---
title: "A Typesafe Representation of an Imperative Language"
date: 2020-10-30T17:19:59-07:00
tags: ["Idris"]
draft: true
---

A recent homework assignment for my university's programming languages
course was to encode the abstract syntax for a small imperative language
into Haskell data types. The language consisted of very few constructs, and was very much a "toy".
On the expression side of things, it had three registers (`A`, `B`, and `R`),
numbers, addition, comparison using "less than", and logical negation. It also
included a statement for storing the result of an expression into
a register, `if/else`, and an infinite loop construct with an associated `break` operation.
A sample program in the language which computes the product of two
numbers is as follows:

```
A := 7
B := 9
R := 0
do
  if A <= 0 then
    break
  else
    R := R + B;
    A := A + -1;
  end
end
```

The homework notes that type errors may arise in the little imperative language.
We could, for instance, try to add a boolean to a number: `3 + (1 < 2)`. Alternatively,
we could try use a number in the condition of an `if/else` expression. A "naive"
encoding of the abstract syntax would allow for such errors.

However, assuming that registers could only store integers and not booleans, it is fairly easy to
separate the expression grammar into two nonterminals, yielding boolean
and integer expressions respectively. Since registers can only store integers,
the `(:=)` operation will always require an integer expression, and an `if/else`
statement will always require a boolean expression. A matching Haskell encoding
would not allow "invalid" programs to compile. That is, the programs would be
type-correct by construction.

Then, a question arose in the ensuing discussion: what if registers _could_
contain booleans? It would be impossible to create such a "correct-by-construction"
representation then, wouldn't it?
{{< sidenote "right" "haskell-note" "Although I don't know about Haskell," >}}
I am pretty certain that a similar encoding in Haskell is possible. However,
Haskell wasn't originally created for that kind of abuse of its type system,
so it would probably not look very good.
{{< /sidenote >}} I am sure that it _is_ possible to do this
in Idris, a dependently typed programming language. In this post I will
talk about how to do that.

### Registers and Expressions
Let's start by encoding registers. Since we only have three registers, we
can encode them using a simple data type declaration, much the same as we
would in Haskell:

{{< codelines "Idris" "typesafe-imperative/TypesafeImp.idr" 1 1 >}}

Now that registers can store either integers or booleans (and only those two),
we need to know which one is which. For this purpose, we can declare another
data type:

{{< codelines "Idris" "typesafe-imperative/TypesafeImp.idr" 3 3 >}}

At any point in the (hypothetical) execution of our program, each
of the registers will have a type, either boolean or integer. The
combined state of the three registers would then be the combination
of these three states; we can represent this using a 3-tuple:

{{< codelines "Idris" "typesafe-imperative/TypesafeImp.idr" 5 6 >}}

Let's say that the first element of the tuple will be the type of the register
`A`, the second the type of `B`, and the third the type of `R`. Then,
we can define two helper functions, one for retrieving the type of
a register, and one for changing it:

{{< codelines "Idris" "typesafe-imperative/TypesafeImp.idr" 8 16 >}}

Now, it's time to talk about expressions. We know now that an expression
can evaluate to either a boolean or an integer value (because a register
can contain either of those types of values). Perhaps we can specify
the type that an expression evaluates to in the expression's own type:
`Expr IntTy` would evaluate to integers, and `Expr BoolTy` would evaluate
to booleans. Then, we could constructors as follows:

```Idris
Lit : Int -> Expr IntTy
Not : Expr BoolTy -> Expr BoolTy
```

Sounds good! But what about loading a register?

```Idris
Load : Reg -> Expr IntTy -- no; what if the register is a boolean?
Load : Reg -> Expr BoolTy -- no; what if the register is an integer?
Load : Reg -> Expr a -- no; a register access can't be either!
```

The type of an expression that loads a register depends on the current
state of the program! If we last stored an integer into a register,
then loading from that register would give us an integer. But if we
last stored a boolean into a register, then reading from it would
give us a boolean. Our expressions need to be aware of the current
types of each register. To do so, we add the state as a parameter to
our `Expr` type. This would lead to types like the following:

```Idris
-- An expression that produces a boolean when all the registers
-- are integers.
Expr (IntTy, IntTy, IntTy) BoolTy

-- An expression that produces an integer when A and B are integers,
-- and R is a boolean.
Expr (IntTy, IntTy, BoolTy) IntTy
```

In Idris, the whole definition becomes:

{{< codelines "Idris" "typesafe-imperative/TypesafeImp.idr" 18 23 >}}

The only "interesting" constructor is `Load`, which, given a register `r`,
creates an expression that has `r`'s type in the current state `s`.

### Statements
Statements are a bit different. Unlike expressions, they don't evaluate to
anything; rather, they do something. That "something" may very well be changing
the current state. We could, for instance, set `A` to be a boolean, while it was
previously an integer. This suggests equipping our `Stmt` type with two
arguments: the initial state (before the statement's execution), and the final
state (after the statement's execution). This would lead to types like this:

```Idris
-- Statement that, when run while all registers contain integers,
-- terminates with registers B and R having been assigned boolean values.
Stmt (IntTy, IntTy, IntTy) (IntTy, BoolTy, BoolTy)
```

However, there's a problem with `loop` and `break`. When we run a loop,
we will require that the state at the end of one iteration is the
same as the state at its beginning. Otherwise, it would be possible
for a loop to keep changing the types of registers every iteration,
and it would become impossible for us to infer the final state
without actually running the program. In itself, this restriction
isn't a problem; most static type systems require both branches
of an `if/else` expression to be of the same type for a similar
reason. The problem comes from the interaction with `break`.

By itself, the would-be type of `break` seems innocent enough. It
doesn't change any registers, so we could call it `Stmt s s`.
But consider the following program:

```
A := 0
B := 0
R := 0
do
  if 5 <= A then
    B := 1 <= 1
    break
    B := 0
  else
    A := A + 1
  end
end
```

The loop starts with all registers having integer values.
As per our aforementioned loop requirement, the body
of the loop must terminate with all registers _still_ having
integer values. For the first five iterations that's exactly
what will happen. However, after we increment `A` the fifth time,
we will set `B` to a boolean value -- using a valid statement --
and then `break`. The `break` statement will be accepted by
the typechecker, and so will the whole `then` branch. After all,
it seems as though we reset `B` back to an integer value.
But that won't be the case. We will have jumped to the end
of the loop, where we are expected to have an all-integer type,
which we will not have.

The solution I came up with to address this issue was to
add a _third_ argument to `Stmt`, which contains the "context"
type. That is, it contains the type of the innermost loop surrounding
the statement. A `break` statement would only be permissible
if the current type matches the loop type. With this, we finally
write down a definition of `Stmt`:

{{< codelines "Idris" "typesafe-imperative/TypesafeImp.idr" 26 30 >}}

The `Store` constructor takes a register `r` and an expression producing some type `t` in state `s`.
From these, it creates a statement that starts in `s`, and finishes
in a state similar to `s`, but with `r` now having type `t`. The loop
type `l` remains unaffected and unused; we are free to assign any register
any value.

The `If` constructor takes a condition `Expr`, which starts in state `s` and _must_ produce
a boolean. It also takes two programs (sequences of statements), each of which
starts in `s` and finishes in another state `n`. This results in
a statement that starts in state `s`, and finishes in state `n`. Conceptually,
each branch of the `if/else` statement must result in the same final state (in terms of types);
otherwise, we wouldn't know which of the states to pick when deciding the final
state of the `If` itself. As with `Store`, the loop type `l` is untouched here.
Individual statements are free to modify the state however they wish.

The `Loop` constructor is very restrictive. It takes a single program (the sequence
of instructions that it will be repeating). As we discussed above, this program
must start _and_ end in the same state `s`. Furthermore, this program's loop
type must also be `s`, since the loop we're constructing will be surrounding the
program. The resulting loop itself still has an arbitrary loop type `l`, since
it doesn't surround itself.

Finally, `Break` can only be constructed when the loop state matches the current
state. Since we'll be jumping to the end of the innermost loop, the final state
is also the same as the loop state. 

These are all the constructors we'll be needing. It's time to move on to
whole programs!

### Programs
A program is simply a list of statements. However, we can't use a regular Idris list,
because a regular list wouldn't be able to represent the relationship between
each successive statement. In our program, we want the final state of one
statement to be the initial state of the following one, since they'll
be executed in sequence. To represent this, we have to define our own
list-like GADT. The definition of the type turns out fairly straightforward:

{{< codelines "Idris" "typesafe-imperative/TypesafeImp.idr" 32 34 >}}

The `Nil` constructor represents an empty program (much like the built-in `Nil` represents an empty list).
Since no actions are done, it creates a `Prog` that starts and ends in the same state: `s`.
The `(::)` constructor, much like the built-in `(::)` constructor, takes a statement
and another program. The statement begins in state `s` and ends in state `n`; the program after
that statement must then start in state `n`, and end in some other state `m`.
The combination of the statement and the program starts in state `s`, and finishes in state `m`.
Thus, `(::)` yields `Prog s m`. None of the constructors affect the loop type `l`: we
are free to sequence any statements that we want, and it is impossible for us
to construct statements using `l` that cause runtime errors.

This should be all! Let's try out some programs.

### Trying it Out
The following (type-correct) program compiles just fine:

{{< codelines "Idris" "typesafe-imperative/TypesafeImp.idr" 36 47 >}}

First, it loads a boolean (`True`, to be exact) into register `A`; then,
inside the `if/else` statement, it stores an integer into `A`. Finally,
it stores another integer into `B`, and adds them into `R`. Even though
`A` was a boolean at first, the type checker can deduce that it
was reset back to an integer after the `if/else`, and the program is accepted.
On the other hand, had we forgotten to set `A` to a boolean first:

```Idris
  [ If (Load A)
    [ Store A (Lit 1) ]
    [ Store A (Lit 2) ]
  , Store B (Lit 2)
  , Store R (Add (Load A) (Load B))
  ]
```

We would get a type error:

```
Type mismatch between getRegTy A (IntTy, IntTy, IntTy) and BoolTy
```

The type of register `A` (that is, `IntTy`) is incompatible
with `BoolTy`. Our `initialState` says that `A` starts out as
an integer, so it can't be used in an `if/else` right away!
Similar errors occur if we make one of the branches of
the `if/else` empty, or if we set `B` to a boolean.

We can also encode the example program from the beginning
of this post:

{{< codelines "Idris" "typesafe-imperative/TypesafeImp.idr" 49 61 >}}

This program compiles just fine, too! It is a little reminiscent of
the program we used to demonstrate how `break` could break things
if we weren't careful. So, let's go ahead and try `break` in an invalid
state:

```Idris
  [ Store A (Lit 7)
  , Store B (Lit 9)
  , Store R (Lit 0)
  , Loop
    [ If (Load A `Leq` Lit 0)
      [ Store B (Lit 1 `Leq` Lit 1), Break, Store B (Lit 0) ]
      [ Store R (Load R `Add` Load B)
      , Store A (Load A `Add` Lit (-1))
      ]
    ]
  ]
```

Again, the type checker complains:

```
Type mismatch between IntTy and BoolTy
```

And so, we have an encoding of our language that allows registers to
be either integers or booleans, while still preventing
type-incorrect programs!