## Section: New Results

### Proof-theoretical and effectful investigations

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

#### Sequent calculus and computational duality

*Categorical semantics.*

During his collaboration with Marcelo Fiore and Pierre-Louis Curien, Guillaume Munch-Maccagnoni characterised the polarised evaluation order through a categorical structure where the hypothesis that composition is associative is relaxed. Duploid is the name of the structure, as a reference to Jean-Louis Loday's duplicial algebras. The main result, in the lineage of Führmann's [38] direct-style characterisation of monadic models, is a reflection Adj$\to $Dupl where Dupl is a category of duploids and duploid functors, and Adj is the category of adjunctions and pseudo maps of adjunctions. The result suggests that the various biases in denotational semantics: indirect, call-by-value, call-by-name... are a way of hiding the fact that composition is not always associative. This work was accepted for publication in FoSSaCs 2014 [53] .

Pierre-Louis Curien, in connection with his increasing interests in operads and algebraic structures of various kinds, found out that the core syntax of system L (underlying the duality of computation) could be used with profit to describe the wiring structures underlying operads, dioperads, cyclic operads, and more generally Lamarche's structads [48] . He also showed a syntactic equivalence between Munch-Maccagnoni's (pre)duploids and system L syntax. These results were presented in his invited talks at the Loday's Mathematical Legacy workshop in Strasbourg and at the workshop Algebra and Computation in Lyon, in January 2014.

*Duality of construction.*

Paul Downen and Zena Ariola developed a generalized theory of the sequent calculus for understanding the concepts of evaluation strategy and of data (for example, pairs in ML) and co-data (for example, functions) in programming languages. This theory provides a single framework for user-defined data and co-data types as well as a generalized treatment of evaluation strategies, including call-by-value, call-by-name, and call-by-need, that are given as parameters to the theory. In the end, the framework encompasses the previously known duality of call-by-name and call-by-value in the sequent calculus, both by Curien and Herbelin [3] and Wadler [59] , while also including call-by-need and its dual. Additionally, the framework reveals connections with approaches by Zeilberger [60] , Munch-Maccagnoni [6] , and Curien and Munch-Maccagnoni [33] , for using polarization and focalization to provide deterministic strategies for classical computation with structures and pattern matching. This work will be presented at ESOP 2014 [15] .

Luke Maurer and Zena Ariola in collaboration with Daniele Varacca studied the connections between $\pi $-calculus encodings of the $\lambda $-calculus and similar continuation-passing style (CPS) transformations, extending the connections for call-by-value and call-by-name encodings to include the call-by-need $\pi $-calculus encoding as well. This development revealed a better understanding of the computational effect needed in the $\lambda $-calculus to model call-by-need evaluation, which better reflects the way that memoization for call-by-need is implemented. The work is going to be submitted to RTA-TLCA.

*Constructive interpretation of an involutive negation.*

Guillaume Munch-Maccagnoni developped a syntax of delimited control operators that exposes a formulae-as-types correspondence between an involutive negation in classical natural deduction, and the idea that captured contexts, unlike continuations, can be inspected. This decomposes technical artefacts found in call-by-name classical realisability, and simplifies witness extraction from proofs of $\Sigma $ formulae. This work has been submitted and appears in his PhD thesis [11] .

#### Dependent monads

Guillaume Claret and Yann Régis-Gianas are developping a monadic translation from functional programs with effects to Coq that uses a dependent monad. The aim of this work is to allow to reason about effectful programs directly in Coq.

#### Linear dependent types

Pierre-Marie Pédrot developped a dependent version of the Dialectica translation, that gives interesting insights into the possibility to design linear dependent types. Indeed, Dialectica can be decomposed as a translation acting on linear types instead of intuitionistic ones.

#### Delimited continuations, polarity and computational effects

Guillaume Munch-Maccagnoni's polarised decomposition of delimited control calculi appeared in his PhD thesis [11] .

#### Reverse mathematics

Ludovic Patey studied with Laurent Bienvenu and Paul Shafer the deep connections between algorithmic randomness and reverse mathematics by defining formally the ability of computing a solution to a problem by probabilistic means within the framework of reverse mathematics, the No Randomized Algorithm property (NRA). They provided a classification of the whole revese mathematics zoo created by Damir Dzhafarov in terms of having the NRA property or not, answering to some open separation questions.

Ludovic Patey stated two dichotomy theorems about satisfiability problems within reverse mathematics and proved them using clone theory. The corresponding paper is submitted to Computability in Europe 2014. He studied also ramseyan theorems related to the Rainbow Ramsey theorem and provided characterizations in terms of diagonally non-computable functions, algorithmic randomness, and related it to the Erdös Moser theorem and Thin set theorem.

#### Gödel's functional interpretation

Pierre-Marie Pédrot showed that the Dialectica translation could be explained in terms of the Krivine abstract machine, in a way similar to the usual presentation of classical realizability. This opens the door to a better understanding of related translations, as well as adding semi-classical effect into PTS.

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

Alexis Saurin and Pierre-Marie Pédrot developed a structured reconstruction of call-by-need based on linear head reduction which arose in the context of linear logic. This opens new directions both to extend call-by-need to control and to apply linear logic proof-theory (and particularly proof-nets) to call-by-need evaluation.

#### Streams and classical logic

Alexis Saurin and Fanny He have been working on transfinite term rewriting in order to model stream calculi and their connections with lambda-calculi for classical logic.

Jaime Gaspar identified the eight simplest variants (some already known) of the Kuroda negative translation that translate classical logic into minimal logic.