I'm going to put on my math hat for a second, because this blog post was a bit confusing for me at first. I had to read through the post a few times to understa

I'm going to put on my math hat for a second, because this blog post was a bit confusing for me at first. I had to read through the post a few times to understand what the "homophony group" was. For this comment, I don't want to dive into this post from the perspective of someone who's never studied abstract algebra before, so what I'm about to write won't make much sense to anyone who hasn't. If you know a little bit of group theory, though, you should be able to follow along. In math-ese, the group under consideration is <a,b,c,...,z | knight=night, ad=add, arc=ark, ...> Defining a group this way is called a "group presentation". The symbols to the left of the | are called "generators" and the symbols to the right are called "relations." For example, one presentation of the integers modulo 4 with mod-4 addition is <x | x+x+x+x = 1> See http://en.wikipedia.org/wiki/Presentation_of_a_group http://en.wikipedia.org/wiki/Presentation_of_a_group for the gory details. Anyhow, the blog post confused me at first because the "homophony group" is a kind of mathematical double entendre. In algebra one often omits the group operation explicitly, writing "xy" instead of "x.y," where "." is the group operation. This implicit understanding is what makes the "joke" work. So, on the one hand, we write down the relation "ad=add" in our presentation because ad and add are homophones in English. On the other hand, in group-land, we mean a.d = a.d.d, where (again) "." is the group operation. In group-land, however, there's no sense that "a" is anything special. The 26 symbols "a" to "z" are arbitrary and we could easily write, say, <σ,b,c,...,z | knight=night, σd=σdd, ...> or use any other 26 distinct symbols for "a" to "z." The generators tell us what symbols we have at our disposal and the relations tell us how we can reduce combinations of those symbols to the identity element. Math jokes. Oh buddy. For the CS folks among us, there's a computational problem called the "group isomorphism problem" which asks, "Given two group presentations, are they isomorphic?" This problem is provably undecidable: http://en.wikipedia.org/wiki/Group_isomorphism_problem http://en.wikipedia.org/wiki/Group_isomorphism_problem This means you can't write a single algorithm which takes two arbitrary group presentations as inputs and correctly determines whether the groups as presented are isomorphic.