blog-static/content/blog/typesafe_imperative_lang.md

title	date	tags	draft
A Typesafe Representation of an Imperative Language	2020-10-30T17:19:59-07:00	Idris	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 pretty certain 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:

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

Sounds good! But what about loading a register?

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:

-- 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. So, the Stmt type will take two arguments: the initial state and the final state. This leads to the following definition:

{{< codelines "Idris" "typesafe-imperative/TypesafeImp.idr" 26 29 >}}

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 If constructor takes a condition, which starts in state s and must produce a boolean. It also takes two programs (sequences of statements), each of which start in s and finishes in another state n. Then, the If constructor creates 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.

The Loop constructor is even more restrictive: it takes a single program (the sequence of instructions that it will be repeating). This program starts and ends in state s; since the loop can repeat many times, and since we're repeating the same program, we want to make sure that program is always run from the same initial state.

I chose not to encode Break as a statement. This is because we don't want Breaks occurring in the middle of a program! Otherwise, it would be possible to write a program that seems like it will terminate in one state, but, because of a break in the middle, terminates in another! Instead, we'll encode Break as a part of the Prog encoding.

Programs

A program is basically a list of statements. However, we can't use a regular Idris list for two reasons:

1. Our type is not as simple as [Stmt]. We want each statement to begin in the state that the previous statement ended in; we will have to do some work to ensure that.
2. We have two ways of ending the sequence of statements: either with or without a break. Thus, instead of having a single Nil constructor, we'll have two.

The definition of the type turns out fairly straightforward:

{{< codelines "Idris" "typesafe-imperative/TypesafeImp.idr" 31 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 Break constructor is similar; however, it represents a break instruction, and thus, must be distinct from the regular End constructor. Finally, 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.

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:

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