Relational Parametricity and Control

We study the equational theory of Parigot's second-order λμ-calculus in connection with a call-by-name continuation-passing style (CPS) translation into a fragment of the second-order λ-calculus. It is observed that the relational parametricity on the target calculus induces a natural notion of equivalence on the λμ-terms. On the other hand, the unconstrained relational parametricity on the λμ-calculus turns out to be inconsistent with this CPS semantics. Following these facts, we propose to formulate the relational parametricity on the λμ-calculus in a constrained way, which might be called ``focal parametricity''.Comment: 22 pages, for Logical Methods in Computer Scienc

A Complete, Co-Inductive Syntactic Theory of Sequential Control and State

We present a new co-inductive syntactic theory, eager normal form bisimilarity, for the untyped call-by-value lambda calculus extended with continuations and mutable references. We demonstrate that the associated bisimulation proof principle is easy to use and that it is a powerful tool for proving equivalences between recursive imperative higher-order programs. The theory is modular in the sense that eager normal form bisimilarity for each of the calculi extended with continuations and/or mutable references is a fully abstract extension of eager normal form bisimilarity for its sub-calculi. For each calculus, we prove that eager normal form bisimilarity is a congruence and is sound with respect to contextual equivalence. Furthermore, for the calculus with both continuations and mutable references, we show that eager normal form bisimilarity is complete: it coincides with contextual equivalence

Classical logic, continuation semantics and abstract machines

One of the goals of this paper is to demonstrate that denotational semantics is useful for operational issues like implementation of functional languages by abstract machines. This is exemplified in a tutorial way by studying the case of extensional untyped call-by-name λ-calculus with Felleisen's control operator 𝒞. We derive the transition rules for an abstract machine from a continuation semantics which appears as a generalization of the ¬¬-translation known from logic. The resulting abstract machine appears as an extension of Krivine's machine implementing head reduction. Though the result, namely Krivine's machine, is well known our method of deriving it from continuation semantics is new and applicable to other languages (as e.g. call-by-value variants). Further new results are that Scott's D∞-models are all instances of continuation models. Moreover, we extend our continuation semantics to Parigot's λμ-calculus from which we derive an extension of Krivine's machine for λμ-calculus. The relation between continuation semantics and the abstract machines is made precise by proving computational adequacy results employing an elegant method introduced by Pitts

Proofs and Refutations for Intuitionistic and Second-Order Logic

The ?^{PRK}-calculus is a typed ?-calculus that exploits the duality between the notions of proof and refutation to provide a computational interpretation for classical propositional logic. In this work, we extend ?^{PRK} to encompass classical second-order logic, by incorporating parametric polymorphism and existential types. The system is shown to enjoy good computational properties, such as type preservation, confluence, and strong normalization, which is established by means of a reducibility argument. We identify a syntactic restriction on proofs that characterizes exactly the intuitionistic fragment of second-order ?^{PRK}, and we study canonicity results

Categorical combinators

Our main aim is to present the connection between λ-calculus and Cartesian closed categories both in an untyped and purely syntactic setting. More specifically we establish a syntactic equivalence theorem between what we call categorical combinatory logic and λ-calculus with explicit products and projections, with β and η-rules as well as with surjective pairing. “Combinatory logic” is of course inspired by Curry's combinatory logic, based on the well-known S, K, I. Our combinatory logic is “categorical” because its combinators and rules are obtained by extracting untyped information from Cartesian closed categories (looking at arrows only, thus forgetting about objects). Compiling λ-calculus into these combinators happens to be natural and provokes only n log n code expansion. Moreover categorical combinatory logic is entirely faithful to β-reduction where combinatory logic needs additional rather complex and unnatural axioms to be. The connection easily extends to the corresponding typed calculi, where typed categorical combinatory logic is a free Cartesian closed category where the notion of terminal object is replaced by the explicit manipulation of applying (a function to its argument) and coupling (arguments to build datas in products). Our syntactic equivalences induce equivalences at the model level. The paper is intended as a mathematical foundation for developing implementations of functional programming languages based on a “categorical abstract machine,” as developed in a companion paper (Cousineau, Curien, and Mauny, in “Proceedings, ACM Conf. on Functional Programming Languages and Computer Architecture,” Nancy, 1985)

A syntactic approach to continuity of T-definable functionals

We give a new proof of the well-known fact that all functions $(\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$ which are definable in G\"odel's System T are continuous via a syntactic approach. Differing from the usual syntactic method, we firstly perform a translation of System T into itself in which natural numbers are translated to functions $(\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$. Then we inductively define a continuity predicate on the translated elements and show that the translation of any term in System T satisfies the continuity predicate. We obtain the desired result by relating terms and their translations via a parametrized logical relation. Our constructions and proofs have been formalized in the Agda proof assistant. Because Agda is also a programming language, we can execute our proof to compute moduli of continuity of T-definable functions