This paper is so much more than the paper title suggests. From the paper title, I was expecting something about parametricity and non-interference and the conclusion does say

It is folklore that noninterference can be encoded via parametricity but we are unaware of any work that successfully shows how to do that.

What they do is take Abadi et al.’s Dependency Core Calculus (DCC) and show how to translate it to the richer language F-omega while preserving a form of full abstraction so that translated terms can be distinguised if and only if the original terms can be translated. (The “a form of” is because, in the context of non-interference, the level of the observer trying to distinguish the terms is significant.)

(An important omission is that they do not handle recursion.)

The idea of the DCC is to encode the secrecy level of a value in its type. e.g., “T_l bool” is a bool that can be observed at level l and above. In this scheme, each level “l” in the lattice is represented by a type constructor “T_l” and edges in the lattice are represented by coercions. Each “T_l” is a monad and the associated return and bind operations are used to wrap and unwrap values.

An earlier attempt (later proved incorrect) translated the DCC type “T_l s” to “alpha_l -> s+” where “alpha_l” is a type whose values can only be constructed by code at level “l” or above. This reminded me of capabilities: you can only access a value if you have an appropriate capability.

Unfortunately, a counterexample was found. This emphasized the importance of having a “back translation” where every value of the translated F-omega type is equivalent to a value of the original DCC type.

The key idea of this paper’s correct translation is inspired by a similarity between the typing rule for bind and the typing rule for existential types which suggests that the encoding of existential types uses continuations might work. Their encoding is (roughly)

```
T_l s ==> forall b. (("l <= b", (s+ -> b)) -> b
```

where “l <= b” is a proof (witness?) that “b” is high enough in the lattice.

The bulk of the paper is a very detailed proof. It is a hard read but full of helpful comments along the lines of “normally, one would do X here but we do Y because …”. (I love this kind of insight in papers!)

One thing that I had expected from the title of this paper was something like Wadler’s “Theorems for free!” paper that had lots of examples of theorems that were normally laboriously proved by recursion but that turned out to be a simple property of the function’s type. This paper isn’t really that sort of paper.