Abstract syntax with variable binding is known to be characterised as an initial algebra in a presheaf category. This paper extends it to the case of polymorphic typed abstract syntax with binding. We consider two variations, second-order and higher-order polymorphic syntax. The central idea is to apply Fiore’s initial algebra characterisation of typed abstract syntax with binding repeatedly, i.e. first to the type structure and secondly to the term structure of polymorphic system. In this process, we use the Grothendieck construction to combine differently staged categories of polymorphic contexts.


  1. [Acz78]
    Aczel, P.: A general Church-Rosser theorem. Technical report, University of Manchester (1978)Google Scholar
  2. [AHS96]
    Altenkirch, T., Hofmann, M., Streicher, T.: Reduction-free normalisation for a polymorphic system. In: Proc. of LICS 1996, pp. 98–106 (1996)Google Scholar
  3. [dB72]
    de Bruijn, N.: Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Indagationes Mathematicae 34, 381–391 (1972)MathSciNetCrossRefGoogle Scholar
  4. [FH10]
    Fiore, M., Hur, C.-K.: Second-order equational logic. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 320–335. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. [Fio02]
    Fiore, M.: Semantic analysis of normalisation by evaluation for typed lambda calculus. In: Proc. of PPDP 2002, pp. 26–37. ACM Press, New York (2002)Google Scholar
  6. [Fio08]
    Fiore, M.: Second-order and dependently-sorted abstract syntax. In: Proc. of LICS 2008, pp. 57–68 (2008)Google Scholar
  7. [Fio09]
    Fiore, M.: Algebraic meta-theories and synthesis of equational logics (2009), Research ProgrammeGoogle Scholar
  8. [FPT99]
    Fiore, M., Plotkin, G., Turi, D.: Abstract syntax and variable binding. In: Proc. of LICS 1999, pp. 193–202 (1999)Google Scholar
  9. [Gro70]
    Grothendieck, A.: Catégories fibrées et descente (exposé VI). In: Grothendieck, A. (ed.) Revêtement Etales et Groupe Fondamental (SGA1). Lecture Notes in Mathematics, vol. 224, pp. 145–194. Springer, Heidelberg (1970)Google Scholar
  10. [GTW76]
    Goguen, J., Thatcher, J., Wagner, E.: An initial algebra approach to the specification, correctness and implementation of abstract data types. Technical Report RC 6487, IBM T. J. Watson Research Center (1976)Google Scholar
  11. [GUH06]
    Ghani, N., Uustalu, T., Hamana, M.: Explicit substitutions and higher-order syntax. Higher-Order and Symbolic Computation 19(2/3), 263–282 (2006)CrossRefGoogle Scholar
  12. [Ham04]
    Hamana, M.: Free Σ-monoids: A higher-order syntax with metavariables. In: Chin, W.-N. (ed.) APLAS 2004. LNCS, vol. 3302, pp. 348–363. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. [Ham05]
    Hamana, M.: Universal algebra for termination of higher-order rewriting. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 135–149. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. [Ham07]
    Hamana, M.: Higher-order semantic labelling for inductive datatype systems. In: Proc. of PPDP 2007, pp. 97–108. ACM Press, New York (2007)Google Scholar
  15. [Ham10]
    Hamana, M.: Initial algebra semantics for cyclic sharing tree structures. Logical Methods in Computer Science 6(3) (2010)Google Scholar
  16. [Hof99]
    Hofmann, M.: Semantical analysis of higher-order abstract syntax. In: Proc. of LICS 1999, pp. 204–213 (1999)Google Scholar
  17. [Kat04]
    Katsumata, S.: A generalisation of pre-logical predicates to simply typed formal systems. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 831–845. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. [MA09]
    Morris, P., Altenkirch, T.: Indexed containers. In: LICS 2009, pp. 277–285 (2009)Google Scholar
  19. [Mic08]
    Miculan, M.: A categorical model of the Fusion calculus. In: Proc. of MFPS XXIV. ENTCS, vol. 218, pp. 275–293. Elsevier, Amsterdam (2008)Google Scholar
  20. [MS03]
    Miculan, M., Scagnetto, I.: A framework for typed HOAS and semantics. In: Proc. of PPDP 2003, pp. 184–194. ACM Press, New York (2003)Google Scholar
  21. [SP82]
    Smyth, M.B., Plotkin, G.D.: The category-theoretic solution of recursive domain equations. SIAM J. Comput 11(4), 763–783 (1982)MathSciNetCrossRefGoogle Scholar
  22. [TP08]
    Tanaka, M., Power, J.: Category theoretic semantics for typed binding signatures with recursion. Fundam. Inform. 84(2), 221–240 (2008)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Makoto Hamana
    • 1
  1. 1.Department of Computer ScienceGunma UniversityJapan

Personalised recommendations