Inside 206-105

Existential Pontification and Generalized Abstract Digressions

Math

Picturing binomial coefficient identities

Guys, I have a secret to admit: I’m terrified of binomials. When I was in high school, I had a traumatic experience with Donald Knuth’s The Art of Computer Programming: yeah, that book that everyone recommends but no one has actually read. (That’s not actually true, but that subject is the topic for another blog […]

  • February 14, 2011

My first proof in Isabelle

One of the distinctive differences between academic institutions in the United States and in Great Britain is the supplementary learning outside of lectures. We have recitations in the US, which are something like extra lectures, while in the UK they have tutorials, or supervisions as they are called in Cambridge parlance. As always, they are […]

  • November 29, 2010

Purpose of proof: semi-formal methods

In which the author muses that “semi-formal methods” (that is, non computer-assisted proof writing) should take a more active role in allowing software engineers to communicate with one another. C++0x has a lot of new, whiz-bang features in it, one of which is the atomic operations library. This library has advanced features that enable compiler […]

  • October 20, 2010

Existential type-curry

This post is for those of you have always wondered why we have a forall keyword in Haskell but no exists keyword. Most of the existing tutorials on the web take a very operational viewpoint to what an existential type is, and show that placing the forall in the “right place” results in the correct behavior. […]

  • October 15, 2010

Abstraction without a concrete concept

Hoare logic, despite its mathematical sounding name, is actually a quite practical way of reasoning about programs that most software engineers subconsciously employ in the form of preconditions and postconditions. It explicitly axiomatizes things that are common sense to a programmer: for example, a NOP should not change any conditions, or if a line of […]

  • October 6, 2010

Flipping arrows in coBurger King

Category theory crash course for the working Haskell programmer. A frequent question that comes up when discussing the dual data structures—most frequently comonad—is “What does the co- mean?” The snippy category theory answer is: “Because you flip the arrows around.” This is confusing, because if you look at one variant of the monad and comonad […]

  • July 14, 2010

Little’s law

A short thought from standing in line at the World Expo: Little’s law is a remarkable result that relates the number of people in a queue, the arrival rate of people to the queue, and the time spent waiting in the queue. It seems that it could be easily applied to a most universal feature […]

  • July 5, 2010

Well-founded recursion in Agda

Last Tuesday, Eric Mertens gave the Galois tech talk Introducing Well-Founded Recursion. I have to admit, most of this went over my head the first time I heard it. Here are some notes that I ended up writing for myself as I stepped through the code again. I suggest reading the slides first to get […]

  • June 16, 2010

Databases are categories

Update. The video of the talk can be found here: Galois Tech Talks on Vimeo: Categories are Databases. On Thursday Dr. David Spivak presented Categories are Databases as a Galois tech talk. His slides are here, and are dramatically more accessible than the paper Simplicial databases. Here is a short attempt to introduce this idea […]

  • June 4, 2010

Bananas, Lenses, Envelopes and Barbed Wire
A Translation Guide

One of the papers I've been slowly rereading since summer began is "Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire", by Erik Meijer, Maarten Fokkinga and Ross Paterson. If you want to know what {cata,ana,hylo,para}morphisms are, this is the paper to read: section 2 gives a highly readable formulation of these morphisms for the […]

  • May 26, 2010