Project-π.r²:Proof-theoretical investigations

Inria / Raweb 2009
Presentation of the Project π.r²

Section: New Results

Proof-theoretical investigations

Participants : Pierre-Louis Curien, Hugo Herbelin, Noam Zeilberger, Alexis Saurin, Guillaume Munch, Vincent Siles, Danko Ilik.

Sequent calculus and Computatonal duality

Axiomatisation of call-by-value

Hugo Herbelin and Stéphane Zimmermann (PPS) designed an original reduction semantics of call-by-value $\lambda$ -calculus (with and without control) that is both complete with respect to the continuation-passing-style semantics of call-by-value and confluent. This has been presented to the 2009 edition of the TLCA conference [14] .

Focalisation

Alexis Saurin has investigated how the focalization theorem of linear logic can be proved by interactive means in Girard's Ludics (in Terui's Computational Ludics setting [63] ) , resulting in [2] , which has since been improved and submitted to an international conference in early november [21] . Connections with algorithmic complexity are discussed since focalization in the framework of computational ludics can be connected with proof methods of the linear-speedup theorem [57] .

In the two months he has been here, Noam Zeilberger worked on different aspects of computational duality and the Curry-Howard interpretation of polarized logic. With Paul-André Melliès and Jonas Frey from PPS, he began developing a categorical semantics of focusing proofs, which will eventually include a categorical account of abstract machines. He has also been studying polarity in intuitionistic logic, with the aim of giving a better explanation of, e.g., Goodman's Theorem, and especially of delimited continuations (an article is in preparation about the latter). Finally, he has been working on an article (with results from his dissertation) on the theory of refinement type systems (intersection and union types, subtyping, etc.) in the presence of effects.

Focalisation and classical realizability

Following the work of Curien and Herbelin [3] and Girard [43] , Guillaume Munch–Maccagnoni gave a term syntax ( $Im5 $\#120235 _foc$$ ) for polarised classical logic ( $Im6 $\#120235 \#120234 _pol$$ ) and linear logic with strategies of reduction based on Girard's classical logic LC. This 'focalising' strategy distinguishes strict (positive) and lazy (negative) programs, and thus encompasses with a single deterministic strategy both call-by-value and call-by-name, in the spirit of Paul Blain Levy's call-by-push-value [52] , however in direct style. Further syntactical investigations on this theme, establishing links with Zeilberger's work and with ludics, have been carried out in joint work between Pierre-Louis Curien and Guillaume Munch–Maccagnoni, and are to be written down for a special issue of the journal HOSC in honour of Peter Landin [12] .

Guillaume Munch–Maccagnoni has extended Krivine's classical realisability [46] to $Im5 $\#120235 _foc$$ , and therefore to CBV in particular. He showed that these tools concisely account for “imperfections” (Zeilberger) of programming languages with effect such as the value restriction required for the polymorphism in CBV to be a sound principle. This work led to a paper which was accepted to the CSL'09 conference during his Master internship [15] .

Guillaume Munch–Maccagnoni then showed during his internship how classical realizability could further be seen as a tool (looking like Pitts' logical relations [60] ) for the study of programming languages:

Provided one accepts the Adequacy Lemma as an argument for type safety (as in [54] ), it gives concise and modular proofs of type safety in the presence of polymorphism, sub-typing and inductive algebraic types, including in CBV.
It can be used to derive certain parametricity results in the manner of Crary [33] .
Properties of 'internal completeness' allow one to prove additional equalities of types (such as the 'shocking equalities' of polymorphism).

Paul-André Melliès (from PPS) and Guillaume Munch–Maccagnoni started to work on the relationship between notions of category theory and features of $Im5 $\#120235 _foc$$ , such as a possible link between dialogue categories and the cautious treatment of bilaterality in $Im5 $\#120235 _foc$$ .

Pure type systems in sequent calculus

Hugo Herbelin and Vincent Siles investigated different formulations of pure type systems in sequent calculus building up on previous works by Hugo Herbelin [5] and Stéphane Lengrand [51] , especially in connection to the problem of Expansion Postponement (see below). Vincent Siles summarised these investigations in a paper under consideration for submission.

On the logical contents of delimited control

Delimited control and $\upper_lambda$ $\mu$ -calculus

In the continuation of his work with Silvia Ghilezan [4] on showing that Saurin's variant $\upper_lambda$ $\mu$ [7] of Parigot's $\lambda$ $\mu$ -calculus [59] 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.

Saurin's calculus has the following surprising feature: its simply-typed version is equivalent to Parigot's $\lambda$ $\mu$ -calculus extended with a $\bottom$ connective but its untyped version is much more expressive because it satisfies a property of completeness (Böhm separation) that $\lambda$ $\mu$ -calculus does not. A precise description of the connection between both calculi and between another call-by-name calculus of delimited control and another variant of $\lambda$ $\mu$ -calculus by de Groote has been conducted by Hugo Herbelin and Alexis Saurin resulting in a conditional acceptance to a special issue of the APAL journal on Computational Classical logic [20] .

Alexis Saurin submitted two articles to international conferences since he joined the team: the first one [22] introduces a hierarchy of calculi for delimited control in call-by-name (that is a CBN correspondent to Danvy-Filinski's CPS hierarchy) and has been accepted to FOSSACS 2010 while the second one [23] establishes a standardization theorem and characterizes solvability for the $\upper_lambda$ $\mu$ -calculus [7] and introduces Böhm trees for $\upper_lambda$ $\mu$ -calculus. Those two works develop previous works by the author alone [7] , [10] , [8] , [11] .

Other current research work of Alexis Saurin, besides the on-going one mentioned above with Ghilezan and Herbelin, concerns a calculus for streams (in collaboration with Marco Gaboardi from Torino), an interactive approach to proof search using ludics [9] , and the logical understanding of intersection types with Simona Ronchi della Rocca.

Control delimiters and Markov's principle

In the last months, Hugo Herbelin discovered a relation between control delimiters and Markov's principle: In an intuitionistic logic extended with classical logic for $\upper_sigma$ ₁⁰ -formulae (i.e. for formulae that correspond to data-types) and a control delimiter, Markov's principle (i.e. the property that ¬¬ $\exists$ xA(x) $\rightarrow$ $\exists$ xA(x) ) becomes provable while still retaining the main property of intuitionism, namely that any proof of $\exists$ xA(x) contains a witness t such that A(t) holds.

Differential linear logic and a logical interpretation of statically bound exceptions

Guillaume Munch–Maccagnoni developed a symmetric syntax for Differential Linear Logic [37] inspired by the linear-non-linear adjunction of linear logic.

This syntax allowed him to give a logical interpretation of statically bound exceptions using primitives from differential linear logic.

In this interpretation, a co-dereliction on the catching context is used to model the fact that an exception binder can catch only one exception. The case where an exception is uncaught thus corresponds to the interaction between a dereliction and a co-weakening in differential linear logic.

As a direct consequence, Herbelin's implementation of Markov's principle on $\forall$ - $\rightarrow$ -free formulae using such exceptions corresponds, through this interpretation, to the validity of co-dereliction on recursively positive formulae.

Kripke semantics for classical logic

Hugo Herbelin and Gyesik Lee (ROPAS Center, Seoul University) presented their work on a simple proof of Kripke completeness for the negative fragment of intuitionistic logic to the 2009 edition of WOLLIC workshop [13] . This work provides an interesting case study on the representation of binders in Coq. This resulted in a paper submitted to the Special Issue on Binding, Substitution and Naming edited by C. Urban and M. Fernandez in the Journal of Automated Reasoning [19] .

Hugo Herbelin, Danko Ilik, and Gyesik Lee gave a new kind of direct semantics for classical logic, similar to Kripke models, and proved constructively that it is sound and complete for first-order logic. They submitted a paper[18] , which was conditionally accepted.

They also formalised the proofs in the Coq proof assistant, allowing them to do experiments on the operational behaviour of the semantics, confirming that the semantics gives rise to call-by-name proof normalisation.

Hugo Herbelin and Danko Ilik gave a constructive proof of completeness of intutionistic logic, with disjunction, with respect to a notion of model dual (call-by-value) to the one above. The proof, formalised in Coq, can be used as a normaliser for $\lambda$ -calculus terms with sums. Work remains to be done in comparing the notion of model introduced to notions of models from the literature.

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