## Section: New Results

### Logical Foundations of Programming Languages

Participants : Rémi Douence, Ambroise Lafont, Étienne Miquey, Xavier Montillet, Guillaume Munch-Maccagnoni, Nicolas Tabareau, Pierre Vial.

#### Classical Logic

##### A sequent calculus with dependent types for classical arithmetic.

In a recent paper, Herbelin developed a calculus $dPA\omega $ in which constructive proofs for the axioms of countable and dependent choices could be derived via the encoding of a proof of countable universal quantification as a stream of it components. However, the property of normalisation (and therefore the one of soundness) was only conjectured. The difficulty for the proof of normalisation is due to the simultaneous presence of dependent dependent types (for the constructive part of the choice), of control operators (for classical logic), of coinductive objects (to encode functions of type $N\to A$ into streams $({a}_{0},{a}_{1},\cdots )$) and of lazy evaluation with sharing (for these coinductive objects).Building on previous works, we introduce in [14], [26] a variant of $dPA\omega $ presented as a sequent calculus. On the one hand, we take advantage of a variant of Krivine classical realisability we developed to prove the normalisation of classical call-by-need. On the other hand, we benefit of $dL$, a classical sequent calculus with dependent types in which type safety is ensured using delimited continuations together with a syntactic restriction. By combining the techniques developed in these papers, we manage to define a realisability interpretation à la Krivine of our calculus that allows us to prove normalisation and soundness.

##### Realisability Interpretation and Normalisation of Typed Call-by-Need $\lambda $-calculus With Control.

In [13], we define a variant of realisability where realisers are pairs of a term and a substitution. This variant allows us to prove the normalisation of a simply-typed call-by-need $\lambda $−calculus with control due to Ariola et al. Indeed, in such call-by-need calculus, substitutions have to be delayed until knowing if an argument is really needed. In a second step, we extend the proof to a call-by-need $\lambda $-calculus equipped with a type system equivalent to classical second-order predicate logic, representing one step towards proving the normalisation of the call-by-need classical second-order arithmetic introduced by the second author to provide a proof-as-program interpretation of the axiom of dependent choice.

#### Lambda Calculus

##### Every $\lambda $-Term is Meaningful for the Infinitary Relational Model.

Infinite types and formulas are known to have really curious and unsound behaviours. For instance, they allow to type $\Omega $, the auto-autoapplication and they thus do not ensure any form of normalisation/productivity. Moreover, in most infinitary frameworks, it is not difficult to define a type R that can be assigned to every $\lambda $-term. However, these observations do not say much about what coinductive (i.e. infinitary) type grammars are able to provide: it is for instance very difficult to know what types (besides R) can be assigned to a given term in this setting. In [17], we begin with a discussion on the expressivity of different forms of infinite types. Then, using the resource-awareness of sequential intersection types (system S) and tracking, we prove that infinite types are able to characterise the arity of every λ-terms and that, in the infinitary extension of the relational model, every term has a " meaning " i.e. a non-empty denotation. From the technical point of view, we must deal with the total lack of guarantee of productivity for typable terms: we do so by importing methods inspired by first order model theory.

##### High-level signatures and initial semantics.

In [9], we present a device for specifying and reasoning about syntax for datatypes, programming languages, and logic calculi. More precisely, we consider a general notion of signature for specifying syntactic constructions. Our signatures subsume classical algebraic signatures (i.e., signatures for languages with variable binding, such as the pure lambda calculus) and extend to much more general examples. In the spirit of Initial Semantics, we define the syntax generated by a signature to be the initial object—if it exists—in a suitable category of models. Our notions of signature and syntax are suited for compositionality and provide, beyond the desired algebra of terms, a well-behaved substitution and the associated inductive/recursive principles. Our signatures are general in the sense that the existence of syntax is not automatically guaranteed. In this work, we identify a large class of signatures which do generate a syntax. This paper builds upon ideas from a previous attempt by Hirschowitz-Maggesi (FICS 2012), which, in turn, was directly inspired by some earlier work of Ghani-Uustalu and Matthes-Uustalu. The main results presented in the paper are computer-checked within the UniMath system.

#### Models of programming languages mixing effects and resources

##### A resource modality for RAII

Systems programming languages C++11 and Rust have developed techniques
and idioms for the safe management of resources called
*“Resource acquisition is initialisation”* (RAII) and
*move semantics*. We have related resources from systems
programming to the notion of resource put forward by linear logic, by
giving a construction in terms of categorical semantics for a resource
modality that model RAII and move semantics. This work was presented
by at the workshop LOLA 2018 in Oxford [20].

##### Resource polymorphism

Thanks to a new logical and semantic understanding of resource-management techniques in systems programming languages, we have proposed [27] a design for an extension of functional programming language towards systems programming, centred on the OCaml language, and based on a notion of resource polymorphism inspired by the C++11 language and by the works on polarisation in proof theory.

#### Distributed Programming

##### Chemical foundations of distributed aspects.

Distributed applications are challenging to program because they have to deal with a plethora of concerns, including synchronisation, locality, replication, security and fault tolerance. Aspect-oriented programming (AOP) is a paradigm that promotes better modularity by providing means to encapsulate cross-cutting concerns in entities called aspects. Over the last years, a number of distributed aspect-oriented programming languages and systems have been proposed, illustrating the benefits of AOP in a distributed setting. Chemical calculi are particularly well-suited to formally specify the behaviour of concurrent and distributed systems. The join calculus is a functional name-passing calculus, with both distributed and object-oriented extensions. It is used as the basis of concurrency and distribution features in several mainstream languages like C# (Polyphonic C#, now C$\omega $), OCaml (JoCaml), and Scala Joins. Unsurprisingly, practical programming in the join calculus also suffers from modularity issues when dealing with crosscutting concerns. We propose the Aspect Join Calculus [8], an aspect-oriented and distributed variant of the join calculus that addresses crosscutting and provides a formal foundation for distributed AOP. We develop a minimal aspect join calculus that allows aspects to advise chemical reactions. We show how to deal with causal relations in pointcuts and how to support advanced customisable aspect weaving semantics.