Programming and Reasoning with Guarded Recursion for Coinductive Types

  • Ranald CloustonEmail author
  • Aleš Bizjak
  • Hans Bugge Grathwohl
  • Lars Birkedal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9034)


We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type-former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program always terminates. We demonstrate the expressiveness of the calculus by showing the definability of solutions to Rutten’s behavioural differential equations. We introduce a program logic with Löb induction for reasoning about the contextual equivalence of programs.


Program Logic Operational Semantic Reduction Rule Natural Deduction Denotational Semantic 
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  1. 1.
    Abel, A., Pientka, B.: Wellfounded recursion with copatterns: A unified approach to termination and productivity. In: ICFP, pp. 185–196 (2013)Google Scholar
  2. 2.
    Abel, A., Vezzosi, A.: A formalized proof of strong normalization for guarded recursive types. In: APLAS, pp. 140–158 (2014)Google Scholar
  3. 3.
    Appel, A.W., Melliès, P.A., Richards, C.D., Vouillon, J.: A very modal model of a modern, major, general type system. In: POPL, pp. 109–122 (2007)Google Scholar
  4. 4.
    Atkey, R., McBride, C.: Productive coprogramming with guarded recursion. In: ICFP, pp. 197–208 (2013)Google Scholar
  5. 5.
    Bierman, G.M., de Paiva, V.C.: On an intuitionistic modal logic. Studia Logica 65(3), 383–416 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Birkedal, L., Møgelberg, R.E., Schwinghammer, J., Støvring, K.: First steps in synthetic guarded domain theory: step-indexing in the topos of trees. LMCS 8(4) (2012)Google Scholar
  7. 7.
    Birkedal, L., Schwinghammer, J., Støvring, K.: A metric model of lambda calculus with guarded recursion. In: FICS, pp. 19–25 (2010)Google Scholar
  8. 8.
    Bizjak, A., Birkedal, L., Miculan, M.: A model of countable nondeterminism in guarded type theory. In: Dowek, G. (ed.) RTA-TLCA 2014. LNCS, vol. 8560, pp. 108–123. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  9. 9.
    Clouston, R., Bizjak, A., Grathwohl, H.B., Birkedal, L.: Programming and reasoning with guarded recursion for coinductive types. arXiv:1501.02925 (2015)Google Scholar
  10. 10.
    Clouston, R., Goré, R.: Sequent calculus in the topos of trees. In: Pitts, A. (ed.) FoSSaCS 2015. LNCS, vol. 9034, pp. 133–147. Springer, Heidelberg (2015)Google Scholar
  11. 11.
    Coquand, T.: Infinite objects in type theory. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993. LNCS, vol. 806, pp. 62–78. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  12. 12.
    Endrullis, J., Grabmayer, C., Hendriks, D.: Mix-automatic sequences. In: Fields Workshop on Combinatorics on Words, contributed talk (2013)Google Scholar
  13. 13.
    Giménez, E.: Codifying guarded definitions with recursive schemes. In: Smith, J., Dybjer, P., Nordström, B. (eds.) TYPES 1994. LNCS, vol. 996, pp. 39–59. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  14. 14.
    Hughes, J., Pareto, L., Sabry, A.: Proving the correctness of reactive systems using sized types. In: POPL, pp. 410–423 (1996)Google Scholar
  15. 15.
    Krishnaswami, N.R., Benton, N.: Ultrametric semantics of reactive programs. In: LICS, pp. 257–266 (2011)Google Scholar
  16. 16.
    McBride, C., Paterson, R.: Applicative programming with effects. J. Funct. Programming 18(1), 1–13 (2008)CrossRefzbMATHGoogle Scholar
  17. 17.
    Milius, S., Moss, L.S., Schwencke, D.: Abstract GSOS rules and a modular treatment of recursive definitions. LMCS 9(3) (2013)Google Scholar
  18. 18.
    Møgelberg, R.E.: A type theory for productive coprogramming via guarded recursion. In: CSL-LICS (2014)Google Scholar
  19. 19.
    Nakano, H.: A modality for recursion. In: LICS, pp. 255–266 (2000)Google Scholar
  20. 20.
    Prawitz, D.: Natural Deduction: A Proof-Theoretical Study. Dover Publ. (1965)Google Scholar
  21. 21.
    Rutten, J.J.M.M.: Behavioural differential equations: A coinductive calculus of streams, automata, and power series. Theor. Comput. Sci. 308(1-3), 1–53 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Severi, P.G., de Vries, F.J.J.: Pure type systems with corecursion on streams: from finite to infinitary normalisation. In: ICFP, pp. 141–152 (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Ranald Clouston
    • 1
    Email author
  • Aleš Bizjak
    • 1
  • Hans Bugge Grathwohl
    • 1
  • Lars Birkedal
    • 1
  1. 1.Department of Computer ScienceAarhus UniversityAarhusDenmark

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