Document Open Access Logo

A Metatheoretic Analysis of Subtype Universes

Authors Felix Bradley , Zhaohui Luo

PDF
Thumbnail PDF

File

LIPIcs.TYPES.2022.9.pdf
  • Filesize: 0.79 MB
  • 21 pages

Document Identifiers

Author Details

Felix Bradley
  • Royal Holloway, University of London, UK
Zhaohui Luo
  • Royal Holloway, University of London, UK

Cite As Get BibTex

Felix Bradley and Zhaohui Luo. A Metatheoretic Analysis of Subtype Universes. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.TYPES.2022.9

Abstract

Subtype universes were initially introduced as an expressive mechanisation of bounded quantification extending a modern type theory. In this paper, we consider a dependent type theory equipped with coercive subtyping and a generalisation of subtype universes. We prove results regarding the metatheoretic properties of subtype universes, such as consistency and strong normalisation. We analyse the causes of undecidability in bounded quantification, and discuss how coherency impacts the metatheoretic properties of theories implementing bounded quantification. We describe the effects of certain choices of subtyping inference rules on the expressiveness of a type theory, and examine various applications in natural language semantics, programming languages, and mathematics formalisation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
Keywords
  • Type theory
  • coercive subtyping
  • subtype universes

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. David Aspinall. Subtyping with power types. In Peter G. Clote and Helmut Schwichtenberg, editors, Computer Science Logic, pages 156-171, Berlin, Heidelberg, 2000. Springer Berlin Heidelberg. Google Scholar
  2. Luca Cardelli. Structural subtyping and the notion of power type. In Proceedings of the 15th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL '88, pages 70-79, New York, NY, USA, 1988. Association for Computing Machinery. URL: https://doi.org/10.1145/73560.73566.
  3. Luca Cardelli and Peter Wegner. On understanding types, data abstraction, and polymorphism. ACM Comput. Surv., 17(4):471-523, December 1985. URL: https://doi.org/10.1145/6041.6042.
  4. Giuseppe Castagna and Benjamin C. Pierce. Decidable bounded quantification. In Proceedings of the 21st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL '94, pages 151-162, New York, NY, USA, 1994. Association for Computing Machinery. URL: https://doi.org/10.1145/174675.177844.
  5. Stergios Chatzikyriakidis and Zhaohui Luo. On the interpretation of common nouns: Types versus predicates. In Stergios Chatzikyriakidis and Zhaohui Luo, editors, Modern Perspectives in Type-Theoretical Semantics, pages 43-70. Springer Cham, 2017. URL: https://doi.org/10.1007/978-3-319-50422-3.
  6. Adriana Compagnoni. Higher-order subtyping and its decidability. Information and Computation, 191(1):41-103, 2004. URL: https://doi.org/10.1016/j.ic.2004.01.001.
  7. Thierry Coquand. An analysis of Girard’s paradox. Technical Report RR-0531, INRIA, May 1986. URL: https://hal.inria.fr/inria-00076023.
  8. Healfdene Goguen. A Typed Operational Semantics for Type Theory. PhD thesis, University of Edinburgh, 1994. Google Scholar
  9. Douglas J. Howe. The computational behaviour of girard’s paradox. Technical report, Cornell University, Ithaca, New York, USA, March 1987. Google Scholar
  10. Antonius J. C. Hurkens. A simplification of Girard’s paradox. In Mariangiola Dezani-Ciancaglini and Gordon Plotkin, editors, Typed Lambda Calculi and Applications, pages 266-278, Berlin, Heidelberg, 1995. Springer Berlin Heidelberg. Google Scholar
  11. DeLesley S. Hutchins. Pure subtype systems. In Proceedings of the 37th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL '10, pages 287-298, New York, NY, USA, 2010. Association for Computing Machinery. URL: https://doi.org/10.1145/1706299.1706334.
  12. Zhaohui Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, London, March 1994. Google Scholar
  13. Zhaohui Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105-130, February 1999. URL: https://doi.org/10.1093/logcom/9.1.105.
  14. Zhaohui Luo. Common nouns as types. In D. Béchet and A. Dikovsky, editors, Proceedings of the 7th International Conference on Logical Aspects of Computational Linguistics (LACL'12), pages 173-185, Berlin, Heidelberg, 2012. Springer-Verlag. Google Scholar
  15. Zhaohui Luo, Sergey Soloviev, and Tao Xue. Coercive subtyping: Theory and implementation. Information and Computation, 223:18-42, 2013. URL: https://doi.org/10.1016/j.ic.2012.10.020.
  16. Harry Maclean and Zhaohui Luo. Subtype Universes. In Ugo de'Liguoro, Stefano Berardi, and Thorsten Altenkirch, editors, 26th International Conference on Types for Proofs and Programs (TYPES 2020), volume 188 of Leibniz International Proceedings in Informatics (LIPIcs), pages 9:1-9:16, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.TYPES.2020.9.
  17. Benjamin C. Pierce. Bounded quantification is undecidable. In Proceedings of the 19th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL '92, pages 305-315, New York, NY, USA, 1992. Association for Computing Machinery. URL: https://doi.org/10.1145/143165.143228.
  18. Richmond H. Thomason, editor. Formal Philosophy: Selected Papers of Richard Montague. Yale University Press, New Haven, 1974. Google Scholar
  19. Kevin Watkins. Hurkens' simplification of Girard’s paradox, July 2004. URL: https://www.cs.cmu.edu/~kw/research/hurkens95tlca.elf.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact