Topic
Union type
About: Union type is a research topic. Over the lifetime, 170 publications have been published within this topic receiving 4178 citations.
Papers published on a yearly basis
Papers
1 May 1991
TL;DR: A type system called refinement types is described, which is an example of a new way to make this tradeoff, as well as a potentially useful system in itself.
Abstract: Programming computers is a notoriously error-prone process. It is the job of the programming language designer to make this process more reliable. One approach to this is to impose some sort of typing discipline on the programs. In doing this, the programming language designer is immediately faced with a tradeoff: if the type system is too simple, it cannot accurately express important properties of the program; if it is too expressive, then mechanically checking or inferring the types becomes impractical. This thesis describes a type system called refinement types, which is an example of a new way to make this tradeoff, as well as a potentially useful system in itself.
Refinement type inference requires programs to have types in two type systems: an expressive type inference system (intersection types with subtyping) and a relatively simple type system (basic polymorphic type inference). Refinement type inference inherits some properties from each of these: as in intersection types with subtyping, we can use the type system to do abstract interpretation; as in basic polymorphic type inference, refinement type inference is decidable (preliminary experiments suggest refinement type inference may be practical as well).
We have implemented refinement type inference for a subset of Standard ML to test these ideas. We have added new syntax, called rectype declarations, to allow the programmer to specify relevant domains for the abstract interpretation. A prototype implementation of refinement type inference can do some interesting case analysis for Standard ML programs; for example, if the programmer uses a rectype declaration to declare interest in whether a boolean expression is in conjunctive normal form (CNF), refinement type inference can efficiently prove that a function for converting boolean expressions to CNF does indeed always return a boolean expression in CNF. Rectype declarations and refinement type inference seem flexible and efficient enough to practically enforce many other useful program properties as well.
502 citations
TL;DR: A notion of parallel reduction is adopted, which is a refinement of Gross-Knuth reduction, and type theories are introduced to refine the system, among which is the theory called ?
Abstract: Type assignment systems with intersection and union types are introduced. Although the subject reduction property with respect to s-reduction does not hold for a natural deduction-like system, we manage to overcome this problem in two, different ways. The first is to adopt a notion of parallel reduction, which is a refinement of Gross-Knuth reduction. The second is to introduce type theories to refine the system, among which is the theory called ? that induces an assignment system preserving s-reduction. This type assignment system further clarifies the relation with the intersection discipline through the decomposition, first, of a disjunctive type into a set of conjunctive types and, second, of a derivation in the new type assignment system into a set of derivations in the intersection type assignment system. For this system we propose three semantics and prove soundness and completeness theorems.
193 citations
TL;DR: This work shows how to define a subtyped relation semantically in the presence of Boolean connectives, functional types and dynamic dispatch on types, without the complexity of denotational models, and how to derive a complete subtyping algorithm.
Abstract: Subtyping relations are usually defined either syntactically by a formal system or semantically by an interpretation of types into an untyped denotational model. This work shows how to define a subtyping relation semantically in the presence of Boolean connectives, functional types and dynamic dispatch on types, without the complexity of denotational models, and how to derive a complete subtyping algorithm.
172 citations
1 Sep 2000
TL;DR: A solution similar to the value restriction for polymorphism adopted in the revised definition of Standard ML is proposed, which is not tied to let-expressions and requires an additional weakening of the usual subtyping rules.
Abstract: We show that standard formulations of intersection type systems are unsound in the presence of computational effects, and propose a solution similar to the value restriction for polymorphism adopted in the revised definition of Standard ML. It differs in that it is not tied to let-expressions and requires an additional weakening of the usual subtyping rules. We also present a bi-directional type-checking algorithm for the resulting language that does not require an excessive amount of type annotations and illustrate it through some examples. We further show that the type assignment system can be extended to incorporate parametric polymorphism. Taken together, we see our system and associated type-checking algorithm as a significant step towards the introduction of intersection types into realistic programming languages. The added expressive power would allow many more properties of programs to be stated by the programmer and statically verified by a compiler.
157 citations
1 Jan 2004
TL;DR: A decidable formulation for this type assignment system based on bidirectional checking is provided, combining type synthesis and analysis following logical principles.
Abstract: In prior work we introduced a pure type assignment system that encompasses a rich set of property types, including intersections, unions, and universally and existentially quantified dependent types. This system was shown sound with respect to a call-by-value operational semantics with effects, yet is inherently undecidable.In this paper we provide a decidable formulation for this system based on bidirectional checking, combining type synthesis and analysis following logical principles. The presence of unions and existential quantification requires the additional ability to visit subterms in evaluation position before the context in which they occur, leading to a tridirectional type system. While soundness with respect to the type assignment system is immediate, completeness requires the novel concept of contextual type annotations, introducing a notion from the study of principal typings into the source program.
104 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 1 |
| 2020 | 6 |
| 2019 | 6 |
| 2018 | 3 |
| 2017 | 11 |