The Expression Problem

In the previous post I had put forward (rather hastily, I must add) my thoughts on what does expressiveness mean for programming languages. I had this feeling of forgetting something important but couldn’t pinpoint what it was. Today, while sipping a hot cup of coffee, I remembered the expression problem.

The Expression Problem is a well known problem in programming language theory dealing with the expressiveness of a programming language. Here is Philip Wadler’s original statement of the problem.

The Expression Problem is a new name for an old problem. The goal is to define a datatype by cases, where one can add new cases to the datatype and new functions over the datatype, without recompiling existing code, and while retaining static type safety (e.g., no casts).

Philip Wadler

He goes on to say “Whether a language can solve the Expression Problem is a salient indicator of its capacity for expression.

In this post, we will see two examples, one in Java and one in Haskell to understand the problem.

OO Programming

In Object Oriented Programming, it is easy to add new datatypes by defining new (sub)classes. Since classes encapsulate the operations on the data type, no existing code needs to be modified. However, adding new functions (operations) over the datatype is hard. All the datatypes have to be modified and recompiled.

Below is a java version explaining the problem. If you want to add new operation on the expressions (say, to type-check), you will have to add the new operation to each object type (Literal and Add) forcing a recompilation of these classes.

Visitor design pattern in OO programming is a known solution to cases where new operations are added over existing type (hierarchy). However, by using this pattern we will restrict our ability to add new types. Adding new (element) class (to the hierarchy) forces changes to all the visitor classes and needs recompilation.

Functional Programming

In functional programming, we think of functions that operate on values of certain types. It is easy to add new functions that operate on a datatype. To add new case to the datatype, we will have to modify all the functions to cater for the new case.

Typeclasses in Haskell offer a solution. But a naive use of type classes will not work either! In the below code, the expression

if True then Lit 10 else Add (Lit 5) (Lit 5)

does not typecheck even thought both Lit and Add are instances of Exp typeclass.

In the functional pearl Data types a la carte, Wouter Swierstra proposes a solution to the expression problem. Philip Wadler himself calls this the best solution in Haskell.

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