Extending the Quantitative Pattern-Matching Paradigm

TL;DR

A type system based on intersection types is introduced, and it is shown through purely logical methods that the set of terminating terms of the language corresponds exactly to that of well-typed terms.

Abstract

We show how (well-established) type systems based on non-idempotent intersection types can be extended to characterize termination properties of functional programming languages with pattern matching features. To model such programming languages, we use a (weak and closed) $λ$-calculus integrating a pattern matching mechanism on algebraic data types (ADTs). Remarkably, we also show that this language not only encodes Plotkin's CBV and CBN $λ$-calculus as well as other subsuming frameworks, such as the bang-calculus, but can also be used to interpret the semantics of effectful languages with exceptions. After a thorough study of the untyped language, we introduce a type system based on intersection types, and we show through purely logical methods that the set of terminating terms of the language corresponds exactly to that of well-typed terms. Moreover, by considering non-idempotent intersection types, this characterization turns out to be quantitative, i.e. the size of the type derivation of a term t gives an upper bound for the number of evaluation steps from t to its normal form.