Section: New Results

Reasoning and programming with infinite data

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

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

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.

In order to achieve this tour de force (for which her publication at LICS 2017 received the Kleene award of the best student paper [38], see Highlights of the year), she identified several fragments of the linear-time mu-calculus corresponding to various classes of $\omega$-automata and proved completeness of those classes by using circular proof systems and finitisation of the infinite proofs in the Kozen's usual axiomatisation (see paragraph on finitising circular proofs for more details).

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:

• firstly, in the setting of the usual validity condition for circular proofs of $\mu MALL$, Doumane extended in her PhD thesis a translatibility criterion for finitising circular proofs which was first used in joint work with Baelde, Saurin and Hirschi and later applied to the full linear-time mu-calculus in her LICS 2017 paper. Her translatibility criterion abstracts the proof scheme for finitising circular proofs and is not formulated with respect to a specific fragment of the logic, but with respect to conditions allowing finitisation of the cycles.

• Secondly, Nollet, working with Saurin and Tasson, recently proposed 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. 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.

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

In collaboration with Manzonetto, Polonsky and Simonsen, Saurin studied two long-standing conjectures on fixpoints in the $\lambda$-calculus: the “fixpoint property” and the “double-fixpoint conjecture”. 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. Those results are currently submitted to a journal [54].