### 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.

### Decidable Equality in Agda

So I've been playing with typing various things in System-F which previously I had left with auxiliary well-formedness conditions. This includes substitutions and contexts, both of which are interesting to have well typed versions of. Since I've been learning Agda, it seemed sensible to carry out this work in that language, as there is nothing like a problem to help you learn a language.

In the course of proving properties, I ran into the age old problem of showing that equivalence is decidable between two objects. In this particular case, I need to be able to show the decidability of equality over types in System F in order to have formation rules for variable contexts. We'd like a context Γ to have (x:A) only if (x:B) does not occur in Γ when (A ≠ B). For us to have statements about whether two types are equal or not, we're going to need to be able to decide if that's true using a terminating procedure.

And so we arrive at our story. In Coq, equality is som…

### Formalisation of Tables in a Dependent Language

I've had an idea kicking about in my head for a while of making query plans explicit in SQL in such a way that one can be assured that the query plan corresponds to the SQL statement desired. The idea is something like a Curry-Howard in a relational setting. One could infer the plan from the SQL, the SQL from the plan, or do a sort of "type-checking" to make sure that the plan corresponds to the SQL.

The devil is always in the details however. When I started looking at the primitives that I would need, it turns out that the low level table joining operations are actually not that far from primitive SQL statement themselves. I decided to go ahead and formalise some of what would be necessary in Agda in order get a better feel for the types of objects I would need and the laws which would be required to demonstrate that a plan corresponded with a statement.

Dependent types are very powerful and give you plenty of rope to hang yourself. It's always something of…

### Plotkin, the LGG and the MGU

Legend has it that a million years ago Plotkin was talking to his professor Popplestone, who said that unification (finding the most general unifier or the MGU) might have an interesting dual, and that Plotkin should find it. It turns out that the dual *is* interesting and it is known as the Least General Generalisation (LGG). Plotkin apparently described both the LGG for terms, and for clauses. I say apparently because I can't find his paper on-line.

The LGG for clauses is more complicated so we'll get back to it after we look at the LGG of terms. We can see how the MGU is related to the LGG by looking at a couple of examples and the above image. We use the prolog convention that function symbols start with lower case, and variables start with uppercase. The image above is organised as a DAG (Directed Acyclic Graph). DAGs are a very important structure in mathematics since DAGs are lattices.

Essentially what we have done is drawn an (incomplete) Hasse diagram f…