Stone Duality for Nominal Boolean Algebras with И

  • Murdoch J. Gabbay
  • Tadeusz Litak
  • Daniela Petrişan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6859)

Abstract

We define Boolean algebras over nominal sets with a function symbol И mirroring the И ‘fresh name’ quantifier (Banonas), and dual notions of nominal topology and Stone space. We prove a representation theorem over fields of nominal sets, and extend this to a Stone duality.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramsky, S., Ghica, D.R., Murawski, A.S., Luke Ong, C.-H., Stark, I.D.B.: Nominal games and full abstraction for the nu-calculus. In: Proceedings of the 19th IEEE Symposium on Logic in Computer Science, LICS 2004, pp. 150–159. IEEE Computer Society Press, Los Alamitos (2004)CrossRefGoogle Scholar
  2. 2.
    Areces, C., ten Cate, B.: Hybrid logics. In: Blackburn, P., van Benthem, J., Wolter, F. (eds.) Handbook of Modal Logic. Elsevier, Amsterdam (2007)Google Scholar
  3. 3.
    Bengtson, J., Parrow, J.: Formalising the π-Calculus Using Nominal Logic. In: Seidl, H. (ed.) FOSSACS 2007. LNCS, vol. 4423, pp. 63–77. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Bonsangue, M., Kurz, A.: Pi-calculus in logical form. In: Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science, LICS 2007, pp. 303–312. IEEE Computer Society Press, Los Alamitos (2007)CrossRefGoogle Scholar
  5. 5.
    Burris, S., Sankappanavar, H.: A Course in Universal Algebra. Graduate Texts in Mathematics. Springer, Heidelberg (1981)Google Scholar
  6. 6.
    Caires, L., Cardelli, L.: A spatial logic for concurrency (part I). Information and Computation 186(2), 194–235 (2003)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Cardelli, L., Gordon, A.: Logical Properties of Name Restriction. In: Abramsky, S. (ed.) TLCA 2001. LNCS, vol. 2044, pp. 46–60. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Cheney, J.: A simpler proof theory for nominal logic. In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 379–394. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Cheney, J., Urban, C.: Nominal logic programming. ACM Transactions on Programming Languages and Systems (TOPLAS) 30(5), 1–47 (2008)CrossRefGoogle Scholar
  10. 10.
    Cîrstea, C., Kurz, A., Pattinson, D., Schröder, L., Venema, Y.: Modal logics are coalgebraic. The Computer Journal (2009)Google Scholar
  11. 11.
    Dowek, G., Gabbay, M.J.: Permissive Nominal Logic. In: Proceedings of the 12th International ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming, PPDP 2010, pp. 165–176 (2010)Google Scholar
  12. 12.
    Fernández, M., Gabbay, M.J.: Nominal rewriting with name generation: abstraction vs. locality. In: Proceedings of the 7th ACM SIGPLAN International Symposium on Principles and Practice of Declarative Programming, PPDP 2005, pp. 47–58. ACM Press, New York (2005)Google Scholar
  13. 13.
    Gabbay, M.J.: Fresh Logic. Journal of Applied Logic 5(2), 356–387 (2007)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Gabbay, M.J.: Nominal Algebra and the HSP Theorem. Journal of Logic and Computation 19(2), 341–367 (2009)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Gabbay, M.J.: A study of substitution, using nominal techniques and Fraenkel-Mostowski sets. Theoretical Computer Science 410(12-13), 1159–1189 (2009)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Gabbay, M.J.: Foundations of nominal techniques: logic and semantics of variables in abstract syntax. Bulletin of Symbolic Logic (2011) (in press)Google Scholar
  17. 17.
    Gabbay, M.J., Cheney, J.: A Sequent Calculus for Nominal Logic. In: Proceedings of the 19th IEEE Symposium on Logic in Computer Science, LICS 2004, pp. 139–148. IEEE Computer Society, Los Alamitos (2004)CrossRefGoogle Scholar
  18. 18.
    Gabbay, M.J., Ciancia, V.: Freshness and Name-Restriction in Sets of Traces with Names. In: Hofmann, M. (ed.) FOSSACS 2011. LNCS, vol. 6604, pp. 365–380. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  19. 19.
    Gabbay, M.J., Mathijssen, A.: Nominal universal algebra: equational logic with names and binding. Journal of Logic and Computation 19(6), 1455–1508 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Gabbay, M.J., Pitts, A.M.: A New Approach to Abstract Syntax with Variable Binding. Formal Aspects of Computing 13(3-5), 341–363 (2001)CrossRefGoogle Scholar
  21. 21.
    Keenan, E., Westerståhl, D.: Generalized quantifiers in linguistics and logic. In: Van Benthem, J., Ter Meulen, A. (eds.) Handbook of Logic and Language, pp. 837–894. Elsevier, Amsterdam (1996)Google Scholar
  22. 22.
    Kurz, A., Petrişan, D.: On universal algebra over nominal sets. Mathematical Structures in Computer Science 20, 285–318 (2010)MATHCrossRefGoogle Scholar
  23. 23.
    Litak, T.: Algebraization of Hybrid Logic with Binders. In: Schmidt, R. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 281–295. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  24. 24.
    Lane, S.M.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5. Springer, Heidelberg (1971)MATHGoogle Scholar
  25. 25.
    Manzonetto, G., Salibra, A.: Applying universal algebra to lambda calculus. Journal of Logic and Computation 20(4), 877–915 (2010)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    McCune, W.: Solution of the Robbins problem. Journal of Automated Reasoning 19, 263–276 (1997)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Menni, M.: About И-quantifiers. Applied Categorical Structures 11(5), 421–445 (2003)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Pitts, A.M.: Nominal system T. In: Proceedings of the 37th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, POPL 2010, pp. 159–170. ACM Press, New York (2010)Google Scholar
  29. 29.
    Pitts, A.M.: Structural recursion with locally scoped names (September 2010) (submitted for publication)Google Scholar
  30. 30.
    Reed, J.: Hybridizing a logical framework. Electronic Notes in Theoretical Computer Science 174(6), 135–148 (2006); Proceedings of the International Workshop on Hybrid Logic (HyLo 2006)CrossRefGoogle Scholar
  31. 31.
    Shinwell, M.R., Pitts, A.M.: On a monadic semantics for freshness. Theoretical Computer Science 342(1), 28–55 (2005)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Shinwell, M.R., Pitts, A.M., Gabbay, M.J.: FreshML: Programming with Binders Made Simple. In: Proceedings of the 8th ACM SIGPLAN International Conference on Functional Programming, ICFP 2003, vol. 38, pp. 263–274. ACM Press, New York (2003)CrossRefGoogle Scholar
  33. 33.
    Staton, S.: Name-passing process calculi: operational models and structural operational semantics. Technical Report UCAM-CL-TR-688, University of Cambridge, Computer Laboratory (June 2007)Google Scholar
  34. 34.
    Tiu, A.: A logic for reasoning about generic judgments. Electronic Notes in Theoretical Computer Science 174(5), 3–18 (2007)CrossRefGoogle Scholar
  35. 35.
    Turner, D.C.: Nominal Domain Theory for Concurrency. PhD thesis, University of Cambridge (2009)Google Scholar
  36. 36.
    Tzevelekos, N.: Full abstraction for nominal general references. In: Proceedings of the 22nd IEEE Symposium on Logic in Computer Science, LICS 2007, pp. 399–410. IEEE Computer Society Press, Los Alamitos (2007)CrossRefGoogle Scholar
  37. 37.
    Venema, Y.: Algebras and coalgebras. In: Blackburn, P., Van Benthem, J., Wolter, P. (eds.) Handbook of Modal Logic. Studies in logic and practical reasoning, ch. 6, vol. 3. Elsevier, Amsterdam (2007)CrossRefGoogle Scholar
  38. 38.
    Westerståhl, D.: Quantifiers in formal and natural languages. In: Handbook of Philosophical Logic. Synthèse, ch. 2, vol. 4, pp. 1–131. Reidel, Dordrechtz (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Murdoch J. Gabbay
  • Tadeusz Litak
  • Daniela Petrişan

There are no affiliations available

Personalised recommendations