Abstract not found

Plotkin suggested using a polymorphic dual intuitionistic / linear type theory (PILLY) as a metalanguage for parametric polymorphism and recursion. In recent work the first two authors and R.L. Petersen have defined a notion of parametric LAPL-structure, which are models of PILLY, in which one can reason using parametricity and, for example, solve a large class of domain equations, as suggested by Plotkin. In this paper we show how an interpretation of a strict version of Bierman, Pitts and Russo’s language Lily into synthetic domain theory presented by Simpson and Rosolini gives rise to a parametric LAPL-structure. This adds to the evidence that the notion of LAPL-structure is a general notion suitable for treating many different parametric models, and it provides formal proofs of consequences of parametricity expected to hold for the interpretation. Finally, we show how these results in combination with Rosolini and Simpson’s computational adequacy result can be used to prove consequences of parametricity for Lily. In particular we show that one can solve domain equations in Lily up to ground contextual equivalence. 1

The aim of these notes is to give an outline of the categorical structure required for a model of linear logic. That means that we only try to motivate the various conditions imposed on such models; this is by no means intended to be a full account. For intuitionistic linear logic a detailed description of the process of defining such a model can be found in Gavin Bierman’s thesis [Bie94]. There are a number of exercises in these notes, they either aim to give examples for the various notions defined or they state minor results that are used subsequently and which hopefully are not too difficult to establish. If the category theory in these notes is too advanced you may want to try [BS] which does not assume any knowledge in that area at all, but proceeds fairly rapidly to the notions we use here. Originally these notes were written for my PhD students, but since I made the first version available on my webpage a number of people have told me they found them useful. As a result I felt obliged to to add details which were rather sketchy in the original account. In particular the account now contains more proofs and so I am hopeful that the various formulations of the notion of linear exponential comonad are finally correct. I would like to thank Paola Maneggia, Peter Selinger, Robin Houston and Nicola Gambino for useful feedback on these

Abstract. This paper provides a call-by-name and a call-by-value calculus, both of which have a Curry-Howard correspondence to the minimal normal logic K. The calculi are extensions of the λµ-calculi, and their semantics are given by CPS transformations into a calculus corresponding to the intuitionistic fragment of K. The duality between call-by-name and call-by-value with modalities is investigated in our calculi. 1

This note investigates two generic constructions used to produce categorical models of linear logic, the Chu construction and the Dialectica construction, in parallel. The constructions have the same objects, but are rather di#erent in other ways. We discuss similarities and di#erences and prove that the dialectica construction can be done over a symmetric monoidal closed basis. We also point out several interesting open problems concerning the Dialectica construction.

We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the Curry-Howard-Tait paradigm. We then prove logic-independent soundness and completeness theorems. The framework is instantiated with a number of examples: classical, intuitionistic, linear and modal propositional logics. Finally, we speculate how this framework may be extended beyond the propositional case.

This paper presents a sound and complete category-theoretic notion of models for Linear Abadi & Plotkin Logic [Birkedal et al., 2006], a logic suitable for reasoning about parametricity in combination with recursion. A subclass of these called parametric LAPL structures can be seen as an axiomatization of domain theoretic models of parametric polymorphism, and we show how to solve general (nested) recursive domain equations in these. Parametric LAPL structures constitute a general notion of model of parametricity in a setting with recursion. In future papers we will demonstrate this by showing how many different models of parametricity and recursion give rise to parametric LAPL structures, including Simpson and Rosolini’s set theoretic models [Rosolini and Simpson, 2004], a syntactic model based on Lily [Pitts, 2000, Bierman et al., 2000] and a model based on admissible pers over a reflexive domain [Birkedal et al., 2007].