Herbrand's theorem exposes some of the constructive content of classical logic. In its simplest form, it reduces the validity of a first-order purely existential formula to that of a finite disjunction. More generally, it reduces first-order validity to propositional validity, by understanding the structure of the assignment of first-order terms to existential quantifiers, and the causal dependency between quantifiers.

In this talk, we show that Herbrand's theorem in its general form can be elegantly stated and proved as a theorem in the framework of concurrent games, a denotational semantics designed to faithfully represent causality and independence in concurrent systems. Closely related to expansion trees, the causal structure of concurrent strategies, paired with annotations by first-order terms, is used to specify the dependency between quantifiers. As furthermore these strategies can be composed we are able to interpret classical sequent proofs, yielding a compositional proof of Herbrand's theorem.

http://chocola.ens-lyon.fr/events/meeting-2018-12-13/

Implicative Algebras were recently introduced as a unified framework for forcing and realisability, whose particularity is to interpret terms and formulæ uniformly.

In this ongoing work with Alexandre Miquel, we show how linear logic fits in this picture: we present a family of models, where certain computational principles are distinguished, and argue thata it provides a satisfactory meaning explanation of the connectives of (perfective) linear logic.

In this talk I will review G4ip, a simple calculus for intuitionistic propositional logic (IPL) that provides a decision procedure for provability. Its basic mechanisms can be traced back to Vorob'ev in the 50s, and were found again by Hudelmaier (88), Dyckhoff (90), Paulson (91), and (with a linear logic flavour) Lincoln-Scedrov-Shankar (91). In 2015, Claessen and Rosén presented the fastest decision procedure for IPL, based on SMT-solving techniques. We describe their algorithm and show how it can be seen as a variant of the aforementioned calculus, albeit with key variations that provide increased performance. Their algorithm relies on a SAT-solver used as a black box and treats intuitionistic entailment as a theory. We show how the recent framework of model-constructing satisfiability for SMT-solving could further integrate intuitionistic reasoning within the main SAT-solving loop. This would constitute an intuitionistic variant of the main algorithm of SAT-solvers, where Kripke models are built instead of Boolean models.

11:00 – 12:30 Éric Tanter (Universidad de Chile & Inria Paris) - Invited talk: Abstracting Gradual Typing: Principles and Application to Gradual Parametricity

  Gradual typing enables programming languages to seamlessly combine dynamic and static checking. Language researchers and designers have extended a wide variety of type systems to support gradual typing. These efforts consistently demonstrate that designing a satisfactory gradual counterpart to a static type system is challenging, and this challenge only increases with the sophistication of the type system. Gradual type system designers need more formal tools to help them conceptualize, structure, and evaluate their designs. Based on an understanding of gradual types as abstractions of static types, we have developed systematic foundations for gradual typing based on abstract interpretation, called Abstracting Gradual Typing (AGT for short). Gradual languages designed with the AGT approach satisfy, by construction, the established criteria for gradually-typed languages. In this talk, I will give a brief introduction to AGT in a simply-typed setting, and will then discuss recent work on a gradual counterpart of System F that enforces relational parametricity, even in the presence of imprecise types.

14:00 – 15:00 Flavien Breuvart (Université Paris 13) - Invited talk: Graded Types Parametricity : Principles and Application to Abstract Interpretation

  In this talk, we will briefly present the yet-to-start project CoGITARe, which aims at developing further graded types expressivity and inference in order to use them to perform abstract interpretations.
  In a second time, we give an historic on graded type systems and their expressivity. Then, we will see that this expressivity is limited by a lake of dependency/parametricity. We will thus explore two ongoing research directions that adds two very different kind of parametricity to graded types.
  (The provocative title should only be taken as a joke. Despite their apparent similitudes, this work and Éric Tanter's have vastly different objectives and involved different issues and techniques. In particular, notice that gradual and graded types are fundamentally different.)

15:30 – 16:30 Shane Mansfield (Univ. of Oxford) - Invited talk: An overview of empirical models

  Empirical models are a way of formalising data that arises in physical experiments. They were first proposed by Abramsky and Brandenburger as part of a framework to analyse fundamentally non-classical phenomena in quantum systems. Conveniently from a computer science perspective, they abstract away from the mathematical baggage of quantum theory and instead allow the key phenomena to be characterised purely as features of empirical data. After introducing the basic framework I will discuss some more recent results and developments, drawing on joint work with a number of collaborators. In particular: quantum computations can simply be modelled as classical computations with the additional ability to interact with a resource empirical model; quantitative measures of non-classicality can be shown to relate directly to some basic quantum-over-classical computational advantages; and the beginnings of a category-theoretic approach to reasoning about empirical models have emerged.

18:00 – 19:00 Xavier Leroy - Lecture: Leçon inaugurale au Collège de France

  This is not part of the CHoCoLa day, but you might be interested in attending! See https://www.college-de-france.fr/site/xavier-leroy/inaugural-lecture-2018-11-15-18h00.htm

  The meeting will take place in Bâtiment Sophie Germain, here.
  We will first gather at the 3rd floor of the building, for a breakfast.
  We will then move for the talk of the morning (11.00-12.30) to Amphi Turing.
  We will have lunch in room 3052
  In the afternoon (14h-16h30), we will be in room 1016

Journées PPS 2018

Journées PPS 2018

We propose a type-based analysis to infer the session protocols of channels in an ML-like concurrent functional language. Combining and extending well-known techniques, we develop a type-checking system that separates the underlying ML type system from the typing of sessions. Without using linearity, our system guarantees communication safety and partial lock freedom. It also supports provably complete session inference with no programmer annotations. We exhibit the usefulness of our system with interesting examples, including one which is not typable in substructural type systems.

Guarded recursion has emerged as a natural paradigm for programming with infinite data structures in type theory and high-assurance functional languages. In the first part of this talk, I will present some intuitions behind guarded recursion, using programming examples. In a second part, I will discuss ongoing work on a typed lambda-calculus equipped with rich facilities for defining and manipulating guarded recursive types.

(joint work with J. Weinberger)

Abstract: Voevodsky's model of the Univalence Axion in simplicial sets is not constructive as shown by Coquand et al. To avoid this problem Cohen, Coquand, Huber, Moertberg have constructed in Cubical Sets a model of a Cubical Type Theory which has computational meaning.

We show that simplicial sets form a subtopos of cubical sets which allows one to contructively reason in the latter about the former.

I will give a new definition of partial Higher Dimension Automata – a geometric model for true concurrency – using lax functors. This definition is simpler than the original, and more natural from a categorical point of view, but also matches more clearly the intuition that pHDA are Higher Dimensional Automata with some missing faces. I will then focus on trees. Originally, for example in transition systems, trees are defined as those systems that have a unique path property. To understand what kind of unique path property is needed in pHDA, I will start by looking at trees as colimits of paths. From this, I will deduce that trees are exactly the pHDA with the unique path property modulo a notion of homotopy, and without any shortcuts. This property will allow me to prove two interesting characterisations of trees: trees are exactly those pHDA that are the unfolding of another pHDA; trees are exactly the cofibrant objects, much as in the language of Quillen’s model structures. In particular, this last characterisation gives the premisses of a new understanding of concurrency theory using homotopy theory.

In a 2012 paper, Richard Statman exhibited an inference system, based on second order monadic logic and non-terminating rewrite rules, that exactly types all strongly normalizable lambda-terms. We show that this system can be simplified to first-order minimal logic with rewrite rules, along the Deduction Modulo Theory lines.

We show that our rewrite system is terminating and that the conversion rule respects weak versions of invertibility of the arrow and of quantifiers. This requires additional care, in particular in the treatment of the latter. Then we study proof reduction, and show that every typable proof term is strongly normalizable and vice-versa.

On discutera différentes manières de relier syntaxiquement la logique linéaire classique (LL) et sa version intuitionniste (ILL) : d'une part à travers la question de la conservativité de LL sur ILL (contre-exemples et conditions suffisantes), et d'autre part via l'étude de non-non traductions de LL dans ILL. On montrera que ces traductions permettent de représenter des extensions de LL dans ILL, mais aussi qu'elles peuvent être utilisées comme outil pour prouver des propriétés de LL : élimination des coupures et focalisation par exemple. Ces différents résultats ont été formalisés en Coq à l'aide de la bibliothèque Yalla, que nous présenterons brièvement.

There exists a large spectrum of techniques for proving correctness of computer programs. At one end is the Curry-Howard isomorphism, in which a typing procedure automatically checks the correctness of a program, and at the other end are techniques à la Hoare, in which non-trivial reasonning is performed in a logic adapted to the programming language. Realizability can be seen as a mixture of these two approaches: proving that a program realizes a specification is done via a combination of automatic techniques in a Curry-Howard fashion, and manual proofs of correctness of subprograms with respect to some axioms. In my talk I will present some realizability models where the realizability relation is defined semantically: a program realizes a specification if its denotation satisfies a property in the model. The denotational models involved include game semantics, continuation models and Scott domains.

This talk will present SyTeCi, the first general automated tool to check contextual equivalence for programs written in a typed higher-order language with references (i.e. local mutable states), corresponding to a fragment of OCaml.

After introducing the notion of contextual equivalence, we will see on some examples why it is hard to prove such equivalences (reentrant calls, private states). As we will see, such examples can be found in many different programming languages.

Then, we will introduce SyTeCi, a tool to automatically check such equivalences. This tool is based on a reduction of the problem of contextual equivalence of two programs to the problem of reachability of “error states” in transition systems of memory configurations.

Contextual equivalence being undecidable (even in a finitary setting), so does the non-reachability problem for such transition systems. However, one can apply model-checking techniques (predicate abstraction, summarization of pushdown systems) to check non-reachability via some approximations. This allows us to prove automatically many non-trivial examples of the literature, that could only be proved by hand before. We will end this talk by the presentation of a prototype implementing this work.

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

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.

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.

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

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.

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.

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.

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).

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.

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

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.

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.

(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.

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.

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.