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## Section: New Results

### Reasoning and programming with infinite data

Participants : Kostia Chardonnet, Lucien David, Abhishek De, Farzad Jafar-Rahmani, Luc Pellissier, Yann Régis-Gianas, Alexis Saurin.

This theme is part of the ANR project Rapido (see the National Initiatives section) which ended octobre 1st 2019.

#### Proof theory of non-wellfounded and circular proofs

##### Validity conditions of infinitary and circular proofs

In collaboration with David Baelde, Amina Doumane and Denis Kuperberg, Alexis Saurin extended the proof theory of infinite and circular proofs for fixed-point logics in various directions by relaxing the validity condition necessary to distinguish sound proofs from invalid ones. The original validity condition considered by Baelde, Doumane and Saurin in CSL 2016 rules out lots of proofs which are computationally and semantically sound and does not account for the cut-axiom interaction in sequent proofs. In the setting of sequent calculus, Alexis Saurin studied together with David Baelde, Amina Doumane and Denis Kuperberg a relaxed validity condition to allow infinite branches to be supported by threads which may leave the infinite branch, visiting other parts of the proofs and bouncing on axioms and cuts. This allows for a much more flexible criterion, inspired from Girard's geometry of interaction. The most general form of this criterion does not ensure productivity in the sequent calculus due to a discrepancy between the sequential nature of proofs in sequent calculus and the parallel nature of threads. David Baelde, Amina Doumane, Denis Kuperberg and Alexis Saurin provided a slight restriction of the full bouncing validity which grants productivity and validity of the cut-elimination process. This restriction still strictly extends previous notions of validity and is actually expressive enough to be undecidable.

Several directions of research have therefore been investigated from that point:

• Decidability can be recovered by constraining the shapes of bounces (bounding the depth of bounces). They actually exhibited a hierarchy of criteria, all decidable and satisfying the fact that their union corresponds to bouncing validity (which is therefore semi-decidable)

• While the result originaly held only for the multiplicative fragment of linear logic, the result was extended to multiplicative and additive linear logic.

Those results are currently submitted.

##### On the complexity of the validity condition of circular proofs

Alexis Saurin, together with Rémi Nollet and Christine Tasson, characterised the complexity of deciding the validity of circular proofs. While deciding validity was known to be in PSPACE, they proved that, for $\mu MALL$ proof, it is in fact a PSPACE-complete problem.

The proof is based on a deeper exploration of the connection between thread-validity and the size-change termination principle, a standard tool to prove program termination.

##### Proof nets for non-wellfounded proofs

The results of the TABLEAUX paper are limited in that they only address the case of proofs with finitely many cuts inferences. Abhishek De and Alexis Saurin are currently investigating, with Luc Pellissier, the general case of infinitely many cut in order to then lift the results from straight thread validity to bouncing thread validity.

#### On the denotational semantics of non-wellfounded proofs

Farzad Jafar-Rahmani started his PhD under the supervision of Thomas Ehrhard and Alexis Saurin in October 2019. His PhD work will focus on the denotational semantics of circular proofs of linear logic with fixed points. After working on the denotational semantics of finitary proofs for linear logic with fixed points (with Kozen rules) during his master, he is currently working at understanding the denotational counterpart of the validity condition of circular proofs.

#### Towards inductive and coinductive types in quantum programming languages

Kostia Chardonnet started his PhD under the supervision of Alexis Saurin and Benoît Valiron in November 2019. Previously, he did his MPRI Master internship under their joint supervision on designing a calculus of reversible programs with inductive and coinductive types. His research focused on extending a languages of type isomorphisms with inductive and coinductive types and understanding the connections of those reversible programs with $\mu MALL$ type isomorphisms and more specifically with $\mu MALL$ focused circular proof isomorphisms. In his PhD, he shall extend this to the case of a quantum programming language with inductive and coinductive data types.

#### Theory of fixed points in the lambda-calculus

The results of Alexis Saurin in collaboration with Giulio Manzonetto, Andrew Polonsky and Jacob Grue Simonsen, on two long-standing conjectures on fixed points in the $\lambda$-calculus – the “fixpoint property” and the “double-fixpoint conjecture” – have now appeared in the Journal of Logic and Computation [34]. The former asserts that every $\lambda$-term admits either a unique or an infinite number of $\beta$-distinct fixpoints while the second, formulated by Statman, says that there is no fixpoint satisfying $Y\delta =Y$ for $\delta =\lambda y,x.x\left(yx\right)$. They proved the first conjecture in the case of open terms and refute it in the case of sensible theories (instead of $\beta$). Moreover, they provide sufficient conditions for both conjectures in the general case. Concerning the double-fixpoint conjecture, they propose a proof technique identifying two key properties from which the results would follow, while they leave as conjecture to prove that those actually hold.