking relies in most approaches on semantic or type-theoretic tools. In this talk, I will explain how an analysis based on linear logic of a model-checking result of 2009 by Kobayashi and Ong led Melliès and I to the construction of a model for model-checking. This model is such that, when interpreting a term with recursion representing the tree of traces of a functional program, its denotation determines whether it satisfies a MSO property of interest. A related and similar model was obtained independently by Salvati and Walukiewicz. In the second part of the talk, I will discuss the verification of termination for functional programs with recursion and probabilistic choice. Dal Lago and I defined recently a type system which is such that typable programs terminate with probability 1. In other terms, their set of diverging executions is negligible. If time allows, I will sketch ideas towards an extension of the model-checking results of the deterministic case to quantitative logics and functional programs with recursion.

The starting point of the talk will be the identification of structure common to tree-like combinatorial objects, exemplifying the situation with abstract syntax trees~(as used in formal languages) and with opetopes (as used in higher-dimensional algebra). The emerging mathematical structure will be then formalized in a categorical setting, unifying the algebraic aspects of the theory of abstract syntax of [2,3] and the theory of opetopes of [5]. This realization conceptually allows one to transport viewpoints between these, now bridged, mathematical theories and I will explore it here in the direction of higher-dimensional algebra, giving an algebraic combinatorial framework for a generalisation of the slice construction of [1] for generating opetopes. The technical work will involve setting up a microcosm principle for near-semirings and subsequently exploiting it in the cartesian closed bicategory of generalised species of structures of [4]. Connections to (cartesian and symmetric monoidal) equational theories, lambda calculus, and algebraic combinatorics will be mentioned in passing.

[1] J.Baez and J.Dolan. Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes. Advances in Mathematics 135, pages 145-206, 1998.

[2] M.Fiore, G.Plotkin and D.Turi. Abstract syntax and variable binding. In 14th Logic in Computer Science Conf. (LICS'99), pages 193-202. IEEE, Computer Society Press, 1999.

[3] M.Fiore. Second-order and dependently-sorted abstract syntax. In Logic in Computer Science Conf. (LICS'08), pages 57–68. IEEE, Computer Society Press, 2008.

[4] M.Fiore, N.Gambino, M.Hyland, and G.Winskel. The cartesian closed bicategory of generalised species of structures. In J. London Math. Soc.}, 77:203-220, 2008.

[5] S.Szawiel and M.Zawadowski. The web monoid and opetopic sets. In arXiv:1011.2374 [math.CT], 2010.

Dynamic programming is about exploiting sharing opportunities in recursive call trees of a function. If the pattern of sharing opportunities is known statically, a call dag can be built instead of a tree from the start.

In this talk, we present a generic framework for doing so and demonstrate it in action on two textbook examples of dynamic programming - Fibonacci and edit distance. We describe sharing opportunity patterns with systems of equations between node addresses. A technique from term rewriting systems, namely Knuth-Bendix completion, gives us unique normal forms for node addresses. A dag is represented as a spanning tree whose nodes are defined by addresses in normal form. Mathematically, the function taking an input for our function of interest into a call tree is a comonad coalgebra, node addresses are coterms and equations are coequations.

A Haskell implementation of this framework uses circular programming and a generic implementation of derivatives of functors.

We view this work as a showcase of how basic algorithmics and more advanced category theory can interact in a mutually beneficial way.

This joint work with Nicolas Wu, University of Bristol.