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
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
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
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
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.