Proof theory of infinitary and circular proofs

In collaboration with David Baelde and Guilhem Jaber, Amina Doumane and Alexis Saurin extended the proof theory of infinite proofs for fixpoint logics by relaxing the validity condition necessary to distinguish sound proofs from invalid ones. In CSL 2016, Baelde, Doumane and Saurin proved cut-elimination and focalisation for infinite proofs for $\mu MALL$ with a validity condition inspired from the acceptance condition of parity automata (or the winning condition of parity games). However, this validity condition rules out lots of proofs which are computational sound and does not account for the cut-axiom interaction in sequent proofs.

With Jaber, they relaxed the 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, approximating productivity. If the decidability of the validity condition in the most general case is still open, it allows for decidable restrictions which are still useful in the sense they allow for a much more flexible writing of circular proofs (or, through the proofs-as-programs bridge, circular progams). Cut-elimination is obtained in two steps, combining CSL 2016 result with a technique for “straightening” bouncing threads, that is perfoming just the necessary amount of cut-elimination to recover straight threads, the two results are combined thanks to a compression lemma, a standard result from infinitary rewriting ensuring that a transfinite strongly converging sequence can be turned into an $\omega$-indexed strongly converging sequence. Preliminary results were presented at the Types 2017 conference.

Automata theory meets proof theory: completeness of the linear time mu-calculus.

Amina Doumane extended her previous results with David Baelde, Lucca Hirschi and Alexis Saurin proving a constructive completeness theorem for the full linear-time $\mu$-calculus, while the previous results only captured a fragment of the linear-time mu-calculus expressing all inclusions of Büchi automata suitably encoded as formulas.

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.

In the setting of fixpoint logics with circular proofs, several investigations were carried on in the team:

Co-patterns

In collaboration with Paul Laforgue (Master 2, University Paris 7), Yann Régis-Gianas developed an extension of OCaml with copatterns. Copatterns generalize standard ML patterns for algebraic datatypes: While a pattern-matching destructs a finite value defined using a constructor, a copattern-matching creates an infinite computation defined in terms of its answers to observations performed by the evaluation context. They exploits the duality between functions defined by pattern matching and functions that define codata by copattern-matching, going from the second to the first by introducing a well-typed inversion of control which is a purely local syntactic transformation. This result shows that copattern-matching can be added with no effort to any programming language equipped with second-order polymorphism and generalized algebraic datatypes. This work has been published in the proceeding of PPDP'17. A short paper has also been accepted at JFLA'18.

Streams, classical logic and the ordinal $\lambda$-calculus

Polonsky and Saurin defined an extension of infinitary $\lambda$-calculi allowing transfinite iteration of abstraction and ordinal sequences of applications, ${\Lambda }^o$, and established a standardisation theorem for this calculus. The $\Lambda \mu$-calculus can be embedded in this calculus, as well as Saurin's full Stream hierarchy: as a consequence, they obtain a uniform framework to investigate this family of calculi and provide uniform proofs of important results such a standardisation.

Theory of fixpoints in the lambda-calculus