The CS 6120 Course Blog

Type Inference for Bril

by Christopher Roman


The goal of this project is to build a type inferrer for Bril. Ultimately, we want to take a Bril program that has some (or none) of the types specified on variables and produce a new Bril program that has all the correct type annotations. For example, this program:

main {
  v0 = const 1;
  v1 = const 2;
  v2 = add v0 v1;
  print v2;

should be transformed into the program:

main {
  v0: int = const 1;
  v1: int = const 2;
  v2: int = add v0 v1;
  print v2;

This makes it quicker to write Bril programs because we don't have to worry about correctly typing the variables.

We also will implement type checking through this type inference.


For our workflow, we'd like to take a semi-typed Bril program and convert it between JSON and text representations. This requires changing the grammar so that type annotations are optional.

Because Bril only supports two types, int and bool, the type inference is very straightforward. For example, consider arithmetic operations, i.e., add, mul, sub, and div. The arguments to this operations must be ints, and the result is an int. Therefore, if we see a statement like x = add a b, we know x, a, and b are all ints. If at any point we inferred that one of these variables are not ints, then we have a type unification error, and the program is not well-typed.


Modifying the Grammar

We want to make type annotations optional. So first I modified the grammar to have a rule for type annotations:

type_decl.5: ":" type
const.4: IDENT [type_decl] "=" "const" lit ";"

This is an issue however because there is an ambiguity between labels and assignments. Consider these two Bril programs:

main {
  x = const 5;


main {
  l: x = const 5;

These programs consist of the same tokens. However, the first one intuitively means that we have a label l and some int x with a value of 5. The second one means that we have a variable called l of type x with a value of 5. It is incorrect to simply have one rule as a higher priority, because semantically, both of these should be allowed as separate programs. So we have two options:

  1. Force labels to be on its own line
  2. Only allow fixed type names, like "int" and "bool"

I decided to go with (2) because I didn't know how to easily do (1), but in retrospect (1) may have been better, because some other students' projects allow for user-defined types.

Type Inference

As noted in the design, inferring the types of variables for a single statement is straightforward. Here is an example snippet showing how types for comparison ops are inferred:

elif instr["op"] in COMPARISON_OPS:
    for arg in instr["args"]:
        type_var(gamma, arg, "int", i)
    type_var(gamma, instr["dest"], "bool", i)

Here, we keep track of a typing context gamma which maps variables to their type. Then, we check that each argument is either untyped or already has the type int; otherwise we'll throw a type unification error. Finally, we do the same for the destination, making sure that it is typed as bool.

For our implementation, we simply iterate through each instruction and determine the types of our variables. The one case where this presents issues is with id. In general, id sets the type of the destination to the type of the variable on the right hand side. However, what happens if we don't know the type of the variable being copied? For example, consider this Bril program:

main {
  jmp later;
  x = id y;
  y = const 5;
  jmp earlier;

This is a valid program that should typecheck. Specifically, x and y are both ints. However, if we naively go sequentially through each instruction, we don't know the type of y until we have y = const 5, at which point it is too late to type x. To resolve this, we have two options:

  1. Rerun the type inference algorithm, stopping when no more variables have been inferred
  2. Keep track of which variables must have the same type, and after 1 pass, setting their types to be the same.

For simplicity sake, we choose (1). This means in the worst case, type inference will take O(n^2) time, where n is the number of instructions. This happens when there are multiple such id assignments as in the example above.

Type Checking

With type inference, type checking is relatively simple. After type inference, we have the original Bril program and the fully typed Bril program. We then go through the original Bril program and make sure that for any variable that has a type annotation, the type matches the inferred type. For completeness, I also check to make sure that variables aren't being used as labels, and vice versa.


To properly evaluate that everything presented here is correct, we have to test the parser and the type inferrer. To do these simultaneously, we can run tests as follows:

command = "cat {filename} | bril2json | python ../../ | bril2txt"

We are taking Bril text programs, turning them into JSON, generating a new equivalent typed program, and turning it back to text. First I started with simple Bril programs that already existed to build some confidence. Then to gain full confidence, I wrote tests that use every kind of operations, e.g., arithmetic ops, comparison ops, logical ops, effect ops, and misc. ops.

I tested for both positive and negative results. In other words, I ensured Bril programs that should typecheck were correctly type-inferred, and programs that shouldn't typecheck in fact could not be type-inferred. For example:

main {
  F1 = const false;
  T1 = const 1;
  b1 = and F1 T1;

Unfortunately I couldn't get Turnt to validate the error message that was output, so instead I manually checked to see that the error was what I expected. The exception raised for the above program is:

Exception: (stmt 3) Expected "T1" to have type "bool" but found "int"

I also made sure to test Bril programs that contained some type annotations and made sure that didn't interfere with the inference. This same process was repeated for testing the typechecker, which was run by passing the -t flag to

command = "cat {filename} | bril2json | python ../../ -t | bril2txt"

Through this process, I uncovered some bugs when implementing the type inferrer:

By rigorously testing all possible language features, we can be confident that the type inference is correct.

Hardest Parts to Get Right

The hardest parts to get right were the parser and full correctness of the type inferrer. I didn't realize the issue with the parser towards the beginning because I arbitrarily set a priority for the type annotation rule; debugging this took a while. Additionally, it was really easy to get a type inferrer that seemed to work for majority of cases. However, small bugs like the ones mentioned previously were only fixed through intense testing. Comparatively, the actual type inference and typechecking was relatively straightforward, mostly because there are only two types in Bril and no function calls.

Possible Extensions

Other groups are working on adding function calls to the language. By doing type inference on the function arguments and checking the return type, we can determine the type of a particular function. From this, we can relatively easily infer types for assignments to function calls.