Intersection and union types are becoming more popular by the day, entering the mainstream in programming languages like TypeScript and Scala 3. Yet, no language so far has managed to combine these powerful types with principal polymorphic type inference. We present a solution to this problem in MLstruct, a language with subtyped records, equirecursive types, first-class unions and intersections, class-based instance matching, and ML-style principal type inference. While MLstruct is mostly structurally typed, it contains a healthy sprinkle of nominality for classes, which gives it desirable semantics, enabling the expression of a powerful form of extensible variants that does not need row variables. Technically, we define the constructs of our language using conjunction, disjunction, and negation connectives, making sure they form a Boolean algebra, and we show that the addition of a few nonstandard but sound subtyping rules gives us enough structure to derive a sound and complete type inference algorithm. With this work, we hope to foster the development of better type inference for present and future programming languages with expressive subtyping systems.

/pdf/mlstruct-principal-type-inference-in-a-boolean-algebra-of-17q2s6zo.pdf

MLstruct: principal type inference in a Boolean algebra of structural types

The typed merge operator offers the promise of a compositional style of statically-typed programming in which solutions to the expression problem arise naturally. This approach, dubbed compositional programming, has recently been demonstrated by Zhang et al. Unfortunately, the merge operator is an unwieldy beast. Merging values from overlapping types may be ambiguous, so disjointness relations have been introduced to rule out undesired nondeterminism and obtain a well-behaved semantics. Past type systems using a disjoint merge operator rely on intersection types, but extending such systems to include union types or overloaded functions is problematic: naively adding either reintroduces ambiguity. In a nutshell: the elimination forms of unions and overloaded functions require values to be distinguishable by case analysis, but the merge operator can create exotic values that violate that requirement. This paper presents F⋈, a core language that demonstrates how unions, intersections, and overloading can all coexist with a tame merge operator. The key is an underlying design principle that states that any two inhabited types can support either the deterministic merging of their values, or the ability to distinguish their values, but never both. To realize this invariant, we decompose previously studied notions of disjointness into two new, dual relations that permit the operation that best suits each pair of types. This duality respects the polarization of the type structure, yielding an expressive language that we prove to be both type safe and deterministic.

A Bowtie for a Beast: Overloading, Eta Expansion, and Extensible Data Types in F⋈

Erlang is a functional programming language with dynamic typing. The language offers great flexibility for destructing values through pattern matching and dynamic type tests. Erlang also comes with a type language supporting parametric polymorphism, equi-recursive types, as well as union and a limited form of intersection types. However, type signatures only serve as documentation; there is no check that a function body conforms to its signature. Set-theoretic types and semantic subtyping fit Erlang’s feature set very well. They allow expressing nearly all constructs of its type language and provide means for statically checking type signatures. This article brings set-theoretic types to Erlang and demonstrates how existing Erlang code can be statically type checked without or with only minor modifications to the code. Further, the article formalizes the main ingredients of the type system in a small core calculus, reports on an implementation of the system, and compares it with other static type checkers for Erlang.

Set-theoretic Types for Erlang

The typed merge operator offers the promise of a compositional style of statically-typed programming in which solutions to the expression problem arise naturally.This approach, dubbed compositional programming, has recently been demonstrated by Zhang, Sun, and Oliveira [2021]. Unfortunately, the merge operator is an unwieldy beast. Merging values from overlapping types may be ambiguous, so disjointness relations have been introduced to rule out undesired nondeterminism and obtain a well-behaved semantics. Past type systems using a disjoint merge operator rely on intersection types, but extending such systems to include union types or overloaded functions is problematic: naively adding either reintroduces ambiguity. In a nutshell: the elimination forms of unions and overloaded functions require values to be distinguishable by case analysis, but the merge operator can create exotic values that violate that requirement. This paper presents F⊲⊳, a core language that demonstrates how unions, intersections, and overloading can all coexist with a tame merge operator. The key is an underlying design principle that states that any two inhabited types can support either the deterministic merging of their values, or the ability to distinguish their values, but never both. To realize this invariant, we decompose previously studied notions of disjointness into two new, dual relations that permit the operation that best suits each pair of types. This duality respects the polarization of the type structure, yielding an expressive language that we prove to be both type safe and deterministic.

A Bowtie for a Beast (Technical Appendix)

In this essay, I present the advantages and, I dare say, the beauty of programming in a language with set-theoretic types, that is, types that include union, intersection, and negation type connectives. I show by several examples how set-theoretic types are necessary to type some common programming patterns, but also how they play a key role in typing several language constructs-from branching and pattern matching to function overloading and type-cases-very precisely. I start by presenting the theory of types known as semantic subtyping and extend it to include polymorphic types. Next, I discuss the design of languages that use these types. I start by defining a theoretical framework that covers all the examples given in the first part of the presentation. Since the system of the framework cannot be effectively implemented, I then describe three effective restrictions of this system: (i) a polymorphic language with explicitly-typed functions, (ii) an implicitly-typed polymorphic language{\`a} la Hindley-Milner, and (iii) a monomorphic language that, by implementing classic union-elimination, precisely reconstructs intersection types for functions and implements a very general form of occurrence typing. I conclude the presentation with a short overview of other aspects of these languages, such as pattern matching, gradual typing, and denotational semantics.

Programming with union, intersection, and negation types.

We extend classic union and intersection type systems with a type-case construction and show that the
combination of the union elimination rule of the former and the typing rules for type-cases of our extension
encompasses occurrence typing. To apply this system in practice, we define a canonical form for the expressions
of our extension, called MSC-form. We show that an expression of the extension is typable if and only if its
MSC-form is, and reduce the problem of typing the latter to the one of reconstructing annotations for that
term. We provide a sound algorithm that performs this reconstruction and a proof-of-concept implementation

https://hal.archives-ouvertes.fr/hal-03426711/document

Matthew Lutze

Papers

On Type-Cases, Union Elimination, and Occurrence Typing

Restrictable Variants: A Simple and Practical Alternative to Extensible Variants (Artifact)