Superdeduction in Lambda-Bar-Mu-Mu-Tilde

Superdeduction is a method specially designed to ease the use of first-order theories in predicate logic. The theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. A proof-term language and a cut-elimination reduction already exist for superdeduction, both based on Christian Urban's work on classical sequent calculus. However the computational content of Christian Urban's calculus is not directly related to the (lambda-calculus based) Curry-Howard correspondence. In contrast the Lambda bar mu mu tilde calculus is a lambda-calculus for classical sequent calculus. This short paper is a first step towards a further exploration of the computational content of superdeduction proofs, for we extend the Lambda bar mu mu tilde calculus in order to obtain a proofterm langage together with a cut-elimination reduction for superdeduction. We also prove strong normalisation for this extension of the Lambda bar mu mu tilde calculus.Comment: In Proceedings CL&C 2010, arXiv:1101.520

Classical realizability in the CPS target language

AbstractMotivated by considerations about Krivine's classical realizability, we introduce a term calculus for an intuitionistic logic with record types, which we call the CPS target language. We give a reformulation of the constructions of classical realizability in this language, using the categorical techniques of realizability triposes and toposes.We argue that the presentation of classical realizability in the CPS target language simplifies calculations in realizability toposes, in particular it admits a nice presentation of conjunction as intersection type which is inspired by Girard's ludics

From delimited CPS to polarisation

Appeared in the author's PhD thesis (Chapter III) along with more details. See (and cite) Guillaume Munch-Maccagnoni, Syntax and Models of a non-Associative Composition of Programs and Proofs, Université Paris-Diderot - Paris VII, 2013, .The understanding of continuation-passing style (CPS) translations, an historical source of denotational semantics for programming languages, benefits from notions brought by linear logic, such as focalisation, polarities and the involutive negation. Here we aim to show how linear logic helps as well when continuations are delimited, i.e. return and can be composed, in the sense of Danvy and Filinski. First we provide a polarised calculus with delimited control (first-class delimited continuations) which is, at the level of computation, a variant of Girard's polarised classical logic LC. It contains variants of delimited control calculi which spawned as answers to the question "what order of evaluation can we consider with delimited control?". Thus our polarised calculus is one answer which is unifying to some degree. Subsequently we decompose the calculus through polarised linear logic. The only difference with non-delimited continuations is the use of specific exponentials, that account for the specific semantics of the target of delimited CPS translation, as well as annotations of type-and-effect systems. As a by-product, we obtain an explanation of CPS translations through a factoring, each step of which accounts for distinct phenomena of CPS translations. Although the factoring also holds for non-delimited CPS translations, it did not appear in its entirety before

Monadic translation of classical sequent calculus

International audienceWe study monadic translations of the call-by-name (cbn) and call-by-value (cbv) fragments of the classical sequent calculus ${\overline{\lambda}\mu\tilde{\mu}}$ due to Curien and Herbelin, and give modular and syntactic proofs of strong normalisation. The target of the translations is a new meta-language for classical logic, named monadic λμ. This language is a monadic reworking of Parigot's λμ-calculus, where the monadic binding is confined to commands, thus integrating the monad with the classical features. Also, its μ-reduction rule is replaced by a rule expressing the interaction between monadic binding and μ-abstraction.Our monadic translations produce very tight simulations of the respective fragments of ${\overline{\lambda}\mu\tilde{\mu}}$ within monadic λμ, with reduction steps of ${\overline{\lambda}\mu\tilde{\mu}}$ being translated in a 1–1 fashion, except for β steps, which require two steps. The monad of monadic λμ can be instantiated to the continuations monad so as to ensure strict simulation of monadic λμ within simply typed λ-calculus with β- and η-reduction. Through strict simulation, the strong normalisation of simply typed λ-calculus is inherited by monadic λμ, and then by cbn and cbv ${\overline{\lambda}\mu\tilde{\mu}}$, thus reproving strong normalisation in an elementary syntactical way for these fragments of ${\overline{\lambda}\mu\tilde{\mu}}$, and establishing it for our new calculus. These results extend to second-order logic, with polymorphic λ-calculus as the target, giving new strong normalisation results for classical second-order logic in sequent calculus style.CPS translations of cbn and cbv ${\overline{\lambda}\mu\tilde{\mu}}$ with the strict simulation property are obtained by composing our monadic translations with the continuations-monad instantiation. In an appendix to the paper, we investigate several refinements of the continuations-monad instantiation in order to obtain in a modular way improvements of the CPS translations enjoying extra properties like simulation by cbv β-reduction or reduction of administrative redexes at compile time

Focalisation and Classical Realisability (version with appendices)

The original publication is avalaible at : www.springerlink.comInternational audienceWe develop a polarised variant of Curien and Herbelin's lambda-bar-mu-mu-tilde calculus suitable for sequent calculi that admit a focalising cut elimination (i.e. whose proofs are focalised when cut-free), such as Girard's classical logic LC or linear logic. This gives a setting in which Krivine's classical realisability extends naturally (in particular to call-by-value), with a presentation in terms of orthogonality. We give examples of applications to the theory of programming languages

Polarised Intermediate Representation of Lambda Calculus with Sums

Dec. 2015: see the added footnote on page 7International audienceThe theory of the λ-calculus with extensional sums is more complex than with only pairs and functions. We propose an untyped representation—an intermediate calculus—for the λ-calculus with sums, based on the following principles: 1) Computation is described as the reduction of pairs of an expression and a context; the context must be represented inside-out, 2) Operations are represented abstractly by their transition rule, 3) Positive and negative expressions are respectively eager and lazy; this polarity is an approximation of the type. We offer an introduction from the ground up to our approach, and we review the benefits.A structure of alternating phases naturally emerges through the study of normal forms, offering a reconstruction of focusing. Considering further purity assumption, we obtain maximal multi-focusing. As an application, we can deduce a syntax-directed algorithm to decide the equivalence of normal forms in the simply-typed λ-calculus with sums, and justify it with our intermediate calculus

Which simple types have a unique inhabitant?

International audienceWe study the question of whether a given type has a unique inhabitant modulo program equivalence. In the setting of simply-typed lambda-calculus with sums, equipped with the strong βη-equivalence, we show that uniqueness is decidable. We present a saturating focused logic that introduces irreducible cuts on positive types "as soon as possible". Backward search in this logic gives an effective algorithm that returns either zero, one or two distinct inhabitants for any given type. Preliminary application studies show that such a feature can be useful in strongly-typed programs, inferring the code of highly-polymorphic library functions, or "glue code" inside more complex terms