Type-directed Translation

As usual, before we get into a full discussion of CPS, we'll need to develop some more machinery.

Usually, language translations are "syntax-directed", a translation judgment defined inductively over the syntax tree. Between two types of lambda calculi, for example, we might define our translation like this:

\require{bussproofs} \begin{aligned} \overline{x} &= x \\ \overline{\lambda (x:\tau).e} &= \lambda(x:\overline{\tau}).\overline{e} \\ \overline{e_1\ e_2} &= \overline{e_1}\ \overline{e_2} \end{aligned}

where the the translation can be determined solely by looking at the particular syntactic form.

One problem: This doesn't always work.

For example, consider translating a language with booleans into System $$F_\omega$$. Under a standard Church encoding, we'd translate the type bool as

$\overline{bool} = \forall(\alpha:*).(\alpha \rightarrow \alpha \rightarrow \alpha)$

with the corresponding introduction forms

\begin{aligned} \overline{true} &= \Lambda \alpha.\lambda(x:\alpha).\lambda(y:\alpha).x \\ \overline{false} &= \Lambda \alpha.\lambda(x:\alpha).\lambda(y:\alpha).y \\ \end{aligned}

A problem, however, comes when trying to translate the elimination form:

$\overline{if\ e_1\ then\ e_2\ else\ e_3} = \overline{e_1}[\tau]\ \overline{e_2} \ \overline{e_3}$

What type should we use for $$\tau$$? It should be the type of $$e_1$$ and $$e_2$$, but without actually running a typechecker, we won't know what type that is! To deal with this, we can use a type-directed translation, which couples the translation step with the typechecking/type synthesis process.

Of course, there are really two typing judgments here, one for the source and one for the target language. These are usually notated

\begin{aligned} \Gamma \vdash_S e:\tau \\ \Gamma \vdash_T e:\tau \end{aligned}

for source and target, respectively. However, we will elide the subscripts in cases where the context makes it clear, typically when we know what language the given term is in.

Our translation judgment, then, becomes

$\Gamma \vdash_S e:\tau \rightsquigarrow e'$

where $$e$$ is in the source and $$e'$$ is in the target language.

Note that we can actually use a syntax-directed translation of constructors, kinds and contexts. For a more complex constructor language, we may need to perform a kind-directed translation, but we will avoid that for reasons to be discussed.

A translation, then, should have the following regularity conditions:

1. $$\Gamma \vdash_S \tau: *$$ if and only if $$\overline{\Gamma} \vdash_T \overline{\tau}: *$$.
2. $$\Gamma \vdash_S e:\tau$$ if and only if there exists some $$e'$$ such that $$\Gamma \vdash_S e:\tau \rightsquigarrow e'$$.
3. If $$\Gamma \vdash_S e:\tau \rightsquigarrow e'$$, then $$\overline{\Gamma} \vdash_T e':\overline{\tau}$$ (static correctness).

You might think that, since we have a notion of static correctness, we may also have some form of dynamic correctness to preserve. Loosely, this might be phrased as "If $$\Gamma \vdash e:\tau \rightsquigarrow e'$$, then $$e$$ and $$e'$$ 'do the same thing'". However, it is actually quite difficult to even state this formally! Certainly, we can't state this without having some formal notion of dynamic behavior, which we won't be dealing with in this class. It turns out that these proofs are also quite involved, which is another reason we don't bother.

A type-directed translation rule will always follow the same form of the associated regular typing rule, where the premises are also a translation. The variable translation rule, for example might look like this:

$\begin{prooftree} \AxiomC{\Gamma(x) = \tau} \UnaryInfC{\Gamma \vdash x:\tau \rightsquigarrow x} \end{prooftree}$

In general, it is best not to touch variable names. If we do start messing with variable names, we'll need to ensure freshness, perform substitutions on subterms, and so on -- not worth it at all.

Returning to the if-then-else example, we can now express the correct translation (assuming the usual rule for typing if-then-else):

$\begin{prooftree} \AxiomC{\Gamma \vdash e_1:bool \rightsquigarrow e_1'} \AxiomC{\Gamma \vdash e_2:\tau \rightsquigarrow e_2'} \AxiomC{\Gamma \vdash e_3:\tau \rightsquigarrow e_3'} \TrinaryInfC{\Gamma \vdash if\ e_1\ then\ e_2\ else\ e_3:\tau \rightsquigarrow e_1'[\overline{\tau}]\ e_2'\ e_3'} \end{prooftree}$

Finally, note that these rules will need to be adjusted somewhat to work with the algorithmic type synthesis and type checking rules.

Although we have not given a full formal system, we can sketch out what a proof of the regularity conditions may look like:

1. Depends on the specifics of the kind system, but often follows via induction on the kinding judgment (in both directions).
2. The forward direction is fairly clear, as we design our rules to follow from the source typing judgment. The backwards direction is even easier, as we can simply delete the translations from every premise and conclusion to show what we need.
3. Was not given in lecture; can be proven relatively easily for yourself.

Coherence

An important property of translations in general is coherence, namely that translations are unique.

Suppose that $$\Gamma \vdash_S e:\tau$$, and so $$\Gamma \vdash e:\tau \rightsquigarrow e'$$. What if we also have $$\Gamma \vdash e:\tau \rightsquigarrow e''$$? In our cases, this will generally be impossible, as our translations are based on typing judgments and typing judgments are unique. In real languages, on the other hand, this is not necessarily the case. The statement of coherence, then, in this case, is that $$e' = e''$$. It is very difficult to prove this, so we won't.

For example, what if we're in a language with subtyping? The typical subsumption rule,

$\begin{prooftree} \AxiomC{\Gamma \vdash e:\tau} \AxiomC{\Gamma \vdash \tau' \le \tau} \BinaryInfC{\Gamma \vdash e:\tau'} \end{prooftree}$

actually translates to coercion code in a type-directed setting:

$\begin{prooftree} \AxiomC{\Gamma \vdash e:\tau \rightsquigarrow e'} \AxiomC{\Gamma \vdash \tau' \le \tau \rightsquigarrow f} \BinaryInfC{\Gamma \vdash e:\tau' \rightsquigarrow f\ e'} \end{prooftree}$

where $$f$$ is the function witnessing that $$\tau'$$ subsumes $$\tau$$. Of course, as SML lacks subtyping, we aren't going to bother.

This does, however, bring us back to why we don't perform a kind-directed translation for type constructors -- we do have subkinding! So designing a coherent system for that becomes much more difficult.

Remarks

• In lecture, Prof. Crary chose to use $$\overline{e}$$ as the "output" of the translation judgment, where $$\overline{\cdot}$$ here is not acting as an operator, just taking the symbol $$\overline{e}$$ as a suggestive variable name. I have taken the liberty of rephrasing the rules to use something else for the sake of my own confusion.
• I did a few cases of the proof of the translation from IL-Direct to IL-CPS and it seems to be fairly straightforward. The third condition can be shown by induction over the translation rules, but is muddied by the presence of different syntactic classes.
• Later, there was some discussion about Kleene equivalence, but I zoned out and didn't catch a lot of it (it wasn't particularly relevant to this material). It is a weaker property than full dynamic equivalence; it holds when $$e_1$$ halts iff $$e_2$$ halts.