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# Project 3 - Functions and Elaboration

Project 3 – Functions and Elaboration EECS 662 - Programming Languages

The objective of this project is to add dynamically scoped and statically scoped strict functions to BAE and introduce elaboration to the interpretation process. You will first define an interpreter that adds functions, Booleans, and a conditional to BAE while removing the bind expression. You will then define an interpreter that uses elaboration to define interpretation of bind in terms of function application.

To aid in your quest, the file p3.hs implements Haskell data types and function signatures needed for this project. Look at this file carefully before you start as this project requires two abstract syntaxes. Data types cannot share constructor names, so a tick is used to distinguish constructors. I am not providing a parser for this project because parsers remain evil.

## Exercise 1

In this exercise you will write an interpreter for a modified FBAE language presented in our text that does not include the bind construct, but does include first-class functions. Following is the grammar for this language that we will call FAE:

FAE ::= number | id |
FAE + FAE  | FAE - FAE |
lambda id in FAE | FAE FAE |


CFAE has numbers, dynamically scoped, first-class functions with strict evaluation semantics and no bind. Your interpreter will use deferred substitution for efficiency but will not require closures as it is dynamically scoped. Perform the following:

1. Write a function, evalDynFAE :: Env -> FAE -> (Maybe FAE) that evaluates its second argument using the environment provided in its first and returns a FAE AST structure.

## Exercise 2

In this exercise you will write an interpreter for a modified FAE language from the previous exercise that is statically rather than dynamically scoped. You will need to add closures and values to the interpreter to accomplish this goal.

1. Write a function, evalStatFAE :: Env -> FAE -> (Maybe FAEValue) that interprets its second value using the environment provided in its first. This evaluator needs to return a value rather than a FAE expression to implement static scoping using closures.

## Exercise 3

In this exercise you will write a pair of interpreters for a an extension of the FAE language that includes the bind construct. This new language will be called FBAE. The trick is that for this exercise you will not write another interpreter at all. Instead you will write an elaborator that will translate FBAE language constructs into CFAE constructs, then call the FAE interpreter. The new language CFBAE has the form:

FBAE ::= number | id |
FBAE + FBAE  | FBAE - FBAE |
bind id = FBAE in FBAE |
lambda id in FBAE | FBAE FBAE |

1. Write a function, elabFBAE :: FBAE -> FAE, that takes a FBAE data structure and returns a semantically equivalent FAE structure. Specifically, you must translate the bind construct from CFBAE into constructs from FAE.

2. Write a function, evalFBAE :: Env -> FBAE -> (Maybe FBAEValue), that combines your elaborator and statically scoped FAE interpreter into a single operation that elaborates and interprets a FBAE expression.

The FBAE interpreter introduces elaboration to the FAEinterpreter by using a function that transforms FBAE abstract syntax into FAE syntax before evaluation. Most of this translation is routine - there are shared constructs in the two languages. For bind we have to do a bit more work. Thankfully, not too much more.

As discussed in class, the bind construct can be elaborated to an application of a function. Specifically:

bind id = t1 in t2 == ()(lambda id t1) t0)


Thus, to evaluate a bind expression in FBAE, one need simply translate it into a function application in FAE and execute the result.

## Exercise 4

Now for something completely different. Let’s add Booleans to our language by elaborating Boolean operations to lambdas. We’ll extend the language from the previous exercise leaving bind as is. The new language has the form

FBAEC ::= number | id |
FBAEC + FBAEC  | FBAEC - FBAEC |
true | false
FBAEC && FBAEC | FBAEC || FBAEC | FBAEC <= FBAEC | ~FBAEC
if FBAEC FBAEC FBAEC |
bind id = FBAEC in FBAEC |
lambda id in FBAEC | FBAEC FBAEC |

1. Write a function elabFBAEC :: FBAEC -> FAE that translates a FBAEC term into an equivalent FAE term. Specifically, each list operation must be translarted into an equivalent lambda term and each bind must be translated as above.
2. Write a function evalFBAEC :: Env -> FBAEC -> (Maybe FAEValue) that combines your elaborator and BFA interpreter into a single operation that elaborates and interprets a FBAEC term.

We’re going to use a technique called Church Booleans to implement Boolean values using functions. Let’s start by defining values for true and false:

true = lambda t in lambda f in t
false = lambda t in lambda f in f


true is a two argument function that returns its first value while false returns its second value. Remember that lambdas are values - you cannot interpret these lambda expressions further. They are literally the values for true and false. Kind of weird, but hang with me.

Let’s think about && as an example of how to implement the other problems.

&& = lambda x in lambda y in x y false


The && function is implemented like a short-circuit conjunction. It takes two arguments and applies the first to the second and false. Assume that the first argument is true, then true applied to y and false is y. If y is true, then x && y is true. If y is false, then the the second argument is returned and the result is always false.

||, ~, and if can be implemented similarly, but I will leave those to you. If you come up with something more complicated than && for any of these things you are doing something wrong.

## Notes

This project looks long. It’s really not. Most of the changes aside from the recursive function processing are trivial. Don’t be intimidated and just do things step-by-step. Define the first interpreter with dynamic scoping first, then add static scoping for Exercise 2. The elaborator in Exercise 3 need only translate bind whlle the elaborator in Exercise 4 adds Booleans. Once you get the definitions, the elaborator is not difficult to build.