Section: New Results

Proof-theoretical and effectful investigations

Participants : Federico Aschieri, Pierre Boutillier, Pierre-Louis Curien, Hugo Herbelin, Guillaume Munch-Maccagnoni, Pierre-Marie Pédrot, Alexis Saurin, Arnaud Spiwack.

Sequent calculus and computational duality

System L syntax. Pierre-Louis Curien studied in some detail the differences (and translations) between variants of “system L” syntax for polarised classical logic (developed by Guillaume Munch-Maccagnoni and himself):

• weakly focalised systems (where negatives can be worked on at any moment in a proof) versus focalised systems (where negative and positive phases alternate strictly), versus strongly focalised systems (where furthermore negative phases have to decompose negatives completely);

• systems where changes of polarity are implicit (like in Girard's LC) versus systems where they are explicitly marked using shift operators. These shift operators are formally adjoint, and as a matter of fact a suitable intuitionistic fragment of system L corresponds exactly to Levi's CBPV;

• systems with stoup (which retain only proofs that follow the focalisation discipline) versus (still focalised) systems without stoup (where the focalisation is forced by the dynamics of reduction);

• one-sided systems (with an implicit negation given by De Morgan duality) versus two-sided systems (allowing for explicit negation, and for distinguishing the left/right and the positive/negative dualities).

Pierre-Louis Curien is also currently studying a polarised version of a notion of general connective suggested earlier by Hugo Herbelin (unpublished work), and the composition structure of these connectives (in the spirit of operads).

Categorical semantics. Guillaume Munch-Maccagnoni investigated a notion of “direct style” for adjunction models, inspired by his work on polarisation in the “L” system, in the lineage of Führmann's [47] direct-style characterisation of monadic models. (It is part of joint work with Marcelo Fiore and Pierre-Louis Curien.)

Polarised Peano arithmetic. Guillaume Munch-Maccagnoni investigates the computational contents of polarised classical logic in arithmetic and in natural deduction. This allows him to compare the constructivisation of the principle $\neg \forall \Rightarrow \exists \neg$ based on classical realisability (Krivine) and the one based on delimited control (via “double negation shift”); both of which seem to be simplified by a better understanding of the “formulae-as-types” paradigm for a negation which is involutive in a strong sense.

Guillaume Munch-Maccagnoni investigates how a notion of classical realisability structure (inspired by Krivine's) can be used to prove properties of type systems which are usually regarded as syntactic.

Classical call-by-need and the duality of computation. In 2011, Zena Ariola, Hugo Herbelin and Alexis Saurin characterised the semantics of call-by-need calculus with control in the framework of the duality of computation. The same set of authors extended with Paul Downen and Keiko Nakata worked on abstract machines and continuation-passing-style semantics for call-by-need with control, resulting into a paper presented at FLOPS 2012 [20] .

Further work has been done by Zena Ariola, Hugo Herbelin, Luís Pinto, Keiko Nakata and José Espírito Santo on typing the continuation-passing-style of call-by-need calculus, opening the way to a proof of normalisation of simply-typed call-by-need with control, and from there to a proof of consistency of classical arithmetic with dependent choice.

Zena Ariola also investigated how to formulate a parametric theory which encompasses call-by-value, call-by-name and call-by-need. Each theory is obtained by giving the appropriate definition of what is a value and a co-value. The theory also includes so called lifting axioms which allow one to relax the syntactic restrictions previously imposed on the call-by-value, call-by-name and call-by-need calculi. The theory also allows to include the $\eta$-rules which before were causing confluence to fail. The approach can be applied to natural deduction and this allows to express different embeddings of natural deduction into sequent calculus directly in the theory. The advantage of the new formalisation is that analogously to natural deduction, one can experiment with different strategies starting from the same term. Moreover, the theory is well-suited for continuation passing style transformation and, in particular, it leads to a different and simpler formalisation of classical call-by-need, its abstract machine and continuation passing style.

Dependent monads

Pierre-Marie Pédrot generalised the notion of monad in order to be able to use it in a dependent framework. This new structure allows to write effects in a pure functional language, such as Coq, through a monadic encoding.

This way, the whole monadic apparatus can be lifted to dependent programs, as well as proofs.

Linear dependent types

Arnaud Spiwack continued his investigations on dependently typed linear sequent calculus (based on Curien & Herbelin's $\mu \tilde{\mu }$). The current version of his system resembles Andreoli's focalised linear logic (yet to be published).

Pierre-Marie Pédrot has been working on a delimited CPS translation of the Calculus of Inductive Constructions, seen through the prism of polarised linear logic. Restricting dependencies to positives naturally fits into the scenery of delimited control, while extending negatives to infinitary objects permits to recover some properties of the involutivity of linear double-negation.

Proving with side-effects

Axiom of dependent choice. Hugo Herbelin showed that classical arithmetic in finite types extended with strong elimination of existential quantification proves the axiom of dependent choice. To get classical logic and choice together without being inconsistent is made possible first by constraining strong elimination of existential quantification to proofs that are essentially intuitionistic and secondly by turning countable universal quantification into an infinite conjunction of classical proofs evaluated along a call-by-need evaluation strategy so as to extract from them intuitionistic contents that complies to the intuitionistic constraint put on strong elimination of existential quantification. This work has been presented at LICS 2012 [22] .

Memory assignment, forcing and delimited control. Hugo Herbelin investigated how to extend his work on intuitionistically proving Markov's principle [54] and the work of Danko Ilik on intuitionistically proving the double negation shift (i.e. $\forall x\neg \neg A\to \neg \neg \forall xA$) [15] to other kind of effects. In particular, memory assignment is related to Cohen's forcing as emphasised by Krivine  [58] and by the observation that Cohen's translation of formula $P$ into $\forall y\le x\exists z\le yP\left(z\right)$ is similar to a state-passing-style transformation of type $P$ into $S\to S\times P$.

Hugo Herbelin then designed a logical formalism with memory assignment that allows to prove in direct-style any statement provable using the forcing method, the same way as logic extended with control operators allows to support direct-style classical reasoning. Thanks to the use of delimiters over “small” formulas similar to the notation of ${\Sigma }_1^0$-formulas in arithmetic, the whole framework remains intuitionistic, in the sense that it satisfies the disjunction and existence property.

Two typical applications of proving with side-effects are global-memory proofs of the axiom of countable choice and an enumeration-free proof of Gödel's completeness theorem.

The main ideas of this research program have been communicated during the Logic and Interaction weeks in Marseille in February 2012.

In the continuation of his work with Silvia Ghilezan [4] on showing that Saurin's variant $\Lambda \mu$ [8] of Parigot's $\lambda \mu$-calculus [65] for classical logic was a canonical call-by-name version of Danvy-Filinski's call-by-value calculus of delimited control, Hugo Herbelin studied with Alexis Saurin and Silvia Ghilezan another variant of call-by-name calculus of delimited control. This is leading to a general paper on call-by-name and call-by-value delimited control.

Classical logic, stack calculus and stream calculus. Alexis Saurin studied the connection between the stack calculus recently proposed by Ehrhard et al and $\lambda \mu$-calculus and how the former can be precisely compared to the target of the CPS of the latter. He also investigated separation issues related to the stack calculus. During a visit to UPenn in the spring, Alexis Saurin and Marco Gaboardi investigated type systems for a stream calculus which contains $\Lambda \mu$.

Moreover, Alexis Saurin's paper Böhm theorem and Böhm trees for the $\Lambda \mu$-calculus [16] was published in TCS early 2012.

PTS and delimited control

From the study of one-pass CPS on the one side and of previous presentations of pure type systems with control operators on the other side, Pierre Boutillier and Hugo Herbelin have investigated how splitting terms into categories opens a new way to merge dependent types and monads. A preliminary set of rules has been presented during the third week of Logic and Interaction in Marseille.

It was refined since then but has not reached yet the maturity required to be accepted for publication in an international conference.

Interactive realisability

Thanks to the Curry-Howard correspondence for classical logic, it is possible to extract programs from classical proofs. These programs use control operators as a way to implement backtracking and processes of intelligent learning by trial and error. Unfortunately, such programs tend to be poorly efficient. The reason is that, in a sense, they are designed in order to keep their correctness and termination proofs simple. Each small modification of these programs seems, at best, to require major and difficult adaptations of their correctness proofs. This is due to a lack of understanding and control of the backtracking mechanism that interprets classical proofs. In order to write down more efficient programs, it is necessary to describe exactly: a) what the programs learn, b) how the knowledge of programs varies during the execution.

A first step towards this goal is the theory of Interactive Realisability, a semantics for intuitionistic arithmetic with excluded middle over semi-decidable predicates. It is based on a notion of state, which describes the knowledge of programs coming from a classical proof, and explains how the knowledge evolves during computation.

Federico Aschieri has extended the theory of interactive realisability to a full classical system, namely first-order Peano arithmetic with Skolem axioms. This is a very expressive system, with non-trivial axioms of choice and comprehension. The resulting programs are interpreted as stratified-learning algorithms, which build in a very organised way the approximations of the Skolem functions used in the proofs. The work has appeared in the proceedings of the conference Computer Science Logic 2012. A careful implementation of this extended theory –yet to be developed – will lead to a dramatic efficiency improvement over the already existing computational interpretations.

Federico Aschieri has also showed how to use interactive realisability to provide purely proof-theoretic results. He proved with a new method the conservativity of Peano arithmetic with Skolem axioms over Peano arithmetic alone for arithmetical formulas. In particular, the method can be seen as a constructivisation and substantial refinement of Avigad's forcing. The work has appeared in the proceedings of the workshop Classical Logic and Computation 2012.

Reverse mathematics

Hugo Herbelin explored with Gyesik Lee and Keiko Nakata the constructive content of the big five subsystems of second-order arithmetic considered in the context of (classical) reverse mathematics. They obtained a uniform characterisation of these systems in terms of variants of the comprehension axiom called separation, co-separation and interpolation.

This is the first step in a larger project attempting first to connect to predicative type theory the subsystems of System F underlying the proof-as-program structure of the big five subsystems of second-order arithmetic, and secondly to reformulate these subsystems in terms of pure systems of inductive definitions.

Jaime Gaspar has several projects running simultaneously. For example, in one of his projects he created a small unoptimised automated theorem prover, and he hopes to optimise it and use it to obtain a certain completely formalised proof to which he can apply a proof interpretation in order to extract computational content. As another example, in another project he his trying to show that several classical proof interpretations are instances of a unified proof interpretation, in a parallel way to what is known for intuitionistic proof interpretations.