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).

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). 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.