System F is Amazing - Part II

As I mentioned in my last post System-F is amazing. I haven't written in a very long time because I have been busy ensuring that I graduate. I passed my viva finally! So now that I'm back in the real world I thought I'd talk a bit about things that I found while I was in my cave. I'll put my dissertation up with some notes when I've completed the revisions.

Those who have dealt with proof assistants based on type theories (Such as Agda and Coq) might have noticed that we often require types to have a positivity restriction. This restriction essentially states that you can't have non-positive occurrences of the recursive type variable. As I described earlier, System F avoids this whole positivity restriction by forcing the programmer to demonstrate that constructors themselves (as a Church encoding) can be implemented, avoiding the problem of uninhabited types being inadvertently asserted.

So this means that in some sense positivity is probably the same thing as saying that there is an algorithmic method of translation of a data-type into System-F. I suspect (but can not yet prove) it also means that we should be allowed to write down datatypes as long as we can show that the inductive or coinductive types are inhabited by a Church encoding! If we could also do this trick for the Calculus of Constructions it might give a tricky way to increase the number of (co)inductive types we are allowed to write down.

Now, I'll write down an example of what I'm talking about in System-F, embedded in the propositional fragment of Coq (which is suitably impredicative), so that you can see a very straightforwardly non-positive type which has a Church encoding. This example of the LamMu type came from Andreas Abel's paper on sized types (which one I forget!) where he demonstrates that his system allows the definition.

In the very final entry we attempt to enter LamMu into the Coq Inductive type framework and watch it fail (for reasons of positivity violation). The original type definition gives us three constructors, each of which is proved in turn below by a "constructor".

```Definition LamMu := forall (X : Prop),   (Nat -> X) ->   (Nat -> List X -> X) ->   ((forall (Y : Prop), ((X -> Y) -> Y)) -> X) -> X.Definition var : Nat -> LamMu :=   fun (n : Nat) =>     fun (X : Prop)       (v : Nat -> X)       (f : Nat -> List X -> X)       (m : (forall (Y : Prop), ((X -> Y) -> Y)) -> X) =>       v n.Definition func : Nat -> List LamMu -> LamMu :=   fun (n : Nat) =>     fun (t : List LamMu) =>       fun (X : Prop)         (v : Nat -> X)         (f : Nat -> List X -> X)         (m : (forall (Y : Prop), ((X -> Y) -> Y)) -> X) =>         f n (t (List X) (nil X) (fun (x : LamMu) (y : List X) => cons (x X v f m) y)).Definition mu : (forall (Y : Prop), (LamMu -> Y) -> Y) -> LamMu. Proof.  unfold LamMu.  refine     (fun (zi : (forall (Y : Prop), (LamMu -> Y) -> Y)) =>       fun (X : Prop)         (v : Nat -> X)         (f : Nat -> List X -> X)         (m : (forall (Y : Prop), ((X -> Y) -> Y)) -> X) =>         m (fun (Y : Prop) (g : X -> Y) =>           g (zi X (fun e : LamMu => e X v f m)))).Defined.Inductive LamMu2 : Type :=| Var : Nat -> LamMu2| Fun : Nat -> LamMu2 -> LamMu2| Mu : ((forall Y, ((LamMu2 -> Y) -> Y)) -> LamMu2) -> LamMu2 ```

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…