Nominal techniques are based on the idea of sets with a finitely-supported atoms-permutation action.

We consider the idea of nominal renaming sets, which are sets with a finitely-supported atoms-renaming action; renamings can identify atoms, permutations cannot. We show that nominal renaming sets exhibit many of the useful qualities found in (permutative) nominal sets; an elementary sets-based presentation, inductive datatypes of syntax up to binding, cartesian closure, and being a topos. Unlike is the case for nominal sets, the notion of names-abstraction coincides with functional abstraction. Thus we obtain a concrete presentation of sheaves on the category of finite sets in the form of a category of sets with structure.


Nominal renaming sets nominal sets abstract syntax with binding 


  1. 1.
    Benton, N., Leperchey, B.: Relational reasoning in a nominal semantics for storage. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 86–101. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Brunner, N.: 75 years of independence proofs by Fraenkel-Mostowski permutation models. Mathematica Japonica 43, 177–199 (1996)MathSciNetMATHGoogle Scholar
  3. 3.
    Bucalo, A., Honsell, F., Miculan, M., Scagnetto, I., Hofmann, M.: Consistency of the theory of contexts. Journal of Functional Programming 16(3), 327–395 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Despeyroux, J.: A higher-order specification of the π–calculus. In: IFIP TCS, pp. 425–439 (2000)Google Scholar
  5. 5.
    Despeyroux, J., Felty, A.P., Hirschowitz, A.: Higher-order abstract syntax in COQ. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 124–138. Springer, Heidelberg (2005)Google Scholar
  6. 6.
    Despeyroux, J., Hirschowitz, A.: Higher-order abstract syntax with induction in COQ. In: Pfenning, F. (ed.) LPAR 1994. LNCS, vol. 822, pp. 159–173. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  7. 7.
    Fiore, M.P., Plotkin, G.D., Turi, D.: Abstract syntax and variable binding. In: LICS 1999, pp. 193–202. IEEE, Los Alamitos (1999)Google Scholar
  8. 8.
    Fiore, M.P., Staton, S.: A congruence rule format for name-passing process calculi from mathematical structural operational semantics. In: LICS 2006, pp. 49–58. IEEE, Los Alamitos (2006)Google Scholar
  9. 9.
    Gabbay, M.J.: A Theory of Inductive Definitions with alpha-Equivalence. PhD thesis, Cambridge, UK (2000)Google Scholar
  10. 10.
    Gabbay, M.J.: A General Mathematics of Names. Information and Computation 205, 982–1011 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gabbay, M.J.: Nominal renaming sets. Technical Report HW-MACS-TR-0058, Heriot-Watt University (2007), http://www.gabbay.org.uk/papers.html#nomrs-tr
  12. 12.
    Gabbay, M.J., Mathijssen, A.: Capture-avoiding Substitution as a Nominal Algebra. In: Barkaoui, K., Cavalcanti, A., Cerone, A. (eds.) ICTAC 2006. LNCS, vol. 4281, pp. 198–212. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Gabbay, M.J., Pitts, A.M.: A New Approach to Abstract Syntax with Variable Binding (journal version). Formal Aspects of Computing 13(3–5), 341–363 (2001)MATHGoogle Scholar
  14. 14.
    Gabbay, M.J., Pitts, A.M.: A new approach to abstract syntax involving binders. In: 14th Annual Symposium on Logic in Computer Science, pp. 214–224. IEEE Computer Society Press, Los Alamitos (1999)Google Scholar
  15. 15.
    Hirschkoff, D.: A full formalization of pi-calculus theory in the Calculus of Constructions. In: Gunter, E.L., Felty, A.P. (eds.) TPHOLs 1997. LNCS, vol. 1275, pp. 153–169. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  16. 16.
    Hofmann, M.: Semantical analysis of higher-order abstract syntax. In: 14th Annual Symposium on Logic in Computer Science, pp. 204–213. IEEE, Los Alamitos (1999)Google Scholar
  17. 17.
    Honsell, F., Miculan, M., Scagnetto, I.: An axiomatic approach to metareasoning on nominal algebras in HOAS. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 963–978. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    McKinna, J., Pollack, R.: Some lambda calculus and type theory formalized. Journal of Automated Reasoning 23(3-4), 373–409 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Pfenning, F., Schürmann, C.: System description: Twelf - a meta-logical framework for deductive systems. In: Ganzinger, H. (ed.) CADE 1999. LNCS, vol. 1632, pp. 202–206. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  20. 20.
    Shinwell, M.R.: The Fresh Approach: Functional Programming with Names and Binders. PhD thesis, Computer Laboratory, University of Cambridge (December 2004)Google Scholar
  21. 21.
    Shinwell, M.R., Pitts, A.M., Gabbay, M.J.: FreshML: Programming with binders made simple. In: ICFP 2003. SIGPLAN Not., vol. 38(9), pp. 263–274. ACM Press, New York (2003)Google Scholar
  22. 22.
    Shinwell, M.R., Pitts, A.M.: On a monadic semantics for freshness. Theoretical Computer Science 342(1), 28–55 (2005)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Shinwell, M.R., Pitts, A.M.: Fresh objective Caml user manual. Technical Report UCAM-CL-TR-621, University of Cambridge (2005)Google Scholar
  24. 24.
    Staton, S.: Name-passing process calculi: operational models and structural operational semantics. PhD thesis, University of Cambridge (2007)Google Scholar
  25. 25.
    Urban, C., Tasson, C.: Nominal techniques in Isabelle/HOL. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS, vol. 3632, pp. 38–53. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Murdoch J. Gabbay
    • 1
  • Martin Hofmann
    • 2
  1. 1.UK
  2. 2.Germany

Personalised recommendations