### More on I/O, effects and logic programming

I've been having some talks on #prolog about the best way to implement effects in logic programming languages and have come across some really good information, due in large part to "ski" who hangs out on #prolog.

There seem to be quite a few ways to implement effects for I/O. I'm not sure exactly at this point how many of them will be appropriate to implementing non-monotonic logic, by which I mean allowing the data store to grow/shrink.

Apparently the Monad strategy can move into the logic programming scene without too much difficulty. There are even a few papers about using monads in lambda-prolog that I've come across lately. This method has a lot going for it, as it seems to have worked fairly well in haskell and clean.

Another way is using linearity. This has a lot of appeal since linear logic has a well understood declarative semantics and it doesn't force us to use higher order constructs in order to achieve I/O. I'm not sure how this would interact with changing the data store though.

Mercury uses the linearity approach. First we declare the main predicate to be of a determinate mode, meaning that there is no way that we can have other choices. now the I/O resource is exhausted when used and we don't have to worry about backtracking over the use of the resource.

All of this new information makes the end goal of a fully declarative system where effects can take place seem more feasible.

### 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…