Recently, the authors identified the “logical essence” of the call-by-value CPS translation to be its decomposition into two translations: a map into a fragment of the sequent calculus LJQ already determining the same semantics as CPS; and a later, optional, “negative” encoding of the referred fragment into the CPS target, responsible for (and requiring) the introduction of double negations. In this paper we analyze how much of this architecture applies to call-by-name CPS translations. We take the most common translation, essentially due to Plotkin, and use the sequent calculus LJT. A similar decomposition holds again, with the CPS semantics already obtained by the translation into the sequent calculus. The perfect match, obtained by the negative encoding, between the sequent calculus and the CPS target is somewhat stronger than the call-by-value case, since it holds both before and after optimizing the CPS translation (by doing administrative reduction on the fly). The optimized CPS target is isomorphic to the source language: there is no call-by-name administrative normal form. Finally, we insert in the picture Hofmann-Streicher CPS translation. In the “positive” translation of the CPS target, back to the negation-free sequent calculus, the HS target can be used as a stepping stone: first, some negations are spared by the use of conjunction; next, the right introduction of conjunction is encoded by the left introduction of implication.

We introduce a call-by-name lambda-calculus $\lambda Jn$ with generalized applications which is equipped with distant reduction. This allows to unblock $\beta$-redexes without resorting to the standard permutative conversions of generalized applications used in the original $\Lambda J$-calculus with generalized applications of Joachimski and Matthes. We show strong normalization of simply-typed terms, and we then fully characterize strong normalization by means of a quantitative (i.e. non-idempotent intersection) typing system. This characterization uses a non-trivial inductive definition of strong normalization --related to others in the literature--, which is based on a weak-head normalizing strategy. We also show that our calculus $\lambda Jn$ relates to explicit substitution calculi by means of a faithful translation, in the sense that it preserves strong normalization. Moreover, our calculus $\lambda Jn$ and the original $\Lambda J$-calculus determine equivalent notions of strong normalization. As a consequence, $\lambda J$ inherits a faithful translation into explicit substitutions, and its strong normalization can also be characterized by the quantitative typing system designed for $\lambda Jn$, despite the fact that quantitative subject reduction fails for permutative conversions.

The essence of compiling with continuations is that conversion to continuation-passing style (CPS) is equivalent to a source language transformation converting to administrative normal form (ANF). Taking as source language Moggi's computational lambda-calculus (lbc), we define an alternative to the CPS-translation with target in the sequent calculus LJQ, named value-filling style (VFS) translation, and making use of the ability of the sequent calculus to represent contexts formally. The VFS-translation requires no type translation: indeed, double negations are introduced only when encoding the VFS target language in the CPS target language. This optional encoding, when composed with the VFS-translation reconstructs the original CPS-translation. Going back to direct style, the "essence" of the VFS-translation is that it reveals a new sublanguage of ANF, the value-enclosed style (VES), next to another one, the continuation-enclosing style (CES): such an alternative is due to a dilemma in the syntax of lbc, concerning how to expand the application constructor. In the typed scenario, VES and CES correspond to an alternative between two proof systems for call-by-value, LJQ and natural deduction with generalized applications, confirming proof theory as a foundation for intermediate representations.

In the context of intuitionistic sequent calculus, “naturality” means permutation-freeness (the terminology is essentially due to Mints). We study naturality in the context of the lambda-calculus with generalized applications and its multiary extension, to cover, under the Curry-Howard correspondence, proof systems ranging from natural deduction (with and without general elimination rules) to a fragment of sequent calculus with an iterable left-introduction rule, and which can still be recognized as a call-by-name lambda-calculus. In this context, naturality consists of a certain restricted use of generalized applications. We consider the further restriction obtained by the combination of naturality with normality w.r.t. the commutative conversion engendered by generalized applications. This combination sheds light on the interpretation of naturality as a vectorization mechanism, allowing a multitude of different ways of structuring lambda-terms, and the structuring of a multitude of interesting fragments of the systems under study. We also consider a relaxation of naturality, called weak naturality: this not only brings similar structural benefits, but also suggests a new “weak” system of natural deduction with generalized applications which is exempt from commutative conversions. In the end, we use all of this evidence as a stepping stone to propose a computational interpretation of generalized application (whether multiary or not, and without any restriction): it includes, alongside the argument(s) for the function, a general list – a new, very general, vectorization mechanism, that structures the continuation of the computation.

Abstract We introduce a call-by-name lambda-calculus $$\lambda J$$ λJ with generalized applications which integrates a notion of distant reduction that allows to unblock $$\beta $$ β -redexes without resorting to the permutative conversions of generalized applications. We show strong normalization of simply typed terms, and we then fully characterize strong normalization by means of a quantitative typing system. This characterization uses a non-trivial inductive definition of strong normalization –that we relate to others in the literature–, which is based on a weak-head normalizing strategy. Our calculus relates to explicit substitution calculi by means of a translation between the two formalisms which is faithful, in the sense that it preserves strong normalization. We show that our calculus $$\lambda J$$ λJ and the well-know calculus $$\varLambda J$$ ΛJ determine equivalent notions of strong normalization. As a consequence, $$\varLambda J$$ ΛJ inherits a faithful translation into explicit substitutions, and its strong normalization can be characterized by the quantitative typing system designed for $$\lambda J$$ λJ , despite the fact that quantitative subject reduction fails for permutative conversions.

The lambda-calculus with generalized applications is the Curry-Howard counterpart to the system of natural deduction with generalized elimination rules for intuitionistic implicational logic. In this paper we identify a call-by-value variant of the system and prove confluence, strong normalization, and standardization. In the end, we show that the cbn and cbv variants of the system simulate each other via mappings based on extensions of the "protecting-by-a-lambda" compilation technique.

If we consider as “member” of a simple type the outcome of any successful (possibly infinite) run of bottom-up proof search that starts from the type, then several concepts of “finiteness” for simple types are possible: the finiteness of the search space, the finiteness of any member, or the finiteness of the number of finite members (in other words, the inhabitants). In this paper we show that these three concepts are instances of the same parameterized notion of finiteness, and that a single, parameterized proof shows the decidability of all of them. One instance of this result means that termination of proof search is decidable. A separate result is that emptiness is also decidable (where emptiness is absence of “members” as above, not just absence of inhabitants). This fact is an ingredient of the main decidability result, but it also has a different application, the definition of the pruned search space - the one where branches leading to failure are chopped off. We conclude with our version of König’s lemma for simple types: a simple type has an infinite member exactly when the pruned search space is infinite.

A new approach to inhabitation problems in simply typed lambda-calculus is shown, dealing with both decision and counting problems. This approach works by exploiting a representation of the search space generated by a given inhabitation problem, which is in terms of a lambda-calculus for proof search that the authors developed recently. The representation may be seen as extending the Curry–Howard representation of proofs by lambda terms. Our methodology reveals inductive descriptions of the decision problems, driven by the syntax of the proof-search expressions, and produces simple, recursive decision procedures and counting functions. These allow to predict the number of inhabitants by testing the given type for syntactic criteria. This new approach is comprehensive and robust: based on the same syntactic representation, we also derive the state-of-the-art coherence theorems ensuring uniqueness of inhabitants.

We study strong normalization in a lambda calculus of proof-terms with co-control for the intuitionistic sequent calculus. In this sequent lambda calculus, the management of formulas on the left hand side of typing judgements is "dual" to the management of formulas on the right hand side of the typing judgements in Parigot's lambdamu calculus - that is why our system has first-class "co-control". The characterization of strong normalization is by means of intersection types, and is obtained by analyzing the relationship with another sequent lambda calculus, without co-control, for which a characterization of strong normalizability has been obtained before. The comparison of the two formulations of the sequent calculus, with or without co-control, is of independent interest. Finally, since it is known how to obtain bidirectional natural deduction systems isomorphic to these sequent calculi, characterizations are obtained of the strongly normalizing proof-terms of such natural deduction systems.

We 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.

This paper gives a characterisation, via intersection types, of the strongly normalising proof-terms of an intuitionistic sequent calculus (where LJ easily embeds). The soundness of the typing system is reduced to that of a well known typing system w

Multiary sequent terms were originally introduced as a tool for proving termination of permutative conversions in cut-free sequent calculus. This work develops the language of multiary sequent terms into a term calculus for the computational (Curry-Howard) interpretation of a fragment of sequent calculus with cuts and cut-elimination rules. The system, called generalized multiary λ-calculus , is a rich extension of the λ-calculus where the computational content of the sequent calculus format is explained through an enlarged form of the application constructor. Such constructor exhibits the features of multiarity (the ability to form lists of arguments) and generality (the ability to prescribe a kind of continuation). The system integrates in a modular way the multiary λ-calculus and an isomorphic copy of the λ-calculus with generalized application, Λ J (in particular, natural deduction is captured internally up to isomorphism). In addition, the system: (i) comes with permutative conversion rules, whose role is to eliminate the new features of application; (ii) is equipped with reduction rules — either the μ-rule, typical of the multiary setting, or rules for cut-elimination, which enlarge the ordinary β-rule. This article establishes the metatheory of the system, with emphasis on the role of the μ-rule, and including a study of the interaction of reduction and permutative conversions.