INRIA project-team $\pi r^2$

Thematic team Algebra and computation

Thematic team Analysis and conception of systems

Thematic team Proofs and programs

## Proofs, programs and systems

#### Day, hour and place

Thursday at 10:30am, room 3052

#### Contact(s)

#### Future talks

#### Past talks

Preuves, programmes et systèmes

jeudi 22 février 2018, 10h30, Salle 3052

**Pierre-Marie Pédrot** (Max Planck Institute for Software Systems, Saarbrücken, Germany) *Failure is Not an Option, or The Curry-Howard-Shadok correspondence*

No one is without knowing the famous Shadok principle « the more it fails, the more likely it will eventually succeed. » Taking inspiration from this unfathomable wisdom, we came up with a dependent type theory which allows failure, so that we could succeed more in proving things. Such a Shadok theory is justified by translating it away into vanilla dependent theory, just as a mundane compiler would, where failing can be interpreted as a call-by-name exception mechanism. En passant, this gives the first full syntactical model of the Calculus of Inductive Constructions introducing effects.

Alas! The right to fail succeeds a tad too much, insofar as the resulting Shadock type theory is logically inconsistent. Not being impressed in any way, we put order into this madness by requiring that no exception should ever reach toplevel, thanks to a clever use of a variant of Bernardy-Lasson syntactic parametricity. While the former model can be thought of as Friedman A-translation applied to CIC, the latter is no more than a principled variant of Kreisel's modified realizability that scales to dependent types. In particular, it readily gives a model of CIC that still has canonicity, strong normalization and decidable type-checking, while featuring new principles typical of modified realizability such as the independence of premises and unprovability of Markov's principle.

https://www.pédrot.fr/articles/exceptional.pdf

Preuves, programmes et systèmes

jeudi 15 février 2018, 10h30, Salle 3052

**Prakash Panangaden** (McGill University) *Topology, Order and Causal Structure*

The causal structure of spacetime defines a (pair of) natural order structures on the underlying set of events. Much of the analysis of cauasl structure involves a delicate interplay between order, topology and geometry. In view of the fundamental role of the causal order in certain approaches to quantum gravity as well as its fundamental role in concurrency theory one can ask whether the topology can be derived from pure order theoretic considerations.

In a remarkable example of serendipity, order theory has been developed by computer scientists and mathematicians in order to capture computability concepts. Dana Scott developed a notion of a continuous lattice or continuous poset with a view to capturing computability as continuity with a suitable topology that has come to be known as the Scott topology. This subject has acquired the name of ``domain theory.''

We applied domain theory to the problem of reconstructing the spacetime topology from the order and came up with a number of results about reconstruction of spacetime structure from just a countable dense set.

We prove that a globally hyperbolic spacetime with its causality relation is a bicontinuous poset whose interval topology is the manifold topology. From this one can show that from only a countable dense set of events and the causality relation, it is possible to reconstruct a globally hyperbolic spacetime in a purely order theoretic manner. The ultimate reason for this is that globally hyperbolic spacetimes belong to a category that is equivalent to a special category of domains called interval domains.

This was joint work with Keye Martin of Naval Research Laboratories.

Preuves, programmes et systèmes

jeudi 08 février 2018, 10h30, Salle 3052

**Vincent Laporte** (IMDEA Software) *Provably secure compilation of side-channel countermeasures: the case of cryptographic “constant-time”*

Software-based countermeasures provide effective mitigation against side-channel attacks, often with minimal efficiency and deployment overheads. Their effectiveness is often amenable to rigorous analysis: specifically, several popular countermeasures can be formalized as information flow policies, and correct implementation of the countermeasures can be verified with state-of-the-art analysis and verification techniques. However, in absence of further justification, the guarantees only hold for the language (source, target, or intermediate representation) on which the analysis is performed.

We consider the problem of preserving side-channel countermeasures by compilation for cryptographic “constant-time”, a popular countermeasure against cache-based timing attacks. We present a general method, based on the notion of 2-simulation, for proving that a compilation pass preserves the constant-time countermeasure. Using the Coq proof assistant, we verify the correctness of our method and of several representative instantiations.

Preuves, programmes et systèmes

jeudi 08 février 2018, 10h30, Salle 3052

**Séminaire Chocola** (ENS Lyon) *Rencontres Chocola de Février: Prakash Panangaden, Justin Hsu & Thomas Ehrhard*

http://chocola.ens-lyon.fr/events/meeting-2018-02-08

10:30 – 12:00

Prakash Panangaden (Mc Gill University, Canada) Quantitative Equational Logic

14:00 – 15:00

Justin Hsu (University College London, UK) From Couplings to Probabilistic Relational Program Logics

15:30 – 16:30

Thomas Ehrhard (IRIF, Univ. Paris Diderot) Stable and measurable functions on positive cones: a model of probabilistic functional languages with continuous types

Preuves, programmes et systèmes

jeudi 25 janvier 2018, 10h30, Salle 3052

**Justin Hsu** (University College of London) *From Couplings to Probabilistic Relational Program Logics*

Many program properties are relational, comparing the behavior of a program (or even two different programs) on two different inputs. While researchers have developed various techniques for verifying such properties for standard, deterministic programs, relational properties for probabilistic programs have been more challenging. In this talk, I will survey recent developments targeting a range of probabilistic relational properties, with motivations from privacy, cryptography, machine learning. The key idea is to meld relational program logics with an idea from probability theory, called a probabilistic coupling. The logics allow a highly compositional and surprisingly general style of analysis, supporting clean proofs for a broad array of probabilistic relational properties.

Bio: Justin Hsu is a post-doctoral researcher at the University College of London. He obtained his graduate degree from the University of Pennsylvania. His research interests span formal verification and theoretical computer science, including verification of randomized algorithms, differential privacy, and game theory.

Preuves, programmes et systèmes

jeudi 14 décembre 2017, 10h30, ENS Lyon

**Séminaire Chocola** () *ENS Lyon*

Preuves, programmes et systèmes

jeudi 14 décembre 2017, 10h30, Salle 3052

**Juliusz Chroboczek** (IRIF, Université Paris Diderot) *Homenet, l'IETF, et le processus de normalisation*

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.

Preuves, programmes et systèmes

jeudi 07 décembre 2017, 10h30, Salle 3052

**Charles Grellois** (Université Aix-Marseille) *Linearity in Higher-Order Recursion Schemes*

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.

Preuves, programmes et systèmes

jeudi 23 novembre 2017, 10h30, Salle 3052

**Simon Castellan** (Imperial College) *The parallel intensionally fully abstract model for PCF*

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.

Preuves, programmes et systèmes

jeudi 09 novembre 2017, 10h30, ENS Lyon

**Séminaire Chocola** () *ENS LYon*

Preuves, programmes et systèmes

jeudi 19 octobre 2017, 10h30, Salle 3052

**Martin Hyland** (DPMMS, University of Cambridge) *Understanding Computation in Game Semantics*

Preuves, programmes et systèmes

jeudi 15 juin 2017, 10h30, Salle 3052

**Samuele Giraudo** (Paris-Est Marne-la-Vallée) *Découpage d'associativité généralisé*

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.

Preuves, programmes et systèmes

mardi 09 mai 2017, 14h00, Salle 3052

**Valentin Blot** (Queen Mary University, London) *An interpretation of system F through bar recursion*

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.

Preuves, programmes et systèmes

jeudi 27 avril 2017, 10h30, Salle 3052

**Bérénice Delcroix-Oger** (Institut de Mathématiques de Toulouse) *Des arbres sans ambiguités*

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.

Preuves, programmes et systèmes

jeudi 27 avril 2017, 14h00, Salle 3052

**Jeremy Siek** (Indiana University) *The state of the art in gradual typing*

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.

Preuves, programmes et systèmes

jeudi 20 avril 2017, 10h30, Salle 3052

**Jamie Vicary** (Univ. of Oxford) *Formalizing Compositional Proofs*

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

Preuves, programmes et systèmes

jeudi 30 mars 2017, 10h00, Salle 3052

**Giovanni Bernardi** (IRIF) *Un, personne et cent mille: a meta theory for testing equivalences?*

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.

Preuves, programmes et systèmes

jeudi 30 mars 2017, 11h15, Salle 3052

**Daniela Petrisan** (IRIF) *Hybrid set-vector automata from a category-theoretic perspective*

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.

Preuves, programmes et systèmes

jeudi 23 mars 2017, 10h30, Salle 3052

**Thomas Leventis** (Institut de Mathématiques de Marseille) *Full Abstraction of the Probabilistic Böhm Trees*

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.

Preuves, programmes et systèmes

jeudi 16 mars 2017, 10h30, Salle 3052

**Charles Grellois** (INRIA - Univ. Bologna (Italie)) *Verifying properties of functional programs: from the deterministic to the probabilistic case*

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.

Preuves, programmes et systèmes

jeudi 23 février 2017, 10h30, Salle 3052

**Marcelo Fiore** (University of Cambridge) *An Algebraic Combinatorial Approach to the Abstract Syntax of Opetopic Structures*

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.

Preuves, programmes et systèmes

jeudi 26 janvier 2017, 10h30, Salle 3052

**Tarmo Uustalu** (Tallinn University of Technology) *Dynamic programming and coalgebras with sharing*

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.

Preuves, programmes et systèmes

mardi 10 janvier 2017, 11h00, Salle 3052

**Camell Kachour** (IRIF) *Sur des modèles algébriques d'infini-n-catégories faibles cubiques*

Preuves, programmes et systèmes

jeudi 01 décembre 2016, 10h30, Salle 3052

**Julien Lange** (Imperial College) *Building Graphical Choreographies From Communicating Machines: Principles and Applications*

Graphical choreographies, or global graphs, are general multiparty session specifications featuring expressive constructs such as forking, merging, and joining for representing application-level protocols. Global graphs can be directly translated into modelling notations such as BPMN and UML. In the first part of the talk, I will first present an algorithm whereby a global graph can be constructed from asynchronous interactions represented by communicating finite-state machines (CFSMs); and a sound characterisation of a subset of safe CFSMs from which global graphs can be constructed. In the second part, I will outline a few recent applications of this work to communicating timed automata and the Go programming language.

Preuves, programmes et systèmes

jeudi 24 novembre 2016, 10h30, Salle 3052

**Thibaut Balabonski** (LRI, Université Paris Sud) *Optimisation de programmes C++ concurrents*

Preuves, programmes et systèmes

jeudi 17 novembre 2016, 10h30, Salle 3052

**Bruno Barras** () *Exposé repoussé à début 2017*

Preuves, programmes et systèmes

jeudi 03 novembre 2016, 10h30, Salle 3052

**Guilhem Jaber** (IRIF) *Operational Nominal Game Semantics*

We introduce a fully-abstract intensional model for a polymorphic call-by-value language with higher-order references. As in game semantics, the denotation of a term is represented as a set of plays between the term and its environment. But rather than building it compositionally, by induction over the term, we generate it using a labelled transition system representing all the possible interactions between the term and any environment. Names, and more generally the theory of nominal sets, are crucially used to represent locations (i.e. memory addresses) and functional and polymorphic values. Freshness of such names then control the observational power of environments. Thanks to the presence of references, the observational power of environments is strong enough to avoid the need of performing a quotient of the model to be fully abstract. This gives us new principles to reason on effectul and polymorphic programs. This work has been done in collaboration with Nikos Tzevelekos (QMUL).

Preuves, programmes et systèmes

jeudi 06 octobre 2016, 10h30, Salle 3052

**Giulio Manzonetto** (LIPN) *New Results on Morris's Observational Theory — The Benefits of Separating the Inseparable*

We study the theory of contextual equivalence in the untyped lambda-calculus, generated by taking the normal forms as observables. Introduced by Morris in 1968, this is the original extensional lambda theory H+ of observational equivalence. On the syntactic side, we show that this lambda-theory validates the omega-rule, thus settling a long-standing open problem. On the semantic side, we provide sufficient and necessary conditions for relational graph models to be fully abstract for H+. We show that a relational graph model captures Morris's observational pre-order exactly when it is extensional and lambda-König. Intuitively, a model is lambda-König when every lambda-definable tree has an infinite path which is witnessed by some element of the model.

Preuves, programmes et systèmes

jeudi 29 septembre 2016, 10h30, Salle 3052

**Vincent Danos** (ENS) *Bayesian inversion by ω-complete cone duality*

The process of inverting Markov kernels relates to the important subject of Bayesian modelling and learning. In fact, Bayesian update is exactly kernel inversion. We investigate how and when Markov kernels (aka stochastic relations, or probabilistic mappings) can be inverted. We address the question both directly on the category of measurable spaces, and indirectly by interpreting kernels as Markov operators: For the direct option, we introduce a typed version of the category of Markov kernels and use the so-called ‘disintegration of measures’. Here, one has to specialise to measurable spaces borne from a simple class of topological spaces -e.g. Polish spaces (other choices are possible). Our method and result greatly simplify a recent development by Culbertson and Sturz. For the operator option, we use a cone version of the category of Markov operators (aka kernels seen as predicate transformers). That is to say, our linear operators are not just continuous, but are required to satisfy the stronger condition of being ω-chain-continuous. Prior work shows that one obtains an adjunction in the form of a pair of contravariant and inverse functors between the categories of L1- and L∞-cones. Inversion, seen through the operator prism, is just adjunction. No topological assumption is needed. We show that both categories (Markov kernels and ω-chain-continuous Markov operators) are related by a family of contravariant functors Tp for 1 ≤ p ≤ ∞. The Tp’s are Kleisli extensions of (duals of) conditional expectation functors introduced before. We prove that both notions of inversion agree when both defined: if f is a kernel, and f† its direct inverse, then T∞(f)† = T1(f†). This is a joint work with Fredrik Dahlqvist UCL, Ohad Kammar Oxford, Ilias Garnier ENS

Preuves, programmes et systèmes

jeudi 02 juin 2016, 10h30, Salle 3052

**Ugo Dal Lago** (Univ. Bologna) *Infinitary Lambda Calculi from a Linear Perspective*

We introduce a linear infinitary lambda-calculus in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted coinductively. The obtained calculus embeds the infinitary applicative lambda-calculus and is universal for computations over infinite strings. What is particularly interesting about the calculus is that the refinement induced by linear logic allows to restrict both modalities so as to get calculi which are terminating inductively and productive coinductively. We exemplify this idea by isolating a fragment of the calculus around the principles of SLL and 4LL. Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary lambda-calculi.

Preuves, programmes et systèmes

jeudi 21 avril 2016, 10h30, Salle 3052

**Silvia Ghilezan** (Université de Novi Sad) *Preciseness of Subtyping on Intersection and Union Types*

The notion of subtyping has gained an important role both in theoretical and applicative domains: in lambda and concurrent calculi as well as in programming languages. The soundness and the completeness, together referred to as the preciseness of subtyping, can be considered from two different points of view: denotational and operational. The former preciseness is based on the denotation of a type which is a mathematical object that describes the meaning of the type in accordance with the denotations of other expressions from the language. The latter preciseness has been recently developed with respect to type safety, i.e. the safe replacement of a term of a smaller type when a term of a bigger type is expected.

We propose a technique for formalising and proving operational preciseness of the subtyping relation in the setting of a concurrent lambda calculus with intersection and union types. The key feature is the link between typings and the operational semantics. We prove then sound- ness and completeness getting that the subtyping relation of this calculus enjoys both denotational and operational preciseness.

This is a joint work with Mariangiola Dezani-Ciancaglini.

Preuves, programmes et systèmes

mercredi 30 mars 2016, 10h30, Salle 3052

**Guillaume Munch-Maccagnoni** (Cambridge Computer lab) *Enriched-adjunction models and polarisation for modelling effects and resources*

(joint work with Marcelo Fiore and Pierre-Louis Curien)

I will present “linear call-by-push-value” enriched-adjunction models refining call-by-push-value models of effects and linear/non-linear models of linear logic, with higher-order functions, sums, and resource modalities, together with a theorem lifting linear models into cartesian ones. I will also present computational interpretations of these models as intuitionistic (linear) logics (LJ/ILL) where the order of evaluation matters, in the form of polarised calculi that satisfy usual properties of Barendregt-style lambda-calculus, and that have sound and coherent interpretations in the previous models. This suggests an approach to modelling proofs and programs with direct models and calculi.

Preuves, programmes et systèmes

jeudi 24 mars 2016, 10h30, Salle 3052

**Pawel Sobocinski** (University of Southampton) *TBA*

Preuves, programmes et systèmes

jeudi 10 mars 2016, 10h30, Salle du Conseil

**Thomas Seiller** (Department of Computer Science, University of Copenhagen - DIKU) *Complexity Constraints as Group Actions*

The purpose of this talk is to explain a new approach to complexity theory based on (dynamic) semantics of linear logic, whose aim is to enable techniques and invariants from ergodic theory (e.g. l^2-Betti numbers of a countable Borel equivalence relation) to be used in computational complexity.

The origins of the techniques can be traced to Girard's “geometry of interaction” (GoI) program using von Neumann algebras and the recent GoI-inspired results in complexity. However, this approach reaches its full strength when using the more combinatorial setting of Interaction Graphs models of (fragments of) linear logic. Using techniques akin to game semantics (with a bit of measure theory), we are able to characterise (predicate) complexity classes as the set of programs/proofs interpretations of type Pred[m] := Nat ⇒ Bool. These models are parametrised by a group of measure-preserving maps m (equivalently, by a measurable group action) and provide the first sketches of a conjectured correspondence between measurable group actions and complexity constraints.

Preuves, programmes et systèmes

jeudi 14 janvier 2016, 10h30, Salle 3052

**Amar Hadzihasanovic** (Oxford University) *String diagrams and the algebra of entanglement*

The GHZ and W states are two entangled quantum states of three qubits, that are inequivalent in the sense that one cannot be turned into the other by local (single-qubit) operations; this is reflected in their different communicational properties and use in cryptographic protocols. A few years ago, Coecke and Kissinger showed that one can associate, to the two states, two Frobenius algebras in the category of Hilbert spaces - a type of algebra with a well-understood string diagram representation, which could hopefully provide a bridge between algebraic, computational and topological aspects of quantum entanglement. We present a complete graphical axiomatisation of the relations between the GHZ and W states/algebras: the ZW calculus. This calculus refines the pre-existing ZX calculus, while keeping its most desirable characteristics, such as the undirectedness of diagrams; comes with an explicit normalisation procedure; provides an original decomposition of the category of qubits, with a prominent “fermionic” fragment; and hints at a topological explanation of its components and axioms.