Discussing 'Nominal Unification'

UPDATE - Due to upcoming teaching commitments and, after that, paternity leave, this blog will not be updated again in the near future.

(Link).

Authors: Christian Urban (University of Cambridge), Andrew M. Pitts (University of Cambridge), and Murdoch J. Gabbay (INRIA).

Reference: Theoretical Computer Science vol. 323(1-3), pp. 473-497 (2004)

Why this paper? Cited by 4 (of a possible 28) of the papers in my Google Scholar Updates recommendations.

Comments: In the first iteration of this blog I reviewed recent papers recommended to me to by Google Scholar Updates; one of the reasons I abandoned this is that many of these new papers were quite strongly linked to older work, and I ended up having to explain all that older work before writing about the new. Note that being strongly linked to older work is no bad thing - on the contrary, such incremental research is the unglamorous but essential heart of scientific progress. But my new focus for this blog, on 'classic' (which is to say, highly cited) papers, allows me to focus more on foundational work, which tends to be more original and so may be described with less reference to prior literature.

The reason for this discussion is that I have already discussed this week's paper at some length in a post for the first iteration of this blog. Please feel free to look up the post; the only correction I'd make is that is that a quadratic time (which is to say, feasibly fast) algorithm for nominal unification is in fact sketched in the conclusion of this paper, whereas I had previously thought that credit for this should go to Levy and Villaret alone.

Let me instead discuss something slightly orthogonal to this paper. The phrase 'nominal techniques', which I discussed for example here, actually refers to two things - syntax and semantics. I have previously emphasised the semantical (which is to say, mathematical) side because that is where my interests lie; but here is a paper that is clearly nominal techniques yet makes explicit use of neither Fraenkel-Mostowski set theory nor nominal sets. What brings the syntactic and semantical work together under the heading 'nominal techniques' is their commonality of jargon and notation, their treatment of names as 'first-class' entities rather than as embarassments to be hidden away, and their use of name permutations (name-for-name swappings) as the main technical engine. These links mean that for a trained reader it is simple to see how this paper should be understood semantically, although my workshop paper 'Nominal Logic with Equations Only' is the only place I know of where this is done explicitly (see Section 6).