On the Confluence of the Graphic Calculus with Penrose Diagrams (I)
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.
KeywordsMonoidal Category Penrose Diagram Symmetric Monoidal Category Solid Theoretical Foundation Interactive Theorem Prove
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- [Bonante & Guiraud, 2009]Bonfante, G., Guiraud, Y.: Polygraphic programs and polynomial–time functions. LMCS 5 (2009)Google Scholar
- [Bertot & Casteran, 2004]
- [Coecke, 2010]Coecke, R.: Course lecture notes HT. Oxford Computer Laboratory (2010)Google Scholar
- [Coquand, 1989]Coquand, T.: An Introduction to Type Theory. Notes of the FPCL summer school, Glasgow (1989)Google Scholar
- [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
- [Coquand & Pauling–Mohring, 1990]
- [Guiraud & Malbos, 2009]Guiraud, Y., Malbos, P.: Higher-dimensional categories with finite derivation type. Theory and Applications of Categories 22(18) (2009)Google Scholar
- [Lafont, 2010]Lafont, I.: Diagram Rewriting and Operads (2010), http://iml.univ-mrs.fr/~lafont/recherche.html
- [Mimram, 2009]Mimram, S.: Computing Critical Pairs in 3-Polygraphs. In: CEA. CAM-CAD Workshop (2009)Google Scholar
- [Pauling–Mohring, 1993]
- [Molinelli et al, 1998]
- [Pfenning & Pauling–Mohring, 1990]
- [Simpson, 2010]Simpson, C.: http://ncatlab.org/nlab/show/strict+2-category
- [Tabareau, 2010]Tabareau, N.: Aspect Oriented Programming: a language for 2-categories (2010), http://hal.archives-ouvertes.fr/
- [Werner & Paulin–Mohring, 1995]Werner, B., Paulin–Mohrin, C.: ENSTA Course Notes (1995)Google Scholar