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Abstract Generalized Algebraic Data Types(GADTs) are a syntactic generalization of the usual algebraic data types (ADTs), such as lists, trees, etc. ADTs’ standard initial algebra semantics (IAS) in the category$$\mathit{Set}$$of sets justify critical syntactic constructs – such as recursion, pattern matching, and fold – for programming with them. In this paper, we show that semantics for GADTs that specialize to the IAS for ADTs are necessarily unsatisfactory. First, we show that the functorial nature of such semantics for GADTs in$$\mathit{Set}$$introducesghostelements, i.e., elements not writable in syntax. Next, we show how such ghost elements break parametricity. We observe that the situation for GADTs contrasts dramatically with that for ADTs, whose IAS coincides with the parametric model constructed via their Church encodings in System F. Our analysis reveals that the fundamental obstacle to giving a functorial IAS for GADTs is the inherently partial nature of their map functions. We show that this obstacle cannot be overcome by replacing$$\mathit{Set}$$with other categories that account for this partiality. 

Relational parametricity was first introduced by Reynolds for System F. Although System F provides a strong model for the type systems at the core of modern functional programming languages, it lacks features of daily programming practice such as complex data types. In order to reason parametrically about such objects, Reynolds’ seminal ideas need to be generalized to extensions of System F. Here, we explore such a generalization for the extension of System F by Generalized Algebraic Data Types (GADTs) as found in Haskell. Although GADTs generalize Algebraic Data Types (ADTs) — i.e., simple recursive types such as lists, trees, etc. — we show that naively extending the parametric treatment of these recursive types is not enough to tackle GADTs. We propose a tentative workaround for this issue, borrowing ideas from the categorical semantics of GADTs known as “functorial completion”. We discuss some applications, as well as some limitations, of this solution.