## December 6, 2006

### Guiraud on Higher-Dimensional Rewrite Rules

#### Posted by John Baez

In this quarter’s last class on Classical versus Quantum Computation, I hinted that one could use 3-dimensional structures to understand the process of computation. A key figure in this line of work is Yves Guiraud, and he just put some of his papers on this topic on the arXiv:

Even if you don’t understand the math, you’ll enjoy the pictures!

Posted at December 6, 2006 1:08 AM UTC

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### Re: Guiraud on Higher-Dimensional Rewrite Rules

Thank you John for making some publicity for my work! Maybe I should explain a bit what this figure represents for people that do not want to bother reading the full paper. A preliminary remark: the correct figure is there. Since it is hardly readable without being able to turn around or make slices, the one John has chosen is better for explanations.

This figure represents a step-by-step computation on some kind of boolean circuits. For example, the one we see on the foreground computes the formula ((a and b) or (a and b)). The computation is built from axioms of the form (P implies Q), applied in some context. Here, each axiom used is pictured as a big 3D block transforming P into Q and leaving what remains unchanged. The computations are 3-dimensional because the objects on which they act live in a 2-category.

This is exactly the same idea as the last figure of this chapter of Classical versus Quantum Computation. The only difference is that I have chosen to emphasize the computations. If someone wants to see more of these 3-dimensional computations, there are some at the bottom of my research page. I would welcome any comments on all this stuff and answer any question on the subject (or at least try to!).

Posted by: Yves Guiraud on December 6, 2006 1:41 PM | Permalink | Reply to this

### Mark Chu-Carroll; Re: Guiraud on Higher-Dimensional Rewrite Rules

There was brief mention of 3-dimensional programming languages in Mark Chu-Carroll’s “Good Math, Bad Math” blog, the Pathological Programming thread. I’m sure that he would appreciate your papers, ask good questions, and publicize your work on his blog. I am not useful to be in the loop, once I make this recommendation, and John Baez seconds my motion that Mark’s blog is a useful venue, where some category Theory is introduced in a Computer Science context to a very heterogeneous audience.

Posted by: Jonathan Vos Post on December 7, 2006 8:57 PM | Permalink | Reply to this

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