CS 456 Homework: Type Systems

The purpose of this assignment is to help you learn about type systems.

Setup and Guidelines

  git clone data.cs.purdue.edu:/homes/cs456/book-code

As in the ML homework, use function definition by pattern matching. In particular, do not use the functions null, hd, and tl.

Two problems to do by yourself

5. Do Exercise 5 on page 278 of Ramsey: add lists to Typed Impcore. The exercises requires you to design new syntax and to write type rules for lists. Your type rules should be deterministic. Turn in this exercise in PDF file lists.pdf.

   We recommend that you do Exercise 1 first (with your partner). It will give you more of a feel for monomorphic type systems.

   Hint: This exercise is more difficult than it first appears. Be encouraged by scrutinizing the lecture notes for similar cases, and remember that you have to be able to type-check every expression at compile time.

   Here are some things to watch out for:

   • It's easy to conflate syntax, types, and values. In this respect, doing theory is significantly harder than doing implementation, because there's no friendly ML compiler to tell you that you have a type clash among exp, tyex, and value.
   • It's especially easy to get confused about cons. You need to create a new syntax for cons. This syntax needs to be different from the PAIR constructor that is what cons evaluates to.
   • Dealing with the empty list presents a real challenge. Typed Impcore is monomorphic, which implies that any given piece of syntax has at most one type. But you want to allow for empty lists of different types. The easy way out is to design your syntax so that you have many different expressions, of different types, that all evaluate to empty lists.
   • A good mental test case is that if ns is a list of integers, then (car (reverse ns)) is an expression that probably should have a type.
   • You might want to see what happens to an ML program when you try to type-check operations on empty and nonempty lists. For this exercise to be helpful, you really have to understand the phase distinction between a type-checking error and a run-time error.
   • It's easy to write a type system that requires nondeterminism to compute the type of any term involving the empty list. But the problem requires a deterministic type system, because deterministic type systems are much easier to implement. You might consider whether similar problems arise with other kinds of empty data structures or whether lists are somehow unique.

11. Do Exercise 11 on page 279 of Ramsey: write exists? and all? in Typed µScheme. Turn in this exercise in file 11.scm.

Four problems to do with a partner

1. Do Exercise 1 on page 277 of Ramsey: finish the type checker for Typed Impcore by implementing the rules for arrays. Turn in this exercise in file timpcore.sml.
   Our solution to this problem is 21 lines of ML.

16. Do Exercise 16 on page 280 of Ramsey: extend Typed µScheme with a type constructor for queues and with primitives that operate on queues. As it says in the exercise, do not change the abstract syntax, the values, the eval function, or the type checker. If you change any of these parts of the interpreter, your solution will earn No Credit.

    Representing each queue as a list is recommended. If you do this, you will be able to use the following primitive implementation of put:

       let fun put (x, NIL)          = PAIR (x, NIL)
             | put (x, PAIR (y, ys)) = PAIR (y, put (x, ys))
             | put (x, _)            = raise BugInTypeChecking "non-queue passed to put"
       in  put
       end
    

    Turn in the code for this exercise in file tuscheme.sml, which should also include your solution to Exercise 13 below. Please include the answers to parts (a) and(b) in your README file.

    Hint: because empty-queue is not a function, you will have to modify the initialEnvs function on page 274b. You will update: the <primitive functions for Typed μScheme ::> and the <primitives that aren't functions, for Typed μScheme ::>.
    Our solution to this problem, including the implementation of put above, is under 20 lines of ML.

13. Do Exercise 13 on page 279 of Ramsey: write a type checker for Typed µScheme. Turn in this exercise in file tuscheme.sml, which should also include your solution to Exercise 16 above. Don't worry about the quality of your error messages, but do remember that your code must compile without errors or warnings.
    Our solution to this problem is about 120 lines of ML. It is very similar to the type checker for Typed Impcore that appears in the book. If we had given worse error messages, it could have been a little shorter.

20. Create three test cases for Typed µScheme type checkers. In file type-tests.scm, please put three val bindings to names e1, e2, and e3. (A val-rec binding is also acceptable.) After each binding, put in a comment the words "type is" followed by the type you expect the name to have. If you expect a type error, instead put a comment saying "type error". Here is an example (with more than three bindings):

    (val e1 cons)
    ; type is (forall ('a) (function ('a (list 'a)) (list 'a)))
    
    (val e2 (@ car int))
    ; type is (function ((list int)) int)
    
    (val e3 (type-lambda ('a) (lambda (('a x)) x)))
    ; type is (forall ('a) (function ('a) 'a))
    
    (val e4 (+ 1 #t)) ; extra example
    ; type error
    
    (val e5 (lambda ((int x)) (cons x x))) ; another extra example
    ; type error
    

    If you submit more than three bindings, we will use the first three.
    Each binding must be completely independent of all the others. In particular, you cannot use values declared in earlier bindings as part of your later bindings.

How to build a type checker

The only really viable strategy for building the type checker is one piece at a time. Writing the whole type checker before running any of it will make you miserable. Instead, start with small pieces similar to what we've done in class:

1. It's OK to write out a complete case analysis of the syntax, but have every case raise the LeftAsExercise exception. This trick will start to get you a useful scaffolding, in which you can gradually replace each exception with real code. And of course you'll test each time.
2. You can begin by type-checking literal numbers and Booleans.
3. Add IF-expressions as done in class.
4. Implement the VAL rule for definitions, and maybe also the EXP rule. Now you can test a few IF-expressions with different types, but you'll need to disable the initial basis as shown below.
5. Implement the rule for function application. You should be able to test all the arithmetic and comparisons from class.
6. Implement LET binding. The Scheme version is slightly more general than we covered in class. Be careful with your contexts. Implement VAR.
7. Once you've got LET working, LAMBDA should be quite similar. To create a function type, use the funtype function in the book.
8. Knock off SET, WHILE, BEGIN.
9. Because of the representation of types, function application is a bit tricky. Study the funtype function and make sure you understand how to pattern match against its representation.
10. There are a couple of different ways to handle LET-STAR. As usual, the simplest way is to treat it as syntactic sugar for nested LETs.
11. Knock off the definition forms VALREC and DEFINE. (Remember that DEFINE is syntactic sugar for VALREC.)
12. Save TYAPPLY and TYLAMBDA for after the last class lecture on the topic. (Those are the only parts that have to wait until the last lecture; you can have your entire type checker, except for those two constructs, finished before the last class.)

Finally, don't overlook section 6.6.4 in the textbook. It is only a few pages, but it is chock full of useful functions and representations that are already implemented for you.

We suggest that you replace the line in the source code

      val basis   = (* ML representation of initial basis *)

      val basis_included = false
      val basis = if not basis_included then [] else

Before submitting, make sure to turn the basis back on.

Other advice

1. Compile early.
2. Compile often.
3. Use an editor that jumps straight to the location of the error.
4. Come up with examples in Typed µScheme.
5. Figure out how those examples are represented in ML.
6. Keep in mind the distinction between the term language (values of queue type, values of function type, values of list type) and the type language (queue types, function types, list types).
7. If you're talking about a thing in the term language, you should be able to give its type.
8. If you're talking about a thing in the type language, you should be able to give its kind.

Avoid common mistakes

• In Exercise 5, it's a relatively common mistake to try to create a type system that prevents programmers from applying car or cdr to the empty list. Don't do this! It can be done, but by the standards of CS 456, such type systems are insanely complicated. As in ML, taking car or cdr of the empty list should be a well-typed term that causes an error at run time.
• In Exercise 5, it's quite common to write a nondeterministic type system by accident. The rules, typing context, and syntax have to work together to determine the type of every expression. But you're free to choose whatever rules, context, and syntax you want.
• In Exercise 5, it's inexplicably common to forget to write a typing rule for the construct that tests to see if a list is empty.
• There are already interpreters on your PATH with the same name as the interpreters you are working on. So remember to get the version from your current working directory, as in

    rlwrap ./timpcore
  

  Just plain timpcore will get the system version.
• ML equality is broken! The = sign gives equality of representation, which may or may not be what you want. For example, in Typed µScheme, you must use the eqType function to see if two types are equal. If you use built-in equality, you will get wrong answers.
• The val-rec form requires an extra side condition; in

    (val-rec x e)
  

  it is necessary to be sure that e can be evaluated without evaluating x. Many students forget this side condition, which can be implemented very easily with the help of the function appearsUnprotectedIn, which should be listed in the ``code index'' of your book.
• It's a common mistake to call ListPair.foldr and ListPair.foldl when what was really meant was ListPair.foldrEq or ListPair.foldlEq.

A few words about difficulty and time

Exercise 11, writing exists? and all? in Typed μScheme, requires that you really understand instantiation of polymorphic values. Once you get that, the problem is not at all difficult, but the type checker is very, very persnickety. A little of this kind of programming goes a long way.

Exercise 1, type-checking arrays in Typed Impcore, is probably the easiest exercise on this homework. You need to be able to duplicate the kind of reasoning and programming that we did in class for the language of expressions with LET and VAR.

Exercise 16, adding queues to Typed μScheme, requires you to understand how primitive type constructors and values are added to the initial basis. And it requires you to write ML code that manipulates μScheme representations. Study the way the existing primitives are implemented!

Exercise 13, the full type checker for Typed µScheme, presents two kinds of difficulty:

• You really have to understand the connection between typing judgments, typing rules, and code.
• You have to understand a moderately sophisticated ML program (the interpreter) and then build a relatively big and independent extension of it.

What to submit for your joint work with your partner

turnin -c cs456 -p typesys-pair README timpcore.sml tuscheme.sml type-tests.scm 

What to submit for your individual work

1. Your name
2. The names of any collaborators
3. The number of hours you spent on the assignment

How your work will be evaluated