A relationally parametric model of dependent type theory

Reynolds' theory of relational parametricity captures the invariance of polymorphically typed programs under change of data representation. Reynolds' original work exploited the typing discipline of the polymorphically typed lambda-calculus System F, but there is now considerable interest in extending relational parametricity to type systems that are richer and more expressive than that of System F. This paper constructs parametric models of predicative and impredicative dependent type theory. The significance of our models is twofold. Firstly, in the impredicative variant we are able to deduce the existence of initial algebras for all indexed=functors. To our knowledge, ours is the first account of parametricity for dependent types that is able to lift the useful deduction of the existence of initial algebras in parametric models of System F to the dependently typed setting. Secondly, our models offer conceptual clarity by uniformly expressing relational parametricity for dependent types in terms of reflexive graphs, which allows us to unify the interpretations of types and kinds, instead of taking the relational interpretation of types as a primitive notion. Expressing our model in terms of reflexive graphs ensures that it has canonical choices for the interpretations of the standard type constructors of dependent type theory, except for the interpretation of the universe of small types, where we formulate a refined interpretation tailored for relational parametricity. Moreover, our reflexive graph model opens the door to generalisations of relational parametricity, for example to higher-dimensional relational parametricity.