We present a domain-theoretic model of parametric polymorphism based on admissible per’s over a domain-theoretic model of the untyped lambda calculus. The model is shown to be a model of Abadi & Plotkin’s logic for parametricity, by the construction of an LAPL-structure as defined by the authors in [7, 5]. 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 per’s can be considered a domain theory for (impredicative) polymorphism. By studying various categories of admissible and chain complete per’s and their relations, we discover a picture very similar to that of domain theory. 1

Abstract. We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkin’s system for the intuitionistic case. The calculus has the nonlinear andlinear implications as the basic constructs, andthis design choice allows a technically managable axiomatization without commuting conversions. Despite this simplicity, the calculus is shown to be sound andcomplete for category-theoretic models given by ∗-autonomous categories with linear exponential comonads. 1

Flexibility of programming and efficiency of program execution are two important features of a programming language. Unfortunately, however, these two features conflict each other in design and implementation of a modern statically typed programming language. Flexibility is model of computation, while efficiency requires optimal use of low-level primitives specialized to individual data structures. The motivation of this work is to reconcile these two features by developing a mechanism for specializing polymorphic primitives based on static type information. We analyze the existing methods for compiling a record calculus and an unboxed calculus, extract their common structure, and develop a framework for type-directed specialization of polymorphism. 1

We give a series of glueing constructions for categorical models of fragments of linear logic. Specifically, we consider the glueing of (i) symmetric monoidal closed categories (models of Multiplicative Intuitionistic Linear Logic), (ii) symmetric monoidal adjunctions (for interpreting the modality !) and (iii) -autonomous categories (models of Multiplicative Linear Logic); the glueing construction for -autonomous categories is a mild generalization of the double glueing construction due to Hyland and Tan. Each of the glueing techniques can be used for creating interesting models of linear logic. In particular, we use them, together with the free symmetric monoidal cocompletion, for deriving Kripke-like parameterized logical predicates (logical relations) for the fragments of linear logic. As an application, we show full completeness results for translations between linear type theories. Contents 1 Introduction 3 2 Preliminaries 4 2.1 Symmetric Monoidal Structures . . . . . . . ....

We present a short proof of a folklore result: the Girard translation from the simply typed lambda calculus to the linear lambda calculus is fully complete. The proof makes use of a notion of logical predicates for intuitionistic linear logic. While the main result is of independent interest, this paper can be read as a tutorial on this proof technique for reasoning about relations between type theories.

Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning a state of one physical system into a state of another system — perhaps

We study the glueing constructions (comma objects) on general algebraic structures on a 2-category, described in terms of 2-monads and adjunctions. Specifically, lifting theorems for the comma objects and change-of-base results on both algebras of 2-monads and adjunctions in a 2-category are presented. As a leading example, we take the 2-monad on Cat whose algebras are symmetric monoidal categories, and show that many of the constructions in our previous work on models of linear type theories can be derived within this axiomatics. 1 Introduction In the previous work [2, 3] we have considered a glueing construction for symmetric monoidal (closed) categories, for studying the logical predicates for models of linear type theories. In that construction the glueing functor is supposed to be lax symmetric monoidal, thus preserves the structure only up to a few coherent morphisms, not up to isomorphisms or identity. From a view of the study of categories with algebraic structures [8] (which...

Abstract. This paper presents a novel syntactic logical relation for a polymorphic linear λ-calculus that treats all types as linear and introduces the constructor! to account for intuitionistic terms. We define a logical relation for open values under both open linear and intuitionistic contexts, then extend it for open terms with evaluation and open relation substitutions. Relations that instantiate type quantifiers are for open terms and types. We demonstrate the applicability of this logical relation through its soundness with respect to contextual equivalence, along with free theorems for linearity that are difficult to achieve by closed logical relations. When interpreting types on only closed terms, the model defaults to a closed logical relation that is both sound and complete with respect to contextual equivalence, and is sufficient to reason about isomorphisms of type encodings. The idea of using open logical relations also extends easily to System F ◦ —an extension of System F that uses kinds to distinguish linear from intuitionistic types. All of our results have been mechanically verified in Coq, which are also extensive formalizations for polymorphic linear languages in proof assistants. 1