Module 09: Dynamic types

This module is due at 1:15pm on Tuesday, October 23, though I strongly suggest that you try to have it finished, or at least almost finished, before class on Thursday October 18.

Write your team names here:

In this module, we will start exploring type systems, a central topic in programming language design.

We will continue building on the Arith language from previous weeks.

As the initial driver, pick the person with the most vowels in their name. Break ties by playing rock, paper, scissors.
You may either build upon your solution to last week’s module, or you may use mine. In either case, paste it here:

Extending the syntax

Begin by commenting out the interpreter and the eval function, so we can focus on extending and testing the abstract syntax and parser before we worry about updating the interpreter.
Now, extend the abstract syntax and parser:
- Begin by adding False, True, if, then, and else to the set of reservedNames in the lexer.
- Now add Boolean literals False and True.
- Add comparison operators < and ==. They should have lower precedence than + and - (but the same precedence as each other), and no associativity (AssocNone).
- Finally, add an if-then-else construct like Haskell’s. That is, Arith expressions now include the syntax 'if' <arith> 'then' <arith> 'else' <arith> You should add this to parseArithAtom.
After extending your AST definitions and your parser, you can test it by trying to parse things like
```
parse arith "if (3 == y) < z/2 then (if z then 3 else False) else True"
```
(Optional, you may want to skip this at first and come back to it later.) As a fun extra challenge, try adding other standard Boolean and comparison operators to your language such as || (logical or), ! (logical negation), > (greater than), <= (less than or equal to), >= (greater than or equal to), and != (not equal to). The comparison operators should all have the same precedence as == and <; negation (!) should have higher precedence than comparisons; logical and (&&) should have lower precedence than comparisons and be right-associative; and local or (||) should have lower precedence than && and also be right-associative.

Of course one could again extend the abstract syntax and the parser, and deal with all the extra operations throughout the interpreter, type checker, etc.. However, there is a better way: the set of existing features (False, True, less than, equal to, and if-then-else) was carefully chosen so that it is possible to express equivalent encodings of any of these other operators in terms of the existing ones. For example, x > y can be encoded by simply flipping the order of arguments and using y < x; as another example, x || y can be encoded as if x then True else y.
- One can parse to an AST FullArith containing all these operators, and then write a function desugar :: FullArith -> Arith which translates the language with all operators into the simpler language with only False, True, <, ==, and if.
- Alternatively, one can build the desugaring directly into the parser. For example, to parse the greater-than operator, one could write something like
```
, Infix ((\x y -> Bin Lt y x) <$ reservedOp ">") AssocNone
```
  assuming that Lt represents the less-than operation. Note that (\x y -> Bin Lt y x) represents an anonymous function which takes two Arith arguments x and y and returns the Arith value Bin Lt y x.

Extending the interpreter

ROTATE ROLES and write the name of the new driver here:

Of course, the interesting thing we have done is to introduce the possibility of a third kind of error, namely, type errors. We now have both integers and booleans in our language, and it would be an error, say, to try to add two boolean values; addition only applies to integers.

With type errors we have an interesting choice: should type errors be caught at runtime, or should they be caught in a separate type checking phase before the program is run?

Checking for type errors at runtime is known as a dynamic type system. It is considerably easier to implement, and takes as lenient an approach as possible. Some people also claim that programming in such a language makes them more “agile”. These people are sadly deluded.
Checking for type errors at compile time is known as a static type system. It is more complicated to implement, and necessarily disallows some programs that would have run successfully in a dynamically typed framework, but can lead to better runtime performance and more opportunities for runtime optimization. It also means certain kinds of programmer errors can be caught before the program is run.

Today, you will implement dynamic type checking for our language; in future modules you will implement a static type system.

Integer is no longer a sufficient result type for the interpreter, since programs can now evaluate to either an integer or a boolean. Create a new type called Value with two constructors that encapsulate the possible results of evaluation.
Currently, the InterpError type has two constructors, representing the possibility of encountering an undefined variable at runtime, or dividing by zero. Add another constructor representing the possibility of a type error. For now, this constructor will contain no information. Later, when we implement a type checker, we will add more fields so we can generate better error messages.
Now, the moment you have been waiting for: uncomment your interpreter and change its result type from Either InterpError Integer to Either InterpError Value. Of course, it will no longer type check. You should fix it, and extend it to interpret the new syntactic constructs we added. Some notes/hints:
- I suggest trying to uncomment and fix only one part at a time, so you can get each part to work before moving on to the next.
- As an example of what you will have to fix, note that a line such as interpArith e (Bin Plus e1 e2) = (+) <$> interpArith e e1 <*> interpArith e e2 will no longer work, since (+) expects two Integers, but the recursive calls to interpArith now return Values instead of Integers. Essentially you will have to replace (+) with a function of type Value -> Value -> Either InterpError Value which checks to make sure both values are integers and then adds them, throwing a type error if either value is not an integer. (However, literally replacing (+) with such a function in the code above will not quite work either; consider using the (>>=) operator.)
- You may assume the following:
  - The == and < operators expect two integers and return a boolean. (It is possible to overload them so they also work on booleans, but we will not bother with that for now.)
  - if-then-else expects a boolean test after the if, and does not care about the types of its two branches.
  - if-then-else should not evaluate both branches. It should first evaluate the test, and then decide which branch to evaluate (either then or else) based on the result of the test. For example, if True then x else 3 should result in an error saying that x is undefined, but if False then x else 3 should successfully evaluate to 3.
- Try to abstract out common patterns in order to avoid writing repeated code. As one example, I suggest making a function like interpBool :: Env -> Arith -> Either InterpError Bool (and similarly interpInteger) which interprets an expression and throws a type error if it does not result in a boolean. You may come up with other ways to abstract as well.
Finally, once you get your interpreter working, uncomment the eval function and get it working as well.
Be sure to try your interpreter on a number of examples. For example: if (3 == y) < z/2 then (if z then 3 else False) else True let x = 10 in (if (x < 12) then (if 5 == 5 then x + 19 else 12) else 8 let y = 2 in if 3 < 5 then False else y

Feedback

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