Around 2 years ago, for an earlier project of mine (which has not seen its light yet!) in which I had to build a language with variables and prove its properties, I surveyed a number of ways to handle binders. For some background, people have noticed that, when proving properties about a language with bound [...]
One is often amazed that, once two functions are identified as a Galois connection, a long list of nice and often useful properties follow from one concise, elegant defining equation. But how does one construct a program from a specification given as a Galois connection?
I had a chance to show the students, in 25 minutes, what functional program calculation is about. The student have just been exposed to functional programming a week ago in a three-hour course, and I have talked to them about maximum segment sum way too many times.
Given an array of integers having at least two elements, compute the sum of squares of the difference between all pairs of elements. It is not hard to quickly write up a
O(N²) program using nested loops, which, I have to confess, is what I would do before reading Kaldewaij’s book and realised that it is possible to do the task in linear time using one loop.
In the recent work of Sharon and me on maximally dense segments we needed quite a number of functions to be monotonic, idempotent, etc. It only occurred to me after submitting the paper: could they be defined as Galois connections?
Sharon and I have finally concluded, for now, our work on the maximally dense segment problem (download the draft here), on which we have been working on and off for the past two years. Considering the algorithm itself and its derivation/proofs, I am quite happy with what we have achieved. The algorithm is rather complex, however, and it is a challenge presenting it in an accessible way. Sharon has done a great job polishing the paper, and I do hope more people would be interested in reading it and it would, eventually, inspire more work on interesting program derivations.
In a previous paper of mine, regrettably, I wrongly attributed the origin of the maximum segment sum problem to Dijkstra and Feijen’s Een methode van programmeren. In fact, the story behind the problem was told very well in Jon Bentley’s Programming Pearls.
I learned the function derivation of the maximum segment sum problem from one of Jeremy’s papers and was very amazed. It was perhaps one of the early incident that inspired my interest in program calculation.
If you think you know everything you need to know about binary search, but have not read Netty van Gasteren and Wim Feijen’s note The Binary Search Revisited, you should.
If every solution returned by
D is no better than some solution returned by
X, any optimal solution by
X must be no worse than some optimal solution by
D “What? How could this be true?” It turned out that the reasoning can be correct, and the proof uses indirect equality in an unusual way.
A list of numbers is called steep if each element is larger than the sum of elements to its right. It is an example we often use when we talk about tupling. Can we determine the steepness of a list by a list homomorphism?