June 19, 2017
The Geometric McKay Correspondence (Part 1)
Posted by John Baez
The ‘geometric McKay correspondence’, actually discovered by Patrick du Val in 1934, is a wonderful relation between the Platonic solids and the ADE Dynkin diagrams. In particular, it sets up a connection between two of my favorite things, the icosahedron:
and the $\mathrm{E}_8$ Dynkin diagram:
When I recently gave a talk on this topic, I realized I didn’t understand it as well as I’d like. Since then I’ve been making progress with the help of this book:
- Alexander Kirillov Jr., Quiver Representations and Quiver Varieties, AMS, Providence, Rhode Island, 2016.
I now think I glimpse a way forward to a very concrete and vivid understanding of the relation between the icosahedron and E_{8}. It’s really just a matter of taking the ideas in this book and working them out concretely in this case. But it takes some thought, at least for me. I’d like to enlist your help.
June 10, 2017
Eliminating Binders for Easier Operational Semantics
Posted by John Baez
guest post by Mike Stay
Last year, my son’s math teacher introduced the kids to the concept of a function. One of the major points of confusion in the class was the idea that it didn’t matter whether he wrote $f(x) = x^2$ or $f(y) = y^2$, but it did matter whether he wrote $f(x) = x y$ or $f(x) = x z$. The function declaration binds some of the variables appearing on the right to the ones appearing on the left; the ones that don’t appear on the left are “free”. In a few years when he takes calculus, my son will learn about the quantifiers “for all” and “there exists” in the “epsilon-delta” definition of limit; quantifiers also bind variables in expressions.
Reasoning formally about languages with binders is hard:
“The problem of representing and reasoning about inductively-defined structures with binders is central to the PoplMark challenges. Representing binders has been recognized as crucial by the theorem proving community, and many different solutions to this problem have been proposed. In our (still limited) experience, none emerge as clear winners.” – Aydemir, Bohannon, Fairbairn, Foster, Pierce, Sewell, Vytiniotis, Washburn, Weirich, and Zdancewic, Mechanized metatheory for the masses: The PoplMark challenge. (2005)
The paper quoted above reviews around a dozen approaches in section 2.3, and takes pains to point out that their review is incomplete. However, recently Andrew Pitts and his students (particularly Murdoch Gabbay) developed the notion of a nominal set (introductory slides, book) that has largely solved this problem. Bengston and Barrow use a nominal datatype package in Isabell/HOL to formalize $\pi$-calculus, and Clouston defined nominal Lawvere theories. It’s my impression that pretty much everyone now agrees that using nominal sets to formally model binders is the way forward.
Sometimes, though, it’s useful to look backwards; old techniques can lead to new ways of looking at a problem. The earliest approach to the problem of formally modeling bound variables was to eliminate them.
June 7, 2017
Enriched Lawvere Theories for Operational Semantics
Posted by John Baez
guest post by Mike Stay
Programs are an expression of programmer intent. We want the computer to do something for us, so we need to tell it what to do. We make mistakes, though, so we want to be able to check somehow that the program will do what we want. The idea of semantics for a programming language is that we assign some meaning to programs in such a way that we can reason about the behavior of a program. There are two main approaches to this: denotational semantics and operational semantics. I’ll discuss both below, but the post will focus for the most part on operational semantics.
There’s a long history of using 2-categories and related structures for term rewriting and operational semantics, but Greg Meredith and I are particularly fond of an approach using multisorted Lawvere theories enriched over the category of reflexive directed graphs, which we call Gph. Such enriched Lawvere theories are equal in power to, for instance, Sassone and Sobociński’s reactive systems, but in my opinion they have a cleaner categorical presentation. We wrote a paper on them:
Here I’ll just sketch the basic ideas.