Towards a Convenient Category of Topological Domains

We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models

A Convenient Category of Domains

We motivate and define a category of "topological domains", whose objects are certain topological spaces, generalising the usual $omega$-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains combining such properties

Programming with Algebraic Effects and Handlers

Eff is a programming language based on the algebraic approach to computational effects, in which effects are viewed as algebraic operations and effect handlers as homomorphisms from free algebras. Eff supports first-class effects and handlers through which we may easily define new computational effects, seamlessly combine existing ones, and handle them in novel ways. We give a denotational semantics of eff and discuss a prototype implementation based on it. Through examples we demonstrate how the standard effects are treated in eff, and how eff supports programming techniques that use various forms of delimited continuations, such as backtracking, breadth-first search, selection functionals, cooperative multi-threading, and others

Categories of embeddings

AbstractWe present a categorical generalisation of the notion of domains, which is closed under (suitable) exponentiation. The goal was originally to generalise Girard's model of polymorphism to Fω. If we specialise this notion in the poset case, we get new cartesian closed categories of domains

Relational Graph Models at Work

We study the relational graph models that constitute a natural subclass of relational models of lambda-calculus. We prove that among the lambda-theories induced by such models there exists a minimal one, and that the corresponding relational graph model is very natural and easy to construct. We then study relational graph models that are fully abstract, in the sense that they capture some observational equivalence between lambda-terms. We focus on the two main observational equivalences in the lambda-calculus, the theory H+ generated by taking as observables the beta-normal forms, and H* generated by considering as observables the head normal forms. On the one hand we introduce a notion of lambda-K\"onig model and prove that a relational graph model is fully abstract for H+ if and only if it is extensional and lambda-K\"onig. On the other hand we show that the dual notion of hyperimmune model, together with extensionality, captures the full abstraction for H*

Type‐Preserving CPS Translation of Σ and Π Types is Not Not Possible

International audienceDependently typed languages like Coq are used to specify and prove functional correctness of source programs,but what we ultimately need are guarantees about correctness of compiled code. By preserving dependenttypes through each compiler pass, we could preserve source-level specifications and correctness proofs intothe generated target-language programs. Unfortunately, type-preserving compilation of dependent types isnontrivial. In 2002, Barthe and Uustalu showed that type-preserving CPS is not possible for languages likeCoq. Specifically, they showed that for strong dependent pairs (Σ types), the standard typed call-by-name CPSis not type preserving. They further proved that for dependent case analysis on sums, a class of typed CPStranslations—including the standard translation—is not possible. In 2016, Morrisett noticed a similar problemwith the standard call-by-value CPS translation for dependent functions (Π types). In essence, the problem isthat the standard typed CPS translation by double-negation, in which computations are assigned types of theform (A → ⊥) → ⊥, disrupts the term/type equivalence that is used during type checking in a dependentlytyped language.In this paper, we prove that type-preserving CPS translation for dependently typed languages is not notpossible. We develop both call-by-name and call-by-value CPS translations from the Calculus of Constructionswith both Π and Σ types (CC) to a dependently typed target language, and prove type preservation andcompiler correctness of each translation. Our target language is CC extended with an additional equivalencerule and an additional typing rule, which we prove consistent by giving a model in the extensional Calculus ofConstructions. Our key observation is that we can use a CPS translation that employs answer-type polymorphism,where CPS-translated computations have type ∀α.(A → α) → α. This type justifies, by a free theorem,the new equality rule in our target language and allows us to recover the term/type equivalences that CPStranslation disrupts. Finally, we conjecture that our translation extends to dependent case analysis on sums,despite the impossibility result, and provide a proof sketch