### Total Functional Programming

Recently on Lambda the Ultimate there was a post called "Total Functional Programming".  The title didn't catch my eye particularly, but I tend to read the abstract of any paper that is posted there that doesn't sound terribly boring.  I've found that this is a fairly good strategy since I tend to get very few false positives this way and I'm too busy for false negatives.

The paper touches directly and indirectly on subjects I've posted about here before.  The idea is basically to eschew the current dogma that programming languages should be Turing-complete, and run with the alternative to the end of supplying
"Total Functional Programming"

At first glance this might seem to be a paper aimed at "hair-shirt" wearing functional programming weenies.  My reading was somewhat different.

Most hobbiest mathematicians have some passing familiarity with "Turing's Halting Problem" and the related "Goedel's Incompleteness Theorem".  What this paper is trying to do is say "So what!".  The basic idea is that restriction to a syntacticly restricted language can restrict the semantics in such a way that these problems do not apply.  The resulting (*huge*) class of programs can then solve almost everything that we think is important.

The "Bazooka" of the paper is a line describing how every program for which we have a complexity upper bound, is in fact inside the restricted language.  This realisation is profound.  It means that all of the algorithms that anyone has bothered to find upper bounds for (a huge class of programs mind you!) is accessible to this technique.

The paper even deals with infinite data (co-data) in the framework and mechanisms to properly account for streams and other (possibly) infinite structures.

The idea of working in syntactically restricted languages that are sub-Turing (for any number of reasons) is not new.  In fact I think I've mentioned DataLog (wikipedia entry) previously.  In the case of DataLog however, there are serious deficiencies in the expressive power.

Definitely the best paper I've read this year.  I'm not sure how accessible it is to lay-people, but it should be accessible to anyone with a CS B.S. or people who have some familiarity with functional programming.

### Managing state in Prolog monadically, using DCGs.

Prolog is a beautiful language which makes a lot of irritating rudimentary rule application and search easy. I have found it is particularly nice when trying to deal with compilers which involve rule based transformation from a source language L to a target language L'.

However, the management of these rules generally requires keeping track of a context, and this context has to be explicitly threaded through the entire application, which involves a lot of irritating and error prone sequence variables. This often leads to your code looking something a bit like this:

compile(seq(a,b),(ResultA,ResultB),S0,S2) :- compile(a,ResultA,S0,S1), compile(b,ResultB,S1,S2).
While not the worst thing, I've found it irritating and ugly, and I've made a lot of mistakes with incorrectly sequenced variables. It's much easier to see sequence made explicitly textually in the code.

While they were not designed for this task, but rather for parsing, DCGs turn out to be a convenient …

### Automated Deduction and Functional Programming

I just got "ML for the working programmer" in the mail a few days ago,
and worked through it at a breakneck pace since receiving it. It
turns out that a lot of the stuff from the "Total Functional
Programming" book is also in this one. Paulson goes through the use
of structural recursion and extends the results by showing techniques
for proving a large class of programs to be terminating. Work with
co-algebras and bi-simulation didn't quite make it in, except for a
type: D := D → D which is the type of programs in the untyped lambda
calculus, and hence liable to lead one into trouble.

I have to say that having finished the book, this is the single most