## Archive for November, 2009

### Implementing a Correct Type-Checker for the Simply Typed Lambda Calculus

Wednesday, November 25th, 2009 by Darin Morrison

Last Friday I discussed how to implement a “correct” type-checker for the simply typed lambda calculus in a dependently typed metalanguage – in this case Agda.

What does it mean to implement a correct type-checker? Stepping back for a moment, most people have probably at one point or another encountered a buggy type checker. Such a type checker will occasionally give the wrong result either (1) by saying that an ill-typed term does indeed type-check, or (2) that a well-typed term doesn’t type-check.

For example, if we have a typing judgement $\Gamma \vdash M : \tau$ and a type-checking function $check : Ctx \rightarrow Term \rightarrow Type \rightarrow Bool$, a buggy type-checker could give either of the following:

(1) $\neg (\Gamma \vdash M : \tau) \rightarrow check\ \Gamma\ M\ \tau = \mathtt{true}$

(2) $\Gamma \vdash M : \tau \rightarrow \neg (check\ \Gamma\ M\ \tau = \mathtt{true})$

A correct type-checker should be constructed in such a way that both of these cases are always ruled out.

In order to construct such a type-checker, we will leverage the richness of our type system in order to prevent us from writing buggy code. Looking back at the type of our check function, we see that it returns a $Bool$. The problem is that there is nothing in this type that ensures that the value we return is actually related to our input in any way. Our check function could be a constant function returning $true$ for all we know.

We can fix this by refining $Bool$ to a sum type of proofs:

$check : \Pi \Gamma {:} Ctx.\ \Pi M {:} Term.\ \Pi \tau {:} Type.\ \Gamma \vdash M : \tau\ \vee\ \neg (\Gamma \vdash M :\tau)$

What’s going on here? We’re saying that if type-checking is successful, we should have a proof in the form of a typing derivation. If type-checking fails, we want a proof that such a derivation can’t exist (or we would have a contradiction).

Stepping back again, if we look at this new type for the type-checking function through our classical glasses, we obtain:

$check : \forall \Gamma {:} Ctx.\ \forall M {:} Term.\ \forall \tau {:} Type.\ \Gamma \vdash M : \tau\ \vee\ \neg (\Gamma \vdash M :\tau)$

which is completely trivial by the LEM. However, we are working in an intuitionistic metalanguage and by the disjunction property of intuitionistic logic, an implementation of our well-specified checking function is both a proof that type-checking is decidable and a decision procedure for actually doing so!

This is all quite nice so far, except for the fact that the standard typing rules for the STLC are non-algorithmic. By this, I mean one cannot transcribe the typing rules completely straightforwardly as a functional program. This is most apparent by considering the application case:

$\frac{\Gamma \vdash M : \sigma \rightarrow \tau\qquad \Gamma \vdash N : \sigma}{\Gamma \vdash M\ N : \tau}$

This rule must be read bottom-up to be consistent with an algorithmic interpretation realised as a structurally recursive checking function. But where does $\sigma$ come from? We have to more or less pull this type out of thin air to continue the recursive type-checking. Also, in a real implementation, even if we are able to somehow infer some type, say $\sigma'$, how do we know that $\sigma = \sigma'$?

There are ways to get around this, such as by modifying the check function to also return a type, and implementing equality for types, but the problem with this approach is that we now have a mismatch between how we understand the typing judgments and how we understand type-checking. Furthermore, the specification of the check function becomes more complicated.

A better solution is to realise that we need to change the typing rules themselves to reflect when a term should be analysed against a given type and when a type should be synthesised from a term (as in the case for the recursive call on M).

We do this by splitting the original judgement into separate analytic and synthetic judgments.

$\Gamma \vdash M \Leftarrow \tau$ ($M$ checks against $\tau$)

$\Gamma \vdash M \Rightarrow \tau$ ($M$ synthesises $\tau$)

Now we can rewrite the application rule so that, read bottom up, it provides an algorithm from which a case arm in a functional program is immediately derivable:

$\frac{\Gamma \vdash M \Rightarrow \sigma \rightarrow \tau\quad \Gamma \vdash N \Leftarrow \sigma}{\Gamma \vdash M\ N \Rightarrow \tau}$

(As a side note, there is an interesting philosophical argument described by Zeilberger et al. as to why the synthetic judgement should be read top-down and the analytic judgement bottom-up, but I will skip the details here…)

The full set of rules so far becomes as follows:

$\Gamma \vdash M \Leftarrow \tau$ ($M$ checks against $\tau$)

$\Gamma \vdash M \Rightarrow \tau$ ($M$ synthesises $\tau$)

$\sigma = \tau$ ($\sigma$ is equal to $\tau$)

$\frac{\Gamma[ x ] = \tau}{\Gamma \vdash x \Rightarrow \tau}$

$\frac{\Gamma, x {:} \sigma \vdash M \Leftarrow \tau}{\Gamma \vdash \lambda x {:} \sigma.\ M \Leftarrow \sigma \rightarrow \tau}$

$\frac{\Gamma \vdash M \Rightarrow \sigma \rightarrow \tau\quad \Gamma \vdash N \Leftarrow \sigma}{\Gamma \vdash M\ N \Rightarrow \tau}$

$\Gamma \vdash \langle\rangle \Rightarrow \top$

$\frac{\Gamma \vdash M \Rightarrow \sigma\quad \sigma = \tau}{\Gamma \vdash M \Leftarrow \tau}$

These rules should seem right with the exception of the last one. We need this rule so that we can, for example, check a variable in the body of a lambda. But there’s something wrong with it.

The problem is that, unlike all of the other rules, it is not associated with a particular constructor or destructor, which makes it non-syntactic. If we take it as it is, type-checking will no longer be deterministic and syntax directed which would be bad after all we’ve considered so far.

We need to introduce a new constructor to make this rule syntax directed, let’s call it “syn”, because when we encounter it, we recursively synthesise a type for its argument:

$\frac{\Gamma \vdash M \Rightarrow \sigma\quad \sigma = \tau}{\Gamma \vdash \mathtt{syn}\ M \Leftarrow \tau}$

So we’ve added a constructor to a term language, which makes type-checking syntax directed again. But careful consideration reveals that we’ve created a new problem. Some terms which should be checkable are no longer checkable if they occur under “syn”. Consider $\mathtt{syn}\ \lambda x {:} \sigma.\ M$. If we know the type of the codomain for the lambda term, we should be able to check it, but in this case, our type checking algorithm is going to try and infer the type of the lambda term, but looking up at our rules, we see there is no synthesis rule for lambda, so checking is going to fail.

Luckily, we can fix this too. We need to break terms M up into two syntactic categories, one for checkable terms and one for synthesisable terms:

$M^{\Leftarrow} ::= \lambda x {:} \sigma.\ M\ |\ \mathtt{syn}\ R$

$R^{\Rightarrow} ::= x\ |\ R\ M\ |\ \langle\rangle$

Seems good so far, right? Well, not quite. We’ve successfully broken the original term language up into the appropriate syntactic categories, but in the process we’ve restricted the language in such a way that we cannot write any terms containing redexes. Everything we can write is already in normal form. Oops!

The culprit here is the fact that the only thing we can put in the left hand side of an application is an R term, but lambdas are M terms. We need a way to embed M terms back into R terms. Since the only real checkable term is a lambda term, and we cannot infer types for these, how can we possibly embed them into R terms? By adding a type annotation! Let’s add a constructor of the form $M : \tau$. If we were to put this on a lambda term, the type annotation for the variable would become superfluous, so we might as well drop that entirely and only use the annotation constructor. Let’s consider the final system with all of our changes in place:

$M^{\Leftarrow} ::= \lambda x.\ M\ |\ \mathtt{syn}\ R$

$R^{\Rightarrow} ::= x\ |\ R\ M\ |\ \langle\rangle\ |\ M : \tau$

$\Gamma \vdash M \Leftarrow \tau$ ($M$ checks against $\tau$)

$\Gamma \vdash R \Rightarrow \tau$ ($R$ synthesises $\tau$)

$\sigma = \tau$ ($\sigma$ is equal to $\tau$)

$\frac{\Gamma[ x ] = \tau}{\Gamma \vdash x \Rightarrow \tau}$

$\frac{\Gamma, x {:} \sigma \vdash M \Leftarrow \tau}{\Gamma \vdash \lambda x.\ M \Leftarrow \sigma \rightarrow \tau}$

$\frac{\Gamma \vdash R \Rightarrow \sigma \rightarrow \tau\quad \Gamma \vdash M \Leftarrow \sigma}{\Gamma \vdash R\ M \Rightarrow \tau}$

$\Gamma \vdash \langle\rangle \Rightarrow \top$

$\frac{\Gamma \vdash R \Rightarrow \sigma\quad \sigma = \tau}{\Gamma \vdash \mathtt{syn}\ R \Leftarrow \tau}$

$\frac{\Gamma \vdash M \Leftarrow \tau}{\Gamma \vdash M : \tau \Rightarrow \tau}$

What impact is all of this going to have on our checking function? First, we’re going to have to create a new function corresponding to the synthetic judgment, and refine the types of the arguments slightly to reflect the fact that we now are dealing with two separate syntactic categories:

$check : \Pi \Gamma {:} Ctx.\ \Pi M {:} Chk.\ \Pi \tau {:} Type.\ \Gamma \vdash M \Leftarrow \tau\ \vee\ \neg (\Gamma \vdash M \Leftarrow \tau)$

$synth : \Pi \Gamma {:} Ctx.\ \Pi R {:} Syn.\ (\Sigma \tau {:} Type.\ \Gamma \vdash R \Rightarrow \tau\) \vee\ \neg (\Sigma \tau {:} Type.\ \Gamma \vdash R \Rightarrow \tau)$

With all of these changes in place, it becomes pretty straightforward to write ‘check’ and ‘synth’ as two mutually recursive functions. The types will prevent us from writing buggy code and our type checker should be correct by construction.

If we plan to use this type-checker in the “real world”, we might want to change the negative evidence type to two separate inductive types:

$\Gamma \nvdash M \Leftarrow \tau$

and

$\Gamma \nvdash R \Rightarrow$

Then we would want to show that these new inductive types are sound with respect to the negation of the original judgement:

$\Gamma \nvdash M \Leftarrow \tau$ implies $\neg (\Gamma \vdash M \Leftarrow \tau)$

and

$\Gamma \nvdash R \Rightarrow$ implies $\neg (\Sigma \tau {:} Type.\ \Gamma \vdash R \Rightarrow \tau)$

This would allow us to produce error messages based on the negative evidence, since it is inductive whereas the negation (which is a function type) is not.

**UPDATE 1**

I’ve added some screenshots showing an implementation of what I describe above in action:

Type-checking the S combinator:

Type-checking an incorrect version of the S combinator:

Type-checking (and normalising) an arbitrary term:

** UPDATE 2 **

If you would like a copy of the source code, feel free to email me.

Also, I should point out that this idea of constructing a correct type-checker is not an original idea.  At least one earlier iteration of a very similar idea was described by McBride and McKinna in “The view from the left”.  There are two differences with what I do here.  The first is that I use a bidirectional type system instead of the traditional one.  The other difference is that I show that the judgment for negative evidence is sound with respect to negation of the judgement for positive evidence.  This step allows me to prove that type-checking and type-inference are decidable.

### Scala: An Object-Oriented Language with Functional Programming Features

Wednesday, November 25th, 2009 by Darin Morrison

A couple of weeks ago I gave an overview of the Scala language.  This was originally prompted by a recent discussion about languages for teaching and what might be a nice replacement for Java.  I suggested Scala and since I was familiar with the language I was asked to give an introduction.

Scala is an object-oriented language with a sophisticated and flexible type system.  It also supports quite a few features which are familiar to functional programmers.  Some of the features of Scala include:

* Bounded (upper and lower) parametric polymorphism

* Pattern matching via case classes

* First-class “functions” via functional objects

* Type constructor polymorphism (quantifying over types with higher kinds)

* A generalized comprehension syntax (can be used for sequence comprehension or for monadic computations)

* Existential types

etc.

Although Scala’s type system is rather sophisticated, the basics of the language are easy to learn and there are quite a few tutorials and references available online:

http://www.scala-lang.org/node/197