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Lab 7: Type Analysis


In this lab, you define type constraints for MiniJava in NaBL2. The concepts you are going to use in NaBL2 are described in the following papers:

  1. P. Neron, A. Tolmach, E. Visser, G. Wachsmuth: A Theory of Name Resolution, ESOP 2015
  2. H. van Antwerpen, P. Neron, A. Tolmach, E. Visser, G. Wachsmuth: A Constraint Language for Static Semantic Analysis based on Scope Graphs, PEPM 2016

The official NaBL2 documentation is not entirely up to date. It is better to only use the lab description for now.



Specify type analysis for MiniJava in NaBL2. The specification should include:

  1. Name binding constraints for
    • method calls.
  2. Typing constraints for
    • integer and boolean literals,
    • unary and binary expressions,
    • variable and field references,
    • object creation,
    • this expressions,
    • method calls,
    • and the subtyping relation on classes.


You need to submit your MiniJava project with a pull request against branch assignment7 on GitHub. The Git documentation explains how to file such a request.

The deadline for submission is December 7th, 23:59.


You can earn up to 100 points for the correctness of your type analysis. Therefore, we run several test cases against your implementation. You earn points, when your implementation passes test cases. The total number of points depends on how many test cases you pass in each of the following groups:

  • name binding (20 points)
    • method declarations
    • method calls
  • typing rules (35 points)
    • literals (4 points)
    • unary expressions (5 points)
    • binary expressions (8 points)
    • variable and field references (4 points)
    • object creation (4 point)
    • this expression (5 points)
    • method call (5 points)
  • constraints (45 points)
    • expressions (10 points)
    • statements (10 points)
    • subtyping in assignments, return expressions, method calls (10 points)
    • cyclic inheritance (2 points)
    • method overloading and method overriding (10 points)
    • main class usage (3 points)

Early Feedback

We provide early feedback for your language implementation. This feedback gives you an indication which parts of your name and type analysis might still be wrong. It includes a summary on how many tests you pass and how many points you earn by passing them. You have 3 early feedback attempts.

Detailed Instructions

Git Repository

You continue with your work from the previous assignment. See the Git documentation on how to create the assignment7 branch from your previous work.

Declaring Types

The first thing you need to do, is declare the types that are used in MiniJava. Most type constructors were already given in the previous lab. Types are declared in the signature section of an NaBL2 file. As an example, we show the type declaration for class types:

    // ... more constructors ...

We identify classes by the declaration from the class definition. For example, assume a class A {} with corresponding AST Class("A",_). Therefore, the class type constructor takes an occurrence as argument. Instances of this class would have the type ClassType(Class{"A"}).

You should define the constructors for integers, arrays, and booleans similarly. You also need to add one more type constructor for methods. This type should have two arguments, the return type, and a list of argument types of the method. A type argument is indicated by type. Lists of types are themselves considered types, so you can use type for the argument list as well.

Constraint Generation Rules (revisited)

The constraint generation rules of lab 5 need to be extended with type constraints. Some rules will only need extra constraints, some will need to assign a type to the AST node that it matches on. The latter is the case for expressions, which have a type assigned to them. For example, a literal True() will have type Bool(), and This() will have the type of the surrounding class. For AST nodes that have a type assigned, the rules will now look like this:

[[ <Pattern> ^ (<{Scope ","}*>) : <Type> ]] :=

Similarly, whenever doing a recursive call, the type must now be added as well. Schematically, this looks like this:

[[ ... ]] :=
  [[ <Var> ^ ({Scope ","}*) : <Type> ]],

Only the rules for sorts that have a type, like expressions, should be changed this way.

If you used Map1 to iterate over a list of expressions, you now need to use Map1T, which includes a type as well. Consider the following recursive call over a list of expressions es:

Map1T [[ es ^ (s) : tys ]]

This will apply rules of the form [[ <Expr> ^ (<Scope>) : <Type> ]] := ... to every element of the list. The variable tys is instantiated to the list of types of the expressions. So, if es is [True(),False(),IntValue(_)], then tys will be [Bool(),Bool(),Int()] – assuming you wrote the correct rules for expressions of course.

The same general restrictions for constraint generation rules that applied in lab 5 still apply: 1) Only one rule per constructor, and 2) no implicit traversal of the AST.

Types can be concrete terms, like Bool(), or variables, like ty. You can use concrete types or variables in both the rule match, and the recursive calls. The type that you pass in the recursive call, and the type or variable you match on in the rule that is then applied generate an equality constraint together. These are explained in the next section.

Equality and Inference on Types

Types can be compared for equality using equality constraints. For example, the constraint

Bool() == Bool()

would be satisfied. However, the constraint

Bool() == Int()

would result in an error.

Sometimes the concrete type is not known, in which case you can use a variable for the unknown type. For example, the type of a reference depends on the type of the declaration. We use the variable ty to stand for the (unknown) type:

[[ VarRef(x) ^ (s) : ty ]] :=
    // ... constraints ...

When variables are used in equality constraints, the constraint solver will try to find values for them, such that the constraint is satisfied. Take for example the following three constraints:

ty1 == Bool(),    // (1)
ty2 == ty1,       // (2)
ty2 == Int().     // (3)

The constraint solver will solve constraint (1) by assigning ty1 the value Bool(). Now constraint (2) can be satisfied by assigning ty2 the value of Bool() as well. Such an assignment of values is called a unifier. The unifier is built in such a way that, if possible, substituting all variables for their assigned values will result in equal concrete terms on both sides of the equal sign. In this example, there is no valid unifier, because the third constraint will try to equate Int() with Bool().

The order in which constraints are solved is not fixed – it is not the same order as they are generated by the constraint generation function. This makes it possible to use equality constraints for some forms of non-local type inference.

You should introduce equality constraints in your rules wherever you expect types to be equal.

Types for Declarations

It is also possible to assign types to declarations. This is done using constraints of the following form:

<Occurrence> : <Type>

At references, where you have a resolution constraint <Occurrence> |-> <Var>, you use the same constraint, but with the variable, instead of a concrete occurrence, like this:

<Var> : <Type>

For example, the rule for references uses this pattern:

[[ VarRef(x) ^ (s) : ty ]] :=
    // ...
    Var{x} |-> d,
    d : ty,
    // ...

Note that d will be instantiated to the declaring occurrence, making the constraints the same, except for the types. Consider the case where the types do not match. For example, the declaration is an integer, Var{"x"} : Int(), but the use site expects a boolean, Var{"x"} : Bool(). The constraint solver treats all constraint equally, and will report an error on one of them. But in a language like MiniJava, with type annotations, you want the error to appear at the use site, not the definition site. This can be achieved by adding ! to the constraint at declaration site, for example, Var{x} : ty !. This indicates it is more important, and in case of a conflict with a constraint without the exclamation mark, the error will always be reported on the less important constraint.

Add constraints for the types of variables and references.

Be careful not to set a type on a reference! It will not automatically be unified with the type of the declaration it resolves to.

Type-dependent Name Resolution

If you have done everything described above, your expressions all have types. Now we can implement name resolution of method names. In lab 5 you wrote rules to create a scope graph which probably look like this:

[[ Class(x,_,_,_) ^ (s) ]] :=
    // ...
    Class{x} <- s,
    new cs,
    Class{x} ===> cs,
    // ...

Now, using the class declaration of a class type, we can get the class scope by using <Occurrence> ?===> cs. Note that cs will be a variable that is going to be instantiated during constraint solving. Create the method reference in a new scope ms (any name will do). We connect this scope to the variable scope using an edge, like ms ---> cs. After this, write constraints for resolving the method reference, getting its type, etc.

Remember that you can check if method resolution works, by Ctrl/Cmd + click on a method call in an example program.


Classes in MiniJava support subtyping. Building the subtype relation is done using the following constraint:

<Type> <! <Type>

The relation must be a tree, which is automatically enforced by the constraint solver. Checking if one type is a subtype of the other, can be done using the following constraint:

<Type> <? <Type>

Note that these constraints will not be used to find values for variables that appear in the types. They need to get values from somewhere else. If variables in these types remain free, the constraint solver will report an error.

Build the subtype relation from class definitions, and check in the right places if values are subtypes of the expected type.


Finally, write constraints to check if overriding is done correctly. Use the reference introduced for method declarations, that refers to the super method, or itself. Check that the argument types are the same. Also check that the return type is correct. Be careful, it does not have to be the same, but it needs to be co-variant!