PL: Lecture #22  Tuesday, March 24th
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Types

PLAI §24

In our Toy language implementation, there are certain situations that are not covered. For example,

{< {+ 1 2} 3}

is not a problem, but

{+ {< 1 2} 3}

will eventually use Racket’s addition function on a boolean value, which will crash our evaluator. Assuming that we go back to the simple language we once had, where there were no booleans, we can still run into errors — except now these are the errors that our code raises:

{+ {fun {} 1} 2}

or

{1 2 3}

or

{{fun {x y} {+ x y}} 5}

In any case, it would be good to avoid such errors right from the start — it seems like we should be able to identify such bad code and not even try to run it. One thing that we can do is do a little more work at parse time, and declare the {1 2 3} program fragment as invalid. We can even try to forbid

{bind {{x 1}} {x 2 3}}

in the same way, but what should we do with this? —

{fun {x} {x 2 3}}

The validity of this depends on how it is used. The same goes for some invalid expressions — the above bogus expression can be fine if it’s in a context that shadows <:

{bind {{< *}}
  {+ {< 1 2} 3}}

Finally, consider this:

{+ 3 {if <mystery> 5 {fun {x} x}}}

where mystery contains something like random or read. In general, knowing whether a piece of code will run with no errors is a problem that is equivalent to the halting problem — and because of this, there is no way to create an “exact” type system: they are all either too restrictive (rejecting programs that would run with no errors) or too permissive (accepting programs that might crash). This is a very practical issue — type safety means a lot less bugs in the system. A good type system is still an actively researched problem.

What is a Type?

PLAI §25

A type is any property of a program (or an expression) that can be determined without running the program. (This is different than what is considered a type in Racket which is a property that is known only at run-time, which means that before run-time we know nothing so in essence we have a single type (in the static sense).) Specifically, we want to use types in a way that predicts some aspects of the program’s behavior, for example, whether a program will crash.

Usually, types are being used as the kind of value that an expression can evaluate to, not the precise value itself. For example, we might have two kinds of values — functions and numbers, and we know that addition always operates on numbers, therefore

{+ 1 {fun {x} x}}

is a type error. Note that to determine this we don’t care about the actual function, just the fact that it is a function.

Important: types can discriminate certain programs as invalid, but they cannot discriminate correct programs from incorrect ones. For example, there is no way for any type system to know that this:

{fun {x} {+ x 1}}

is an incorrect decrease-by-one function.

In general, type systems try to get to the optimal point where as much information as possible is known, yet the language is not too restricted, no significant computing resources are wasted, and programmers don’t spend much time annotating their code.

Why would you want to use a type system?

Our Types — The Picky Language

The first thing we need to do is to agree on what types are. Earlier, we talked about two types: numbers and functions (ignore booleans or anything else for now), we will use these two types for now.

In general, this means that we are using the Types are Sets meaning for types, and specifically, we will be implmenting a type system known as a Hindley-Milner system. This is not what Typed Racket is using. In fact, one of the main differences is that in our type system each binding has exactly one type, whereas in Typed Racket an identifier can have different types in different places in the code. An example of this is something that we’ve talked about earlier:

(: foo : (U String Number) -> Number)
(define (foo x)          ; \ these `x`s have a
  (if (number? x)        ; / (U Number String) type
    (+ x 1)              ; > this one is a Number
    (string-length x)))  ; > and this one is a String

A type system is presented as a collection of rules called “type judgments”, which describe how to determine the type of an expression. Beside the types and the judgments, a type system specification needs a (decidable) algorithm that can assign types to expressions.

Such a specification should have one rule for every kind of syntactic construct, so when we get a program we can determine the precise type of any expression. Also, these judgments are usually recursive since a type judgment will almost always rely on the types of sub-expressions (if any).

For our restricted system, we have two rules (= judgments) that we can easily specify:

n : Number  (any numeral `n' is a number)
{fun {x} E} : Function

And what about an identifier? Well, it is clear that we need to keep some form of an environment that will keep an account of types assigned to identifiers (note: all of this is not at run-time). This environment is used in all type judgments, and usually written as a capital Greek Gamma character (in some places G is used to stick to ASCII texts). The conventional way to write the two rules above is:

Γ ⊢ n : Number
Γ ⊢ {fun {x} E} : Function

The first one is read as “Gamma proves that n has the type Number”. Note that this is a syntactic environment, much like DE-ENVs that you have seen in homework.

So, we can write a rule for identifiers that simply has the type assigned by the environment:

Γ ⊢ x : Γ(x)    ; "Γ(x)" is similar to a "lookup(x, Γ)"

We now need a rule for addition and a rule for application (note: we’re using a very limited subset of our old language, where arithmetic operators are not function applications). Addition is easy: if we can prove that both a and b are numbers in some environment Γ, then we know that {+ a b} is a number in the same environment. We write this as follows:

Γ ⊢ A : Number  Γ ⊢ B : Number
———————————————————————————————
    Γ ⊢ {+ A B} : Number

Now, what about application? We need to refer to some arbitrary type now, and the common letter for that is a Greek lowercase tau:

Γ ⊢ F : Function  Γ ⊢ V : τᵥ
—————————————————————————————
    Γ ⊢ {call F V} : ???

that is — if we can prove that f is a function, and that v is a value of some type τₐ, then … ??? Well, we need to know more about f: we need to know what type it consumes and what type it returns. So a simple function is not enough — we need some sort of a function type that specifies both input and output types. We will use the notation that was seen throughout the semester and dump function. Now we can write:

Γ ⊢ F : (τ₁ -> τ₂)  Γ ⊢ V : τ₁
——————————————————————————————
    Γ ⊢ {call F V} : τ₂

which makes sense — if you take a function of type τ₁->τ₂ and you feed it what it expects, you get the obvious output type. But going back to the language — where do we get these new arrow types from? We will modify the language and require that every function specifies its input and output type (and assume we have only one argument functions). For example, we will write something like this for a function that is the curried version of addition:

{fun {x : Number} : (Number -> Number)
  {fun {y : Number} : Number
    {+ x y}}}

So: the revised syntax for the limited language that contains only additions, applications and single-argument functions, and for fun — go back to using the call keyword is. The syntax we get is:

<PICKY> ::= <num>
          | <id>
          | { + <PICKY> <PICKY> }
          | { fun { <id> : <TYPE> } : <TYPE> <PICKY> }
          | { call <PICKY> <PICKY> }

<TYPE>  ::= Number
          | ( <TYPE> -> <TYPE> )

and the typing rules are:

Γ ⊢ n : Number

Γ ⊢ {fun {x : τ₁} : τ₂ E} : (τ₁ -> τ₂)

Γ ⊢ x : Γ(x)

Γ ⊢ A : Number  Γ ⊢ B : Number
———————————————————————————————
    Γ ⊢ {+ A B} : Number

Γ ⊢ F : (τ₁ -> τ₂)  Γ ⊢ V : τ₁
——————————————————————————————
    Γ ⊢ {call F V} : τ₂

But we’re still missing a big part — the current rule for a fun expression is too weak, if we use it, we conclude that these expressions:

{fun {x : Number} : (Number -> Number)
  3}
{fun {x : Number} : Number
  {call x 2}}

are valid, as well concluding that this program:

{call {call {fun {x : Number} : (Number -> Number)
              3}
            5}
      7}

is valid, and should return a number. What’s missing? We need to check that the body part of the function is correct, so the rule for typing a fun is no longer a simple one. Here is how we check the body instead of blindly believing program annotations:

          Γ[x:=τ₁] ⊢ E : τ₂            ; Γ[x:=τ₁] is similar to
——————————————————————————————————————  ;    extend(Γ, x, τ₁)
Γ ⊢ {fun {x : τ₁} : τ₂ E} : (τ₁ -> τ₂)

That is — we want to make sure that if x has type τ₁, then the body expression E has type τ₂, and if we can prove this, then we can trust these annotations.

There is an important relationship between this rule and the call rule for application:

(Side note: Racket comes with a contract system that can identify type errors dynamically, and assign blame to either the caller or the callee — and these correspond to these two sides.)

Note that, as we said, number is really just a property of a certain kind of values, we don’t know exactly what numbers are actually used. In the same way, the arrow function types don’t tell us exactly what function it is, for example, (Number -> Number) can indicate a function that adds three to its argument, subtracts seven, or multiplies it by 7619. But it certainly contains much more than the previous naive function type. (Consider also Typed Racket here: it goes much further in expressing facts about code.)

For reference, here is the complete BNF and typing rules:

<PICKY> ::= <num>
          | <id>
          | { + <PICKY> <PICKY> }
          | { fun { <id> : <TYPE> } : <TYPE> <PICKY> }
          | { call <PICKY> <PICKY> }

<TYPE>  ::= Number
          | ( <TYPE> -> <TYPE> )

Γ ⊢ n : Number

Γ ⊢ x : Γ(x)

Γ ⊢ A : Number  Γ ⊢ B : Number
———————————————————————————————
    Γ ⊢ {+ A B} : Number

          Γ[x:=τ₁] ⊢ E : τ₂
——————————————————————————————————————
Γ ⊢ {fun {x : τ₁} : τ₂ E} : (τ₁ -> τ₂)

Γ ⊢ F : (τ₁ -> τ₂)  Γ ⊢ V : τ₁
—————————————————————————————
    Γ ⊢ {call F V} : τ₂

Examples of using types (abbreviate Number as Num) — first, a simple example:

                {} ⊢ 5 : Num  {} ⊢ 7 : Num
                ———————————————————————————
{} ⊢ 2 : Num        {} ⊢ {+ 5 7} : Num
—————————————————————————————————————————————
          {} ⊢ {+ 2 {+ 5 7}} : Num

and a little more involved one:

    [x:=Num] ⊢ x : Num  [x:=Num] ⊢ 3 : Num
    ———————————————————————————————————————
          [x:=Num] ⊢ {+ x 3} : Num
———————————————————————————————————————————————
{} ⊢ {fun {x : Num} : Num {+ x 3}} : Num -> Num  {} ⊢ 5 : Num
——————————————————————————————————————————————————————————————
      {} ⊢ {call {fun {x : Num} : Num {+ x 3}} 5} : Num

Finally, try a buggy program like

{+ 3 {fun {x : Number} : Number x}}

and see where it is impossible to continue.

The main thing here is that to know that this is a type error, we have to prove that there is no judgment for a certain type (in this case, no way to prove that a fun expression has a Num type), which we (humans) can only do by inspecting all of the rules. Because of this, we need to also add an algorithm to our type system, one that we can follow and determine when it gives up.