The logical principles of focalisation and polarisation can be used to design well-behaved term syntaxes for sequent calculus, which play a role as meta-languages for describing effectful computation. On the semantics side, this corresponds to an axiomatic and polarised notion of model of computation stated in terms of non-associative categories as well as adjunctions between “bare” functors (reflexive graph morphisms) over such non-associative categories.

In this paper, we study the special and delicate cases of resource and effect modalities in a general intuitionistic and linear setting: an exponential comonad ! (refining the necessity modality □) and a strong monad (written ◊). The starting point of our contribution is noticing that the completeness for a polarised syntax for ! and ◊ with respect to (co)monads in linear call-by-push-value models can be achieved if we move to relative (co)monads (Altenkirch, Chapman and Uustalu, 2015; Arkor and McDermott, 2024): more precisely, comonads relative to ↓ (the positive shift functor) for ! and monads relative to ↑ (the negative shift functor) for ◊.

These specialisations of the concept of relative (co)monad to call-by-push-value adjunctions recently appeared in Jiang, Xue and New (2025) and Melliès (2025, 2026). Yet the syntax we present arose from proof-theoretic consideration in Munch-Maccagnoni (2009) and Curien, Fiore and Munch-Maccagnoni (CFMM 2016), without the link with relative (co)monads being noticed at the time. Our first remark and explanation is thus that (co)monads relative to a call-by-push-value adjunction have been motivated previously from a proof-theoretic perspective in the context of focalisation, which also provides a meta-language for these concepts in an effectful setting.

We carry out the study of these modalities from the axiomatic, non-associative point of view. We recall the definition of adjunction between bare functors in this context, and establish correspondence results between this notion of adjunction and that of relative adjunction. This correspondence is then extended to linear-non-linear and strong versions of adjunctions as needed to model ! and ◊.

The concept of ‘undone science’ emerged in the 2010s in research in social sciences at the intersection of studies on social movements and of science and technology studies. It refers to research questions that are neglected, ignored, or left unfunded, even though they deserve to be explored. The aim of this special issue is to apply this concept to computer science, by examining whether the way this discipline is structured (including its sociological, economic, and political dimensions), as well as the paradigms that shape it, make it possible to identify epistemological and ethical questions that are crucial for its development and conception.

Le concept de « science non faite » a émergé à partir des années 2010 dans des travaux en sciences sociales à l’interface entre les études sur les mouvements sociaux et celles des sciences et des techniques. Il désigne des questions de recherche négligées, ignorées ou non financées, bien qu’elles méritent d’être explorées. L’objectif de ce cahier thématique est d’appliquer ce concept à l’informatique, en examinant si la manière dont cette discipline est structurée (y compris sa dimension sociologique, économique et politique), ainsi que les paradigmes qui la façonnent, permettent d’identifier des questions épistémologiques et éthiques cruciales pour son développement et sa conception.

In this short paper, we advocate for the idea that continuation-based intermediate languages correspond to intermediate logics. The goal of intermediate languages is to serve as a basis for compiler intermediate representations, allowing to represent expressive program transformations for optimisation and compilation, while preserving the properties that make programs compilable efficiently in the first place, such as the “stackability” of continuations. Intermediate logics are logics between intuitionistic and classical logic in terms of provability. Second-class continuations used in CPS-based intermediate languages correspond to a classical modal logic S4 with the added restriction that implications may only return modal types. This indeed corresponds to an intermediate logic, owing to the Gödel-McKinsey-Tarski theorem which states the intuitionistic nature of the modal fragment of S4. We introduce a three-kinded polarised sequent calculus for S4, together with an operational machine model that separates a heap from a stack. With this model we study a stackability property for the modal fragment of S4.

We analyse the problem of combining linearity, effects, and exceptions, in abstract models of programming languages, as the issue of providing some kind of strength for a monad T(−⊕E) in a linear setting. We consider in particular for T the allocation monad, which we introduce to model and study resource-safety properties. We apply these results to a series of two linear effectful calculi for which we establish their resource-safety properties. The first calculus is a linear (optionally ordered) call-by-push-value language with two allocation effects new and delete. The resource-safety properties follow from the linear and ordered character of the typing rules.

We then integrate exceptions with linearity and effects by adjoining default destruction actions to types, as inspired by C++/Rust destructors. We see destructors as objects δ:A→TI in the slice category over TI. This construction gives rise to a second calculus, the resouce call-by-push-value, featuring exceptions and destructors, and whose weakening and exchange rules perform side-effects. It is therefore affine at the level of types but ordered at the level of derivations. As in C++/Rust, a “move” operation—the side-effecting exchange rule—is necessary for releasing resources in random order, as opposed to LIFO order.

In the spirit of the Curry-Howard correspondence between proofs and programs, we define and study a syntax and semantics for classical logic equipped with a computationally involutive negation, using a polarised effect calculus, the linear classical L-calculus. A main challenge in designing a denotational semantics for the calculus is to accommodate both call-by-value and call-by-name evaluation strategies, which leads to a failure of associativity of composition. In order to tackle this issue, we define a notion of adjunction between graph morphisms on non-associative categories, which we use to formulate polarized and non-associative notions of symmetric monoidal closed duploid and of dialogue duploid. We show that they provide a direct style counterpart to adjunction models: linear effect adjunctions for the (linear) call-by-push-value calculus and dialogue chiralities for linear continuations, respectively. In particular, we show that the syntax of the linear classical L-calculus can be interpreted in any dialogue duploid, and that it defines in fact a syntactic dialogue duploid. As an application, we establish, by semantic as well as syntactic means, the Hasegawa-Thielecke theorem, which states that the notions of central map and of thunkable map coincide in any dialogue duploid (in particular, for any double negation monad on a symmetric monoidal category).

Destructors, responsible for releasing memory and other resources in languages such as C++ and Rust, can lead to stack overflows when releasing a recursive structure that is too deep. In certain cases, it is possible to generate an efficient destructor (non-allocating and tail recursive) using a typed variant of pointer reversal. We extend this technique by making it more modular, in order to handle abstract types, separate compilation, and unboxed types.

We propose a new interface and implementation for managing garbage collector (GC) roots for the OCaml foreign-function interface (FFI), which offers: • better performance than existing root-registration interfaces, • efficient support for multicore OCaml, thanks to a multicore-friendly design, • a reasoning based on resource-management idioms, enabling an easier OCaml FFI in Rust.

This paper reports the good hypothetical performance, in rigorous practical terms, of embedding (borrowing) linearly-allocated values inside garbage-collected values. A companion submission to the ML workshop reports a symmetrical result: how to efficiently embed (own) garbage-collected values inside linearly-allocated values. Taken together, the broader motivation is to show the feasibility of basing linear allocation with re-use (Lafont 1988; Baker 1992) in languages that would still leverage state-of-art garbage collection for non-linear values.

We propose to bring together a dependently-typed theory (ECC) and polarised sequent calculus, by presenting a calculus Ldep suitable as a vehicle for compilation and representation of effectful computations. As a first step in that direction, we show that Ldep advantageously factorize a dependently typed continuation-passing style translation for ECC+call/cc. To avoid the inconsistency of type theory with control operators, we restrict their interaction. Nonetheless, in the pure case, we obtain an unrestricted translation from ECC to itself, thus opening the door to the definition of dependently typed compilation transformations.

The current multicore OCaml implementation bans so-called “naked pointers”, pointers to outside the OCaml heap unless they follow drastic restrictions. A backwards-incompatible change has been proposed to make way for the new multicore GC in OCaml. I argue that out-of-heap pointers are not an anomaly, but are part of a better systems programming future.

The resource-management model of C++ and Rust relies on compiler-generated destructors called predictably and reliably. In current implementations, the generated destructor consumes stack space proportionally to the depth of the structure it destructs. We describe a way to derive destructors for algebraic data types that consume a constant amount of stack and heap. We discuss applicability to C++ and Rust, and also some implication for anyone wishing to extend an ML-style language with first-class resources.

We model exceptions in a linear, effectful setting by relaxing the notion of monadic strength to contexts that are discardable, in the spirit of C++ destructors. This gives rise to a resource modality reminiscent of unique_ptr and move semantics in C++11. We explore consequences in language design for resource management in functional programming languages.

We present a resource-management model for ML-style programming languages, designed to be compatible with the OCaml philosophy and runtime model. This is a proposal to extend the OCaml language with destructors, move semantics, and resource polymorphism, to improve its safety, efficiency, interoperability, and expressiveness. It builds on the ownership-and-borrowing models of systems programming languages (Cyclone, C++11, Rust) and on linear types in functional programming (Linear Lisp, Clean, Alms). It continues a synthesis of resources from systems programming and resources in linear logic initiated by Baker.

It is a combination of many known and some new ideas. On the novel side, it highlights the good mathematical structure of Stroustrup's “Resource acquisition is initialisation” (RAII) idiom for resource management based on destructors, a notion sometimes confused with finalizers, and builds on it a notion of resource polymorphism, inspired by polarisation in proof theory, that mixes C++'s RAII and a tracing garbage collector (GC). In particular, it proposes to identify the types of GCed values with types with trivial destructor: from this definition it deduces a model in which GC is the default allocation mode, and where GCed values can be used without restriction both in owning and borrowing contexts.

The proposal targets a new spot in the design space, with an automatic and predictable resource-management model, at the same time based on lightweight and expressive language abstractions. It is backwards-compatible: current code is expected to run with the same performance, the new abstractions fully combine with the current ones, and it supports a resource-polymorphic extension of libraries. It does so with only a few additions to the runtime, and it integrates with the current GC implementation. It is also compatible with the upcoming multicore extension, and suggests that the Rust model for eliminating data-races applies.

Interesting questions arise for a safe and practical type system, many of which have already been thoroughly investigated in the languages and prototypes Cyclone, Rust, and Alms.

Following renewed interest in duploids arising from the exponential comonad of linear logic (the construction describing polarised intuitionistic translations into linear logic), I summarise here various remarks: • about a decomposition of Girard's “boring” translation as the expression of call-by-value in call-by-name, dual to how thunks are used to express call-by-name in call-by-value • about the coincidence between linear CPS translations and Girard's translations of intuitionistic logic into linear logic, • about a completeness property of historical models of linear logic in the above context • about a rational reconstruction of these translations with the Linear Call-by-Push-Value.

We present Curry-style calculi for intuitionistic (linear) logic with polarised evaluation order and give self-contained proofs of their main properties and of their interpretation into (Linear) Call-by-Push-Value models: subject reduction, confluence, strong normalisation, coherence, soundness, and focusing.

We consider the Curry-Howard-Lambek correspondence for effectful computation and resource management, specifically proposing polarised calculi together with presheaf-enriched adjunction models as the starting point for a comprehensive semantic theory relating logical systems, typed calculi, and categorical models in this context.

Our thesis is that the combination of effects and resources should be considered orthogonally. Model theoretically, this leads to an understanding of our categorical models from two complementary perspectives: (i) as a linearisation of CBPV (Call-by-Push-Value) adjunction models, and (ii) as an extension of linear/non-linear adjunction models with an adjoint resolution of computational effects. When the linear structure is cartesian and the resource structure is trivial we recover Levy's notion of CBPV adjunction model, while when the effect structure is trivial we have Benton's linear/non-linear adjunction models. Further instances of our model theory include the dialogue categories with a resource modality of Melliès and Tabareau, and the [E]EC ([Enriched] Effect Calculus) models of Egger, Møgelberg and Simpson. Our development substantiates the approach by providing a lifting theorem of linear models into cartesian ones.

To each of our categorical models we systematically associate a typed term calculus, each of which corresponds to a variant of the sequent calculi LJ (Intuitionistic Logic) or ILL (Intuitionistic Linear Logic). The adjoint resolution of effects corresponds to polarisation whereby, syntactically, types locally determine a strict or lazy evaluation order and, semantically, the associativity of cuts is relaxed. In particular, our results show that polarisation provides a computational interpretation of CBPV in direct style. Further, we characterise depolarised models: those where the cut is associative, and where the evaluation order is unimportant. We explain possible advantages of this style of calculi for the operational semantics of effects.

The 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.

Negation is not involutive in the λC calculus because it does not distinguish captured stacks from continuations. We show that there is a formulae-as-types correspondence between the involutive negation in proof theory, and a notion of high-level access to the stacks studied by Felleisen and Clements.

We introduce polarised, untyped, calculi compatible with extensionality, for both of classical sequent calculus and classical natural deduction, with connectives for an involutive negation. The involution is due to the ℓ delimited control operator that we introduce, which allows us to implement the idea that captured stacks, unlike continuations, can be inspected. Delimiting control also gives a constructive interpretation to falsity. We describe the isomorphism there is between A and ¬¬A, and thus between ¬∀ and ∃¬.

We characterise the polarised evaluation order through a categorical structure where the hypothesis that composition is associative is relaxed. Duploid is the name of the structure, as a reference to Jean-Louis Loday's duplicial algebras. The main result is a reflection Adj→Dupl where Dupl is a category of duploids and duploid functors, and Adj is the category of adjunctions and pseudo maps of adjunctions. The result suggests that the various biases in denotational semantics: indirect, call-by-value, call-by-name... are a way of hiding the fact that composition is not always associative.

We review the close relationship between abstract machines for (call-by-name or call-by-value) lambda-calculi (extended with Felleisen's C) and sequent calculus, reintroducing on the way Curien-Herbelin's syntactic kit expressing the duality of computation. We use this kit to provide a term language for a presentation of LK (with conjunction, disjunction, and negation), and to transcribe cut elimination as (non confluent) rewriting. A key slogan here, which may appear here in print for the first time, is that commutative cut elimination rules are explicit substitution propagation rules.

We then describe the focalised proof search discipline (in the classical setting), and narrow down the language and the rewriting rules to a confluent calculus (a variant of the second author's focalising system L). We then define a game of patterns and counterpatterns, leading us to a fully focalised finitary syntax for a synthetic presentation of classical logic, that provides a quotient on (focalised) proofs, abstracting out the order of decomposition of negative connectives.

We 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.

In this version extended with appendices, we in particular give the two-sided formulation of LC with the involutive classical negation. We also show that there is in classical realisability a notion of internal completeness similar to the one of Ludics.

The thesis is a contribution to the understanding of the nature, role, and mechanisms of polarisation in programming languages, proof theory and categorical models. Polarisation corresponds to the idea the associativity of composition of morphisms is an hypothesis that has to be relaxed, and underlies many models of computation.

In our demonstration, we introduce duploids, which model polarisation directly. These compositional structures, which, unlike categories, do not always require composition to be associative, are shown to be in correspondence with certain completed forms of categorical adjunctions (Chapter II). This analysis of polarisation is then extended into a fine-grained decomposition of continuation-passing-style (CPS) transformations, which model and implement higher-order programming languages. As an application we show how polarisation accounts for evaluation order in models of delimited control operators (Chapter III). Finally, we give a formulae-as-types interpretation of an involutive negation, which shows the importance of polarisation for constructiveness in proof theory (Chapter IV).

The cornerstone of our approach is an interactive term-based representation of proofs and programs (L calculi) which exposes the structure of polarities. It is based on the correspondence between abstract machines and sequent calculi, and syntheses various trends: the modelling of control, evaluation order and effects in programming languages, the quest for a relationship between categorical duality and continuations, and the interactive notion of construction in proof theory. We give a gentle introduction to our approach which only assumes elementary knowledge of simply-typed λ calculus and rewriting (Chapter I).