Section: New Results

Proof-theoretical and effectful investigations

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

Proving with side-effects

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 is currently working to get a presentation of this work in Curien-Herbelin's $\mu$-$\tilde{\mu }$-calculus, with the aim of getting in the end a CPS-translation. Such a translation would provide a strong argument of normalisation for the calculus, as well as a better undertanding of the mechanisms of the calculus, especially the side-effect part and the meaning of the existential quantifier restriction.

Hugo Herbelin and Danko Ilik carried on their work on the computational content of completeness proofs and in particular of the computational content of Gödel's completeness theorem. Hugo Herbelin presented their work at the workshop PSC 2014.

Reverse mathematics

Ludovic Patey studied with Laurent Bienvenu and Paul Shafer the provability strength of Ramsey-type versions of theorems like König's lemma. The corresponding paper is submitted to the Jourmal of Mathematical Logic. Ludovic Patey studied with Laurent Bienvenu the constructions of diagonal non-computable functions by probabilistic means. They submitted a paper to Information and Computation. Ludovic Patey worked on the existence of universal instances in reverse mathematics, and submitted a paper to Annals of Pure and Applied Logic. He worked on the relations between diagonal non-computability and Ramsey-type theorems and submitted a paper to the Archive for Mathematical Logic. He studied the links between the iterative forcing framework developed by Lerman, Solomon & Towsner and the notion of preservation of hyperimmunity and submitted a paper to Computability in Europe 2015.

Gödel's functional interpretation

Pierre-Marie Pédrot kept developing the proof-as-program interpretation of Gödel's Dialectica translation, as seen through the prism of classical realisability. This work was presented at TYPES 2014 and later published at LICS 2014 [26] .

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. This work was presented at JFLA 2014 [30] early 2014 and later expanded to the classical case, encompassing $\lambda \mu$-calculus.

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. Their work gave rise to a presentation at the Workshop on Infinitary Rewriting that took place in Vienna last July as part of FLOC 2014.

Alternative syntaxes for proofs

Amina Doumane and Alexis Saurin, in a joint work with Marc Bagnol, studied the structure of several correctness criteria for linear logic proof-nets and could relate them through a new primitive notion of dependency. This work was first presented at JFLA 2014 [29] early 2014 and later at Structure and Deduction in Vienna as part of FLOC 2014. An expanded version has recently been accepted at FOSSACS 2015 [19] .