Integrating Linear and Dependent Types

Author:

Krishnaswami Neelakantan R.1,Pradic Pierre2,Benton Nick3

Affiliation:

1. University of Birmingham, Birmingham, United Kingdom

2. ENS Lyon, Lyon, France

3. Microsoft Research, Cambridge, United Kingdom

Abstract

In this paper, we show how to integrate linear types with type dependency, by extending the linear/non-linear calculus of Benton to support type dependency. Next, we give an application of this calculus by giving a proof-theoretic account of imperative programming, which requires extending the calculus with computationally irrelevant quantification, proof irrelevance, and a monad of computations. We show the soundness of our theory by giving a realizability model in the style of Nuprl, which permits us to validate not only the beta-laws for each type, but also the eta-laws. These extensions permit us to decompose Hoare triples into a collection of simpler type-theoretic connectives, yielding a rich equational theory for dependently-typed higher-order imperative programs. Furthermore, both the type theory and its model are relatively simple, even when all of the extensions are considered.

Publisher

Association for Computing Machinery (ACM)

Subject

Computer Graphics and Computer-Aided Design,Software

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