Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
Book cover

International Conference on Foundations of Software Science and Computational Structures

FoSSaCS 2012: Foundations of Software Science and Computational Structures pp 42–57Cite as

  1. Home
  2. Foundations of Software Science and Computational Structures
  3. Conference paper
Fibrational Induction Meets Effects

Fibrational Induction Meets Effects

  • Robert Atkey18,
  • Neil Ghani18,
  • Bart Jacobs17 &
  • …
  • Patricia Johann18 
  • Conference paper
  • 872 Accesses

  • 4 Citations

  • 1 Altmetric

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7213)

Abstract

This paper provides several induction rules that can be used to prove properties of effectful data types. Our results are semantic in nature and build upon Hermida and Jacobs’ fibrational formulation of induction for polynomial data types and its extension to all inductive data types by Ghani, Johann, and Fumex. An effectful data type μ(TF) is built from a functor F that describes data, and a monad T that computes effects. Our main contribution is to derive induction rules that are generic over all functors F and monads T such that μ(TF) exists. Along the way, we also derive a principle of definition by structural recursion for effectful data types that is similarly generic. Our induction rule is also generic over the kinds of properties to be proved: like the work on which we build, we work in a general fibrational setting and so can accommodate very general notions of properties, rather than just those of particular syntactic forms. We give examples exploiting the generality of our results, and show how our results specialize to those in the literature, particularly those of Filinski and Støvring.

Download conference paper PDF

References

  1. Atkey, R., Johann, P., Ghani, N.: When is a Type Refinement an Inductive Type? In: Hofmann, M. (ed.) FOSSACS 2011. LNCS, vol. 6604, pp. 72–87. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  2. Burstall, R.: Proving Properties of Programs by Structural Induction. Computer Journal 12(1), 41–48 (1969)

    MATH  Google Scholar 

  3. Crole, R., Pitts, A.: New Foundations for Fixpoint Computations: FIX-Hyperdoctrines and the FIX-Logic. Information and Computation 98(2), 171–210 (1992)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Fokkinga, M.: Monadic Maps and Folds for Arbitrary Datatypes. Technical Report, University of Twente (1994)

    Google Scholar 

  5. Filinski, A., Støvring, K.: Inductive Reasoning About Effectful Data Types. In: Proc. International Conference on Functional Programming, pp. 97–110 (2007)

    Google Scholar 

  6. Gill, A., Hutton, G.: The worker/wrapper Transformation. Journal of Functional Programming 19(2), 227–251 (2009)

    CrossRef  MATH  Google Scholar 

  7. Ghani, N., Johann, P., Fumex, C.: Fibrational Induction Rules for Initial Algebras. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 336–350. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  8. Ghani, N., Johann, P., Fumex, C.: Generic Fibrational Induction (2011) (submitted)

    Google Scholar 

  9. Hermida, C., Jacobs, B.: Structural Induction and Coinduction in a Fibrational Setting. Information and Computation 145, 107–152 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141. Elsevier (1999)

    Google Scholar 

  11. Jacobs, B.: Comprehension Categories and the Semantics of Type Dependency. Theoretical Computer Science 107, 169–207 (1993)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Lehmann, D., Smyth, M.: Algebraic Specification of Data Types: A Synthetic Approach. Theory of Computing Systems 14(1), 97–139 (1981)

    MathSciNet  MATH  Google Scholar 

  13. Moggi, E.: Computational Lambda-Calculus and Monads. In: Proc. Logic in Computer Science, pp. 14–23 (1989)

    Google Scholar 

  14. Pardo, A.: Combining Datatypes and Effects. In: Vene, V., Yu, H.-J. (eds.) AFP 2004. LNCS, vol. 3622, pp. 171–209. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Radboud University, The Netherlands

    Bart Jacobs

  2. University of Strathclyde, Scotland

    Robert Atkey, Neil Ghani & Patricia Johann

Authors
  1. Robert Atkey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Neil Ghani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bart Jacobs
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Patricia Johann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. IT University of Copenhagen, Rued Langgaards Vej 7, 2300, Copenhagen, Denmark

    Lars Birkedal

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Atkey, R., Ghani, N., Jacobs, B., Johann, P. (2012). Fibrational Induction Meets Effects. In: Birkedal, L. (eds) Foundations of Software Science and Computational Structures. FoSSaCS 2012. Lecture Notes in Computer Science, vol 7213. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28729-9_3

Download citation

  • .RIS
  • .ENW
  • .BIB
  • DOI: https://doi.org/10.1007/978-3-642-28729-9_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-28728-2

  • Online ISBN: 978-3-642-28729-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Share this paper

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature