Skip to main content

Modal Type Theory Based on the Intuitionistic Modal Logic \(\mathbf{IEL}^{-}\)

  • Conference paper
  • First Online:
Logical Foundations of Computer Science (LFCS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11972))

Included in the following conference series:


The modal intuitionistic epistemic logic \(\mathbf{IEL}^{-}\) was proposed by Artemov and Protopopescu as the intuitionistic version of belief logic. We construct the modal lambda calculus which is Curry-Howard isomorphic to \(\mathbf{IEL}^{-}\) as the type-theoretical representation of applicative computation widely known in functional programming.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others


  1. 1.

    In Haskell, type class is a general interface for some special group of datatypes.


  1. Artemov, S., Protopopescu, T.: Intuitionistic epistemic logic. Rev. Symb. Log. 9(2), 266–298 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  2. Benton, P.N., Bierman, G.M., de Paiva, V.: Computational types from a logical perspective. J. Funct. Program. 8(2), 177–193 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  3. Goldblatt, R.I.: Grothendieck topology as geometric modality. Math. Log. Q. 27(3135), 495–529 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  4. Kakutani, Y.: Call-by-name and call-by-value in normal modal logic. In: Shao, Z. (ed.) APLAS 2007. LNCS, vol. 4807, pp. 399–414. Springer, Heidelberg (2007).

    Chapter  MATH  Google Scholar 

  5. Kavvos, G.A.: The many worlds of modal \(\lambda \)-calculi: I. curry-howard for necessity, possibility and time. CoRR abs/1605.08106 (2016).

  6. Krishnaswami, N.: A computational lambda calculus for applicative functors.

  7. Krupski, V.N., Yatmanov, A.: Sequent calculus for intuitionistic epistemic logic IEL. In: Artemov, S., Nerode, A. (eds.) LFCS 2016. LNCS, vol. 9537, pp. 187–201. Springer, Cham (2016).

    Chapter  Google Scholar 

  8. McBride, C., Paterson, R.: Applicative programming with effects. J. Funct. Program. 18(1), 1–13 (2008).

    Article  MATH  Google Scholar 

  9. Moggi, E.: Notions of computation and monads. Inform. Comput. 93(1), 55–92 (1991)., selections from 1989 IEEE Symposium on Logic in Computer Science

    Article  MathSciNet  MATH  Google Scholar 

  10. Nederpelt, R., Geuvers, H.: Type Theory and Formal Proof: An Introduction, 1st edn. Cambridge University Press, New York (2014)

    Book  Google Scholar 

  11. Pfenning, F., Davies, R.: A judgmental reconstruction of modal logic. Math. Struct. Comput. Sci. 11(4), 511–540 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  12. Sorensen, M.H., Urzyczyn, P.: Lectures on the Curry-Howard Isomorphism. Elsevier Science, Amsterdam (2006)

    MATH  Google Scholar 

Download references


The author is grateful to Neel Krishnaswami, Vladimir Krupski, Valerii Plisko, and Vladimir Vasyukov for consulting and advice.

The author thanks anonymous peer-reviewers for valuable and considerable comments and remarks that improved the paper significantly.

The research described in this paper was supported by Russian Foundation for Basic Research (grant 16-03-00364).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Daniel Rogozin .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Rogozin, D. (2020). Modal Type Theory Based on the Intuitionistic Modal Logic \(\mathbf{IEL}^{-}\). In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2020. Lecture Notes in Computer Science(), vol 11972. Springer, Cham.

Download citation

Publish with us

Policies and ethics