## Section: New Results

### Effects in proof theory and programming

Participants : Guillaume Claret, Pierre-Louis Curien, Hugo Herbelin, Étienne Miquey, Ludovic Patey, Pierre-Marie Pédrot, Yann Régis-Gianas, Alexis Saurin.

#### Axiom of dependent choice in classical arithmetic

In 2012, Hugo Herbelin showed that classical arithmetic in finite types extended with strong elimination of existential quantification proves the axiom of dependent choice. To get classical logic and choice together without being inconsistent is made possible first by constraining strong elimination of existential quantification to proofs that are essentially intuitionistic and secondly by turning countable universal quantification into an infinite conjunction of classical proofs evaluated along a call-by-need evaluation strategy so as to extract from them intuitionistic contents that complies to the intuitionistic constraint put on strong elimination of existential quantification. Étienne Miquey has been working on a sequent-calculus version of this system, using Danvy's methodology of semantic artifacts, to progressively reduce the consistency of such a system to the normalisation of Girard-Reynold's system F. To achieve this goal, he incidentally proposed a way to get a dependently-typed sequent calculus, as well as a method to type a state-and-continuation-passing style translation of call-by-need calculus.

#### The computational contents of completeness proofs

Hugo Herbelin worked on the computational content of Gödel's completeness theorem, developing a proof with side-effects suitable for normalisation-by-evaluation.

#### Gödel's functional interpretation

Pierre-Marie Pédrot extended the proof-as-program interpretation of Gödel's Dialectica translation to the fully dependent setting, including dependent elimination [17] .

#### Logical foundations of call-by-need evaluation

Alexis Saurin and Pierre-Marie Pédrot extended their reconstruction of call-by-need based on linear head reduction with control. They showed how linear head reduction could be adapted to the $\lambda \mu $-calculus. This classical linear head reduction lifts the usual properties of the intuitionistic one (with respect to $\sigma $-equivalence) to the $\lambda \mu $-calculus (and its $\sigma $-equivalence already formulated by Olivier Laurent in his PhD thesis). Moreover, they showed that substitution sequences of the $\lambda \mu $-calculus linear head reduction are in correspondence with the classical Krivine abstract machine substitution sequences, validating the known fact that the KAM implements linear head reduction. In a second step, they could lift to the $\lambda \mu $-calculus their three-step transformation from linear head reduction to call-by-need, and study the correspondence with Ariola, Herbelin and Saurin's classical call-by-need. This work appeared as one of the chapters of Pierre-Marie Pédrot's thesis and has been accepted for publication at ESOP'16 [30] .

#### Call-by-name forcing

Pierre-Marie Pédrot studied variants of the forcing construction by decomposing it through call-by-push-value. In particular, the by-name decomposition behaves much more nicely w.r.t. the computational content of proofs and is a candidate for a dependently-typed extension. This work is partially reported on in his PhD [17] .

#### A theory of effects and resources

In joint work with Marcelo Fiore and Guillaume Munch-Maccagnoni, Pierre-Louis Curien considered the Curry-Howard-Lambek correspondence for effectful computation and resource management, specifically proposing polarised calculi together with presheaf-enriched adjunction models as the starting point for a comprehensive semantic theory relating logical systems, typed calculi, and categorical models in this context. Our thesis is that the combination of effects and resources should be considered orthogonally. Model theoretically, this leads to an understanding of our categorical models from two complementary perspectives: (i) as a linearisation of CBPV (Call-by-Push-Value) adjunction models, and (ii) as an extension of linear/non-linear adjunction models with an adjoint resolution of computational effects. When the linear structure is cartesian and the resource structure is trivial, we recover Levy’s notion of CBPV adjunction model, while when the effect structure is trivial, we have Benton’s linear/non-linear adjunction models. Further instances of our model theory include the dialogue categories with a resource modality of Melliès and Tabareau, and the Enriched Effect Calculus models of Egger, Møgelberg and Simpson. Our development substantiates the approach by providing a lifting theorem of linear models into cartesian ones. To each of our categorical models we systematically associate a typed term calculus, each of which corresponds to a variant of the sequent calculi LJ (Intuitionistic Logic) or ILL (Intuitionistic Linear Logic). The adjoint resolution of effects corresponds to polarisation whereby, syntactically, types locally determine a strict or lazy evaluation order and, semantically, the associativity of cuts is relaxed. In particular, our results show that polarisation provides a computational interpretation of CBPV in direct style. Further, we characterise depolarised models: those where the cut is associative, and where the evaluation order is unimportant. This work will be presented at POPL 2016 [26] .

#### Coq as a programming language with effects

As part of his PhD thesis, Guillaume Claret defined a notion of effectful interactive computation as an embedded DSL in Coq (in the spirit of the works on algebraic effects), and used it to implement a web server. It is equipped with a dual notion of effectful interactive execution context. Using these two notions together, Guillaume Claret is able to specify and reason about interactive programs inside Coq. He submitted several papers about this line of work: one has been published [32] , others will be part of his PhD manuscript.