CS 536 (F03) Homework 9: Type Inference

Due: December 15 in class (hardcopy) AND via turnin (assignment name hwk9).

Part I: Generating Type Constraints

Following the Dec 1 notes, derive type constraints for this language:

   <expr> ::= <num>
            | true
            | false
            | {+ <expr> <expr>}
            | {- <expr> <expr>}
            | {* <expr> <expr>}
            | {iszero <expr>}
            | {bif <expr> <expr> <expr>}

            | <id>
            | {with {<id> <expr>} <expr>}
            | {rec {<id> <expr>} <expr>}
            | {fun {<id>} <expr>}
            | {<expr> <expr>}

            | tempty
            | {tcons <expr> <expr>}
            | {tempty? <expr>}
            | {tfirst <expr>}
            | {trest <expr>}
The only novelty of this language is that the list operations are now polymorphic; that is, you can create lists of values of any type.

Note: The right hand side of the rec binding does not have to be a syntactic function. However, you may assume that the rec-bound identifier only appears under a {fun ...} in the right hand side of the binding. In other words, the following expressions are legal:

     {rec {f {fun {x} {f x}}}

     {rec {f {with {y 4}
               {fun {x} {f y}}}}
while the following are not legal:
     {rec {f f}

     {rec {f {+ 1 f}}

Then, write a function which consumes an expression of this language, and returns a list of constraints (of the type defined in Part II).The correspondence between type constraints and the terms in Part II is as follows:

In some cases, you may need to a fresh identifier when defining constraints. The Scheme function gensym returns a unique identifier on every call.

Part II: Unification

Implement the unification algorithm from the Dec 1 notes. The algorithm should work for a generic term representation, as defined below.

A term is either:

In addition, you will need data types for representing a constraint (a pair of terms) and substitution (a variable and a term). The unification algorithm will consume a list of constraints (as defined in Part I) and produce either a list of substitutions or an error string.

Errors can arise from two situations: when the unification of two terms is impossible, or when the occurs check fails. In both cases, you should return a string with an appropriate error message.

Finally, when comparing variables for equality, use Scheme's built-in eq? function. For symbols, it behaves exactly as symbol=?; for other values, it compares them for identity (like Java's == comparison). We will rely on identical variables being deemed equivalent by eq? when solving the constraints generated in the following section.

Part III: Inferring Types

To infer the type of a program, first parse it, then generate constraints, and finally unify the constraints using the functions written for parts I & II as appropriate. The result will be a list of substitutions; by looking up the subsitution for the entire expression, you can access its type.

To implement this, your code needs to define a function, infer-type, which consumes a concrete representation of the program (as given above), and produces either an error string or a representation of the inferred type. Represent types concretely as:

   <type> ::= number
            | boolean
            | (listof <type>)
            | (<type> -> <type>)
            | <string>
where strings are used to represent type variables. For example, the type of length would be:
   ((listof "a") -> number)

Extra Credit

For a very small amount of extra credit, write a program in this language for which your algorithm infers the type ("a" -> "b"). You shouldn't attempt this problem until you've fully completed the assignment.

What Not To Do

You do not need to implement an interpreter for this language.

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