Section: New Results

Reasoning and programming with infinite data

Participants : Yann Régis-Gianas, Alexis Saurin, Abhishek De, Luc Pellissier, Xavier Onfroy.

This theme is part of the ANR project Rapido (see the National Initiatives section) which goes until end of september 2019.

Proof theory of infinitary and circular proofs

In collaboration with David Baelde, Amina Doumane, Guilhem Jaber 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, Saurin introduced together with Baelde, Doumane and Jaber a relaxed validity condition to allow infinite branches to be supported by threads 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 due to a discrepancy between the sequential nature of proofs in sequent calculus and the parallel nature of threads. Several directions of research have therefore been investigated from that point:

• In sequent calculus, Baelde, Doumane and 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 as proved together with Kuperberg. Decidability can be recovered by constraining the shapes of bounces. Doumane and Saurin were able in the fall 2018 to generalise the CSL proof technique to be applicable to bouncing threads. Those results are currently being written targetting a submission early 2019.

• In the setting of natural deduction, Saurin and Jaber introduced a validity criterion aiming at ensuring productivity of a circular $\lambda$-calculus with inductive and coinductive types.

• In the fall 2018, Abhishek De started his PhD under Saurin's supervision. The first part of his PhD work is dedicated to lifting the proof theory of circular and infinitary proofs to the setting of proof nets, in which the bouncing criterion will be much more convenient to work with since the discrepancy between sequent proofs and parallel threads will be dealt with.

Brotherston-Simpson's conjecture: Finitising circular proofs

An important and most active research topic on circular proofs is the comparison of circular proof systems with usual proof systems with induction and co-induction rules à la Park. This can be viewed as comparing the proof-theoretical power of usual induction reasoning with that of Fermat's infinite descent method. Berardi and Tatsuta, as well as Simpson, obtained in 2017 important results in this direction for logics with inductive predicates à la Martin-Löf. Those frameworks, however, are weaker than those of fixpoint logic which can express and mix least and greatest fixpoints by interleaving $\mu$ and $\nu$ statements. New results on this topics followed in 2018.

In a work with Nollet and Tasson, Saurin published in CSL 2018 a new validity condition which is quite straightfoward to check (it can be checked at the level of elementary cycles of the circular proofs, while the other criteria need to check a condition on every infinite branch) and still capture all circular proofs obtained from $\mu MALL$ finite proofs [46]. The condition for cycling in those proofs is more constrained than that of Baelde, Doumane and Saurin, but the proof contains more information which can be used to exctract inductive invariants. With this validity condition which can be useful for proof search for circular proofs, they obtained partial finitisation results and are currently aiming at solving the most general Brotherston-Simpson's conjecture.

Streams and classical logic

Luc Pellissier started a post-doc in december 2018 funded by the RAPIDO project and started working with Alexis Saurin on the stream interpretation of $\Lambda \mu$-calculi by investigating the connection between $\Lambda \mu$-calculus and the parsimonious $\lambda$-calculus.

Formalising circular proofs and their validity condition

During the spring and summer 2018, Saurin started with Xavier Onfroy a formalisation of circular proofs in Coq. Until now, Onfroy formalised parity-automata and their meta-theory as a first step to capture the decidability condition of circular proofs. Preliminary formalisations of circular proofs have been considered by Onfroy but shall still be pursued in order to fit into the picture.