research-article

Positively dependent types

Authors:

Daniel R. Licata,

Robert HarperAuthors Info & Claims

PLPV '09: Proceedings of the 3rd workshop on Programming languages meets program verification

Pages 3 - 14

https://doi.org/10.1145/1481848.1481851

Published: 20 January 2009 Publication History

Abstract

This paper is part of a line of work on using the logical techniques of polarity and focusing to design a dependent programming language, with particular emphasis on programming with deductive systems such as programming languages and proof theories. Polarity emphasizes the distinction between positive types, which classify data, and negative types, which classify computation. In previous work, we showed how to use Zeilberger's higher-order formulation of focusing to integrate a positive function space for representing variable binding, an essential tool for specifying logical systems, with a standard negative computational function space. However, our previous work considers only a simply-typed language. The central technical contribution of the present paper is to extend higher-order focusing with a form of dependency that we call positively dependent types: We allow dependency on positive data, but not negative computation. Additionally, we present the syntax of dependent pair and function types using an iterated inductive definition, mapping positive data to types, which gives an account of type-level computation. We construct our language inside the dependently typed programming language Agda 2, making essential use of coinductive types and induction-recursion.

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Keywords: LF, refinement types, subtyping, dependent types, intersection types

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Index Terms

Positively dependent types
1. Theory of computation
  1. Semantics and reasoning
    1. Program constructs
      1. Type structures
    2. Program reasoning
      1. Program specifications

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cover image ACM Conferences

PLPV '09: Proceedings of the 3rd workshop on Programming languages meets program verification

January 2009

90 pages

ISBN:9781605583303

DOI:10.1145/1481848

Program Chairs:
Thorsten Altenkirch
University of Nottingham, UK
,
Todd Millstein
University of California, Los Angeles, USA

Copyright © 2009 ACM.

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Published: 20 January 2009

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POPL09: The 36th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

January 20, 2009

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https://dl.acm.org/doi/10.1145/3139398
Kokologiannakis MLahav OSagonas KVafeiadis V(2017)Effective stateless model checking for C/C++ concurrencyProceedings of the ACM on Programming Languages10.1145/31581052:POPL(1-32)Online publication date: 27-Dec-2017
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Bach Poulsen CRouvoet ATolmach AKrebbers RVisser E(2017)Intrinsically-typed definitional interpreters for imperative languagesProceedings of the ACM on Programming Languages10.1145/31581042:POPL(1-34)Online publication date: 27-Dec-2017
https://dl.acm.org/doi/10.1145/3158104
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Casinghino CSjöberg VWeirich S(2014)Combining proofs and programs in a dependently typed languageACM SIGPLAN Notices10.1145/2578855.253588349:1(33-45)Online publication date: 8-Jan-2014
https://dl.acm.org/doi/10.1145/2578855.2535883
Casinghino CSjöberg VWeirich SJagannathan SSewell P(2014)Combining proofs and programs in a dependently typed languageProceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages10.1145/2535838.2535883(33-45)Online publication date: 11-Jan-2014
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