(Deep) induction rules for GADTs

Johann, Patricia; Ghiorzi, Enrico

doi:10.1145/3497775.3503680

Cited by 3 publications

(1 citation statement)

References 21 publications

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“…The main result of this paper shows that the choice of representation of data types in a language supporting GADTs determines whether or not that language can have parametric models [32]. It thus determines whether or not GADTs can enjoy consequences of parametricity such as representation independence [1,6], equivalences between programs [12], deep induction principles [17,20], and useful ("free") theorems about programs derived from their types alone [37]. This result is unintuitive, novel, and surprising.…”

Section: Introductionmentioning

confidence: 99%

GADTs, Functoriality, Parametricity: Pick Two

Johann,

Ghiorzi,

Jeffries

2021

Preprint

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GADTs can be represented either as their Church encodings à la Atkey, or as fixpoints à la Johann and Polonsky. While a GADT represented as its Church encoding need not support a map function satisfying the functor laws, the fixpoint representation of a GADT must support such a map function even to be well-defined. The two representations of a GADT thus need not be the same in general. This observation forces a choice of representation of data types in languages supporting GADTs. In this paper we show that choosing whether to represent data types as their Church encodings or as fixpoints determines whether or not a language supporting GADTs can have parametric models. This choice thus has important consequences for how we can program with, and reason about, these advanced data types.

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Section: Introductionmentioning

confidence: 99%

GADTs, Functoriality, Parametricity: Pick Two

Johann,

Ghiorzi,

Jeffries

2021

Preprint

Self Cite

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GADTs are not (Even partial) functors

Cagne,

Ghiorzi,

Johann

2024

Math. Struct. Comp. Sci.

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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 $\mathit{Set}$ introduces ghost elements, 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.

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The Essence of Generalized Algebraic Data Types

Sieczkowski,

Stepanenko,

Sterling

et al. 2024

Proc. ACM Program. Lang.

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This paper considers direct encodings of generalized algebraic data types (GADTs) in a minimal suitable lambda-calculus. To this end, we develop an extension of System F ω with recursive types and internalized type equalities with injective constant type constructors. We show how GADTs and associated pattern-matching constructs can be directly expressed in the calculus, thus showing that it may be treated as a highly idealized modern functional programming language. We prove that the internalized type equalities in conjunction with injectivity rules increase the expressive power of the calculus by establishing a non-macro-expressibility result in F ω , and prove the system type-sound via a syntactic argument. Finally, we build two relational models of our calculus: a simple, unary model that illustrates a novel, two-stage interpretation technique, necessary to account for the equational constraints; and a more sophisticated, binary model that relaxes the construction to allow, for the first time, formal reasoning about data-abstraction in a calculus equipped with GADTs.

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(Deep) induction rules for GADTs

Cited by 3 publications

References 21 publications

GADTs, Functoriality, Parametricity: Pick Two

GADTs, Functoriality, Parametricity: Pick Two

GADTs are not (Even partial) functors

The Essence of Generalized Algebraic Data Types

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