A Coalgebraic View of Bar Recursion and Bar Induction

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9634)

Abstract

We reformulate the bar recursion and induction principles in terms of recursive and wellfounded coalgebras. Bar induction was originally proposed by Brouwer as an axiom to recover certain classically valid theorems in a constructive setting. It is a form of induction on non-wellfounded trees satisfying certain properties. Bar recursion, introduced later by Spector, is the corresponding function definition principle.

We give a generalization of these principles, by introducing the notion of barred coalgebra: a process with a branching behaviour given by a functor, such that all possible computations terminate.

Coalgebraic bar recursion is the statement that every barred coalgebra is recursive; a recursive coalgebra is one that allows definition of functions by a coalgebra-to-algebra morphism. It is a framework to characterize valid forms of recursion for terminating functional programs. One application of the principle is the tabulation of continuous functions: Ghani, Hancock and Pattinson defined a type of wellfounded trees that represent continuous functions on streams. Bar recursion allows us to prove that every stably continuous function can be tabulated to such a tree, where by stability we mean that the modulus of continuity is its own modulus. Coalgebraic bar induction states that every barred coalgebra is wellfounded; a wellfounded coalgebra is one that admits proof by induction.

References

  1. 1.
    Abbott, M., Altenkirch, T., Ghani, N.: Containers: constructing strictly positive types. Theoret. Comput. Sci. 342(1), 3–27 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Adámek, J., Lücke, D., Milius, S.: Recursive coalgebras of finitary functors. Theoret. Inform. Appl. 41(4), 447–462 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Adámek, J., Milius, S., Moss, L.S., Sousa, L.: Well-pointed coalgebras. Log. Methods Comput. Sci. 9(3), article 2 (2013)Google Scholar
  4. 4.
    Barendregt, H.P., Dekkers, W., Statman, R.: Lambda Calculus with Types: Perspectives in Logic. Cambridge University Press, Cambridge (2013)CrossRefMATHGoogle Scholar
  5. 5.
    Berardi, S., Bezem, M., Coquand, T.: On the computational content of the axiom of choice. J. Symb. Log. 63(2), 600–622 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Berger, U., Oliva, P.: Modified bar recursion. Math. Struct. Comput. Sci. 16(2), 163–183 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Capretta, V., Uustalu, T., Vene, V.: Recursive coalgebras from comonads. Inf. Comput. 204(4), 437–468 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Capretta, V., Uustalu, T., Vene, V.: Corecursive algebras: a study of general structured corecursion. In: Oliveira, M.V.M., Woodcock, J. (eds.) SBMF 2009. LNCS, vol. 5902, pp. 84–100. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Dummett, M.: Elements of Intuitionism, 2nd edn. Oxford Science Publications, Oxford (2000)MATHGoogle Scholar
  10. 10.
    Escardó, M.H., Oliva, P.: Selection functions, bar recursion and backward induction. Math. Struct. Comput. Sci. 20(2), 127–168 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Escardó, M.H., Xu, C.: The inconsistency of a Brouwerian continuity principle with the Curry-Howard interpretation. In: Altenkirch, T. (ed.) 13th International Conference on Typed Lambda Calculi and Applications, TLCA 2015, Leibniz International Proceedings in Informatics, vol. 38, pp. 153–164. Dagstuhl Publishing, Saarbrücken (2015)Google Scholar
  12. 12.
    Gambino, N., Hyland, M.: Wellfounded trees and dependent polynomial functors. In: Berardi, S., Coppo, M., Damiani, F. (eds.) TYPES 2003. LNCS, vol. 3085, pp. 210–225. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Ghani, N., Hancock, P., Pattinson, D.: Continuous functions on final coalgebras. Electron. Notes Theoret. Comput. Sci. 164(1), 141–155 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hancock, P., Pattinson, D., Ghani, N.: Representations of stream processors using nested fixed points. Log. Methods Comput. Sci. 5(3), article 9 (2009)Google Scholar
  15. 15.
    Jacobs, B.: The temporal logic of coalgebras via Galois algebras. Math. Struct. Comput. Sci. 12(6), 875–903 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kleene, S., Vesley, R.: The Foundations of Intuitionistic Mathematics: Especially in Relation to Recursive Functions. North-Holland, Amsterdam (1965)MATHGoogle Scholar
  17. 17.
    Nakata, K., Uustalu, T., Bezem, M.: A proof pearl with the fan theorem and bar induction. In: Yang, H. (ed.) APLAS 2011. LNCS, vol. 7078, pp. 353–368. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Osius, G.: Categorical set theory: a characterization of the category of sets. J. Pure Appl. Algebra 4(1), 79–119 (1974)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Spector, C.: Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics. In: Dekker, F.D.E. (ed.) Recursive Function Theory: Proceedings of Symposia in Pure Mathematics. vol. 5, pp. 1–27. American Mathematical Society, Providence, RI (1962)Google Scholar
  20. 20.
    Taylor, P.: Intuitionistic sets and ordinals. J. Symb. Logic 61(3), 705–744 (1996)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Taylor, P.: Towards a unified treatment of induction, I: The general recursion theorem. Manuscript (1996)Google Scholar
  22. 22.
    Taylor, P.: Practical Foundations of Mathematics. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  23. 23.
    Troelstra, A.: Choice Sequences. Clarendon Press, Oxford (1977)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of NottinghamNottinghamUK
  2. 2.Institute of CyberneticsTallinn University of TechnologyTallinnEstonia

Personalised recommendations