Rasmus Ejlers Møgelberg

First Steps in Synthetic Guarded Domain Theory: Step-Indexing in the Topos of Trees

We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S.

Categorical models for Abadi and Plotkin's logic for parametricity

We propose a new category-theoretic formulation of relational parametricity based on a logic for reasoning about parametricity given by Abadi and Plotkin (Plotkin & Abadi 1993). The logic can be used to reason about parametric models, such that we may prove consequences of parametricity that to our knowledge have not been proved before for existing category-theoretic notions of relational parametricity. We provide examples of parametric models and describe a way of constructing parametric models from given models of the second-order lambda calculus.

Guarded Dependent Type Theory with Coinductive Types

We present guarded dependent type theory, gDTT, an extensional dependent type theory with a `later modality and clock quantifiers for programming and proving with guarded recursive and coinductive types. The later modality is used to ensure the productivity of recursive definitions in a modular, type based, way. Clock quantifiers are used for controlled elimination of the later modality and for encoding coinductive types using guarded recursive types. Key to the development of gDTT are novel type and term formers involving what we call `delayed substitutions. These generalise the applicative functor rules for the later modality considered in earlier work, and are crucial for programming and proving with dependent types. We show soundness of the type theory with respect to a denotational model.

Intensional Type Theory with Guarded Recursive Types qua Fixed Points on Universes

Guarded recursive functions and types are useful for giving semantics to advanced programming languages and for higher-order programming with infinite data types, such as streams, e.g., for modeling reactive systems. We propose an extension of intensional type theory with rules for forming fixed points of guarded recursive functions. Guarded recursive types can be formed simply by taking fixed points of guarded recursive functions on the universe of types. Moreover, we present a general model construction for constructing models of the intensional type theory with guarded recursive functions and types. When applied to the groupoid model of intensional type theory with the universe of small discrete groupoids, the construction gives a model of guarded recursion for which there is a one-to-one correspondence between fixed points of functions on the universe of types and fixed points of (suitable) operators on types. In particular, we find that the functor category Grpdωop from the preordered set of natural numbers to the category of groupoids is a model of intensional type theory with guarded recursive types.

First steps in synthetic guarded domain theory: step-indexing in the topos of trees

We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S.

A type theory for productive coprogramming via guarded recursion

To ensure consistency and decidability of type checking, proof assistants impose a requirement of productivity on corecursive definitions. In this paper we investigate a type-based alternative to the existing syntactic productivity checks of Coq and Agda, using a combination of guarded recursion and quantification over clocks. This approach was developed by Atkey and McBride in the simply typed setting, here we extend it to a calculus with dependent types. Building on previous work on the topos-of-trees model we construct a model of the calculus using a family of presheaf toposes, each of which can be seen as a multi-dimensional version of the topos-of-trees. As part of the model construction we must solve the coherence problem for modelling dependent types in locally cartesian closed categories simulatiously in a whole family of locally cartesian closed categories. We do this by embedding all the categories in a large one and applying a recent approach to the coherence problem due to Streicher and Voevodsky.

Enriching an effect calculus with linear types

We define an enriched effect calculus by extending a type theory for computational effects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational effects; for example, the linear usage of imperative features such as state and/or continuations. n nOur main syntactic result is the conservativity of the enriched effect calculus over a basic effect calculus without linear primitives (closely related to Moggis computational metalanguage, Filinskis effect PCF and Levys call-by-push-value). The proof of this syntactic theorem makes essential use of a category-theoretic semantics, whose study forms the second half of the paper. n nOur semantic results include soundness, completeness, the initiality of a syntactic model, and an embedding theorem: every model of the basic effect calculus fully embeds in a model of the enriched calculus. The latter means that our enriched effect calculus is applicable to arbitrary computational effects, answering in the positive a question of Benton and Wadler (LICS 1996).

Domain-theoretical models of parametric polymorphism

We present a domain-theoretical model of parametric polymorphism based on admissible pers over a domain-theoretical model of the untyped lambda calculus. The model is shown to be a model of Abadi & Plotkins logic for parametricity, by the construction of an LAPL-structure as defined by the authors in [L. Birkedal, R.E. Mogelberg, R.L. Petersen, Parametric domain-theoretical models of polymorphic intuitionistic/linear lambda calculus, in: M. Escardo, A. Jung, M. Mislove (Eds.), Proceedings of Mathematical Foundations of Programming Semantics 2005, vol. 155, 2005, pp. 191-217; L. Birkedal, R.E. Mogelberg, R.L. Petersen, Category theoretical models of linear Abadi & Plotkin logic, 2006 (submitted for publication)]. This construction gives formal proof of solutions to a large class of recursive domain equations, which we explicate. As an example of a computation in the model, we explicitly describe the natural numbers object obtained using parametricity. The theory of admissible pers can be considered a domain theory for (impredicative) polymorphism. By studying various categories of admissible and chain complete pers and their relations, we discover a picture very similar to that of domain theory.

Linearly-used state in models of call-by-value

We investigate the phenomenon that every monad is a linear state monad. We do this by studying a fully-complete state-passing translation from an impure call-by-value language to a new linear type theory: the enriched call-by-value calculus. The results are not specific to store, but can be applied to any computational effect expressible using algebraic operations, even to effects that are not usually thought of as stateful. There is a bijective correspondence between generic effects in the source language and state access operations in the enriched call-byvalue calculus. n nFrom the perspective of categorical models, the enriched call-by-value calculus suggests a refinement of the traditional Kleisli models of effectful call-by-value languages. The new models can be understood as enriched adjunctions.

The enriched effect calculus: syntax and semantics

This paper introduces the enriched eect calculus , which extends established type theories for computational eects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational eects; for example, the linear usage of imperative features such as state and/or continuations. The enriched eect calculus is implemented as an extension of a basic eect calculus without linear primitives, which is closely related to Moggi’s computational metalanguage, Filinski’s eect PCF and Levy’s call-by-push-value. We present syntactic results showing: the delity of the behaviour of the linear connectives of the enriched eect calculus; the conservativity of the enriched eect calculus over its non-linear core (the eect calculus); and the non-conservativity of intuitionistic linear logic when considered as an extension of the enriched eect calculus. The second half of the paper investigates models for the enriched eect calculus, based on enriched category theory. We give several examples of such models, relating them to models of standard eect calculi (such as those based on monads), and to models of intuitionistic linear logic. We also prove soundness and completeness.