Summary: Subtyping with Singleton Types
David Aspinall
Department of Computer Science, University of Edinburgh, U.K.
e­mail: da@dcs.ed.ac.uk
Abstract. We give syntax and a PER­model semantics for a typed –­calculus with
subtypes and singleton types. The calculus may be seen as a minimal calculus of sub­
typing with a simple form of dependent types. The aim is to study singleton types
and to take a canny step towards more complex dependent subtyping systems. Single­
ton types have applications in the use of type systems for specification and program
extraction: given a program P we can form the very tight specification fPg which is
met uniquely by P . Singletons integrate abbreviational definitions into a type system:
the hypothesis x : fMg asserts x = M . The addition of singleton types is a non­
conservative extension of familiar subtyping theories. In our system, more terms are
typable and previously typable terms have more (non­dependent) types.
1 Introducing Singletons and Subtyping
Type systems for current programming languages provide only coarse distinctions amongst
data values: Real, Bool, String, etc. Constructive type theories for program specification
can provide very fine distinctions such as fx 2 Nat j Prime(x)g, but often terms contain non­
computational parts, or else type­checking is undecidable. We want to study type systems
in between where terms do not contain unnecessary codes and, ideally, type­checking is

Source: Aspinall, David - School of Informatics, University of Edinburgh

Collections: Computer Technologies and Information Sciences