On the Confluence of the Graphic Calculus with Penrose Diagrams (I)

  • J. L. Freire Nistal
  • A. Blanco Ferro
  • J. M. Molinelli Barba
  • E. Freire Brañas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6927)

Abstract

In paper Molinelli et al, 1998 a general model allowing the integration of different kinds of calculus with diagrams appearing in several fields of Science and Engineering was introduced. And also, a computer aided system enabling some manipulation of this graphical stuff was presented.

Traditionally most of these diagrams have been used as an aid in the development of complex calculus, although the lack of a solid theoretical foundation has prevent the existence of practical tools.

As a contribution to that necessary background, we present here an implementation of the diagrams using Coq and a first discussion on the confluence of the rewriting based on the interchange law.

Keywords

Monoidal Category Penrose Diagram Symmetric Monoidal Category Solid Theoretical Foundation Interactive Theorem Prove 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Bonante & Guiraud, 2009]
    Bonfante, G., Guiraud, Y.: Polygraphic programs and polynomial–time functions. LMCS 5 (2009)Google Scholar
  2. [Bertot & Casteran, 2004]
    Bertot, I., Casteran, P.: Interactive Theorem Proving and Program Development. Springer, Heidelberg (2004)CrossRefMATHGoogle Scholar
  3. [Coecke, 2010]
    Coecke, R.: Course lecture notes HT. Oxford Computer Laboratory (2010)Google Scholar
  4. [Coquand, 1989]
    Coquand, T.: An Introduction to Type Theory. Notes of the FPCL summer school, Glasgow (1989)Google Scholar
  5. [Dixon & Duncan]
    Dixon, L., Duncan, R.: Graphical Reasoning in Compact Closed Categories for Quantum Computation. Journal Annals of Mathematics and Artificial Intelligence 56(1) (2009)Google Scholar
  6. [Coquand & Pauling–Mohring, 1990]
    Coquand, T., Pauling–Mohring, C.: Inductively defined types. In: Martin-Löf, P., Mints, G. (eds.) COLOG 1988. LNCS, vol. 417, pp. 50–66. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  7. [Guiraud & Malbos, 2009]
    Guiraud, Y., Malbos, P.: Higher-dimensional categories with finite derivation type. Theory and Applications of Categories 22(18) (2009)Google Scholar
  8. [Lafont, 2010]
    Lafont, I.: Diagram Rewriting and Operads (2010), http://iml.univ-mrs.fr/~lafont/recherche.html
  9. [Mimram, 2009]
    Mimram, S.: Computing Critical Pairs in 3-Polygraphs. In: CEA. CAM-CAD Workshop (2009)Google Scholar
  10. [Pauling–Mohring, 1993]
    Pauling-Mohring, C.: Inductive Definitions in the System Coq–Rules and Properties. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 328–345. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  11. [Molinelli et al, 1998]
    Molinelli, J.M., Barja, J.M., Blanco, A., Freire, J.L.: An Automatic Calculator with Penrose Diagrams. In: Moreno-Díaz, R., Pichler, F. (eds.) EUROCAST 1997. LNCS, vol. 1333, pp. 252–269. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  12. [Pfenning & Pauling–Mohring, 1990]
    Pfenning, F., Pauling–Mohring, C.: Inductively defined types in the Calculus of Constructions. In: Schmidt, D.A., Main, M.G., Melton, A.C., Mislove, M.W. (eds.) MFPS 1989. LNCS, vol. 442, pp. 209–228. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  13. [Simpson, 2010]
  14. [Tabareau, 2010]
    Tabareau, N.: Aspect Oriented Programming: a language for 2-categories (2010), http://hal.archives-ouvertes.fr/
  15. [Werner & Paulin–Mohring, 1995]
    Werner, B., Paulin–Mohrin, C.: ENSTA Course Notes (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • J. L. Freire Nistal
    • 1
  • A. Blanco Ferro
    • 1
  • J. M. Molinelli Barba
    • 1
  • E. Freire Brañas
    • 1
  1. 1.Facultad de InformáticaLa CoruñaSpain

Personalised recommendations