A Modular Inference of Linear Types for Multiplicity-Annotated Arrows
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Bernardy et al. [2018] proposed a linear type system λ^q_→ as a core type system of Linear Haskell. In the system, the linearity is represented by multiplicity-annotated arrow types A →_m B, where m denotes the multiplicity of the argument of a function. Thanks to the representation, existing non-linear code typechecks without changes and newly written linear code can be used with the existing non-linear code for many cases. However, little is known for type inference for λ^q_→. Although Linear Haskell equips type inference, its algorithm has not been formalized and often fails to infer principal types, especially for higher-order functions. In this paper, based on OutsideIn(X) [Vytiniotis et al., 2011], we propose an inference system for a rank 1 qualified-typed variant of λ^q_→, which can infer principal types. A technical challenge is how to deal with ambiguous types inferred by naive qualified typing. We address the ambiguity issue by quantifier elimination and demonstrate the effectiveness of the approach by examples.
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