## Section: New Results

### Reasoning and programming with infinite data

Participants : Amina Doumane, Alexis Saurin, Pierre-Marie Pédrot, Yann Régis-Gianas.

This theme is part of the ANR project Rapido (see the National Initiatives section).

#### Interactive semantics for logic fixed-points and infinitary logics.

Amina Doumane and Alexis Saurin, in a joint work with David Baelde published at CSL 2015 [24] , developed a game-semantics of $\mu MALL$ (Multiplicative Additive Linear Logic with least and greatest fixpoints).

This interactive semantics was worked out in computational ludics, benefitting from both the work by Clairambault on a HO style game semantics for an intuitionistic logic with least and greatest fixpoints and from the flexibility of Terui's computational ludics (in particular its ability to consider designs with cuts).

This framework is built around the notion of design, which can be seen as an analogue of the strategies of game semantics. The infinitary nature of designs makes them particularly well suited for representing computations over infinite data. We provided $\mu MALL$ with a denotational semantics (that is invariant by cut-elimination), interpreting proofs by designs and formulas by particular sets of designs called behaviours. Then a completeness result for a specific class of designs is proved, the class of “essentially finite designs”, which are those designs performing a finite computation followed by a copycat. On the way to the previous completeness result, we investigate semantic inclusion, proving its decidability (given two formulas $A$ and $B$, one can decide whether the semantics of $A$ is included in the semantics of $B$) and completeness (if semantic inclusion holds, the corresponding implication is provable in $\mu MALL$).

#### Proof theory of circular proofs

In a collaboration with David Baelde, Amina Doumane and Alexis Saurin developed further the theory of infinite proofs. Studying the proof theory of circular proofs on MALL, they established a result of focalisation for these infinite proofs. The usual result of focalisation for linear logic can actually be extended to circular proofs but, contrarily to $\mu MALL$ where fixed-points operators can be given an arbitrary polarity, the least fixed-point must be set to be a positive construction and the greatest fixed-points to be negative, which is consistent with intuition from programming with inductive and co-inductive datatypes. An interesting phenomenon arising with focalisation is that some infinite but regular proofs may not have any regular focused proofs. This is similar to what happens for cut-elimination of regular proofs.

Works on cut-elimination for circular proofs are still ongoing.

##### Automata theory meets proof theory: proof certificates for Büchi inclusion

In a joint work with David Baelde and Lucca Hirschi, Amina Doumane and Alexis Saurin carried out a proof-theoretical investigation of the linear-time $\mu $-calculus, proposing well-structured proof systems and showing constructively that they are complete for inclusions of Büchi automata suitably encoded as formulas.

They do so in a way that combines the advantages of two lines of previous work: Kaivola gave a proof of completeness for an axiomatisation that amounts to a finitary proof system, but his proof is non-constructive and yields no reasonable procedure. On the other hand, Dax, Hofmann and Lange recently gave a deductive system that is appropriate for algorithmic proof search, but their proofs require a global validity condition and do not have a well understood proof theory.

They work with well-structured proof systems, effectively constructing proofs in a finitary sequent calculus that enjoys local correctness and cut elimination.

This involves an intermediate circular proof system in which one can obtain proofs for all inclusions of parity automata, by adapting Safra's construction. In order to finally obtain finite proofs of Büchi inclusions, a translation result from circular to finite proofs is designed.