Synthetic Domain Theory and Models of Linear Abadi & Plotkin Logic

In a recent article [L. Birkedal, R. E. Møgelberg, and R. L. Petersen. Parametric domain-theoretic models of linear Abadi & Plotkin logic. Technical Report TR-2005-57, IT University of Copenhagen, February 2005] the first two authors and R.L. Petersen have defined a notion of parametric LAPL-structure. Such structures are parametric models of the equational theory PILLY, a polymorphic intuitionistic / linear type theory with fixed points, in which one can reason using parametricity and, for example, solve a large class of domain equations [L. Birkedal, R. E. Møgelberg, and R. L. Petersen. Parametric domain-theoretic models of linear Abadi & Plotkin logic. Technical Report TR-2005-57, IT University of Copenhagen, February 2005, L Birkedal, R. E. Møgelberg, and R. L. Petersen. Parametric domain-theoretic models of polymorphic intuitionistic / linear lambda calculus. In Proceedings of the Twenty-first Conference on the Mathematical Foundations of Programming Semantics, 2005. To appear].Based on recent work by Simpson and Rosolini [G. Rosolini and A. Simpson. Using synthetic domain theory to prove operational properties of a polymorphic programming language based on strictness. Manuscript, 2004] we construct a family of parametric LAPL-structures using synthetic domain theory and use the results of loc. cit. and results about LAPL-structures to prove operational consequences of parametricity for a strict version of the Lily programming language. In particular we can show that one can solve domain equations in the strict version of Lily up to ground contextual equivalence.