What is the expression problem?
Sunday 27 May 2012 21:00
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The problem
The expression is a tricky problem in many languages that asks: given a set of functions that operate over a set of types, how do we allow both the set of functions and the set of types that those functions operate over be extended without losing type safety?
Abstract data types
Let's say we have the abstract syntax tree (AST) for a simple
mathematical language that contains literals and additions. In
ML, we can represent a node with the abstract data type (ADT)
node
which has two data constructors,
LiteralNode
and AddNode
:
datatype node
= LiteralNode of real
 AddNode of node * node
We can then define a function evaluate
that turns
the AST into a single number.
datatype node
= LiteralNode of real
 AddNode of node * node
fun evaluate (LiteralNode value) = value
 evaluate (AddNode (left, right)) = (evaluate left) + (evaluate right)
Note that evaluate
is typesafe since it handles
all possible instances of node
. Now, suppose we
want to add another operation over nodes, say to turn the AST
into a string. Using ADTs, this is simply a case of adding
another function. Importantly, this doesn't require any
modification to the existing source code.
datatype node
= LiteralNode of real
 AddNode of node * node
fun evaluate (LiteralNode value) = value
 evaluate (AddNode (left, right)) = (evaluate left) + (evaluate right)
fun nodeToString (LiteralNode value) = Real.toString value
 nodeToString (AddNode (left, right)) =
"(" ^ (nodeToString left) ^ " + " ^ (nodeToString right) ^ ")"
The problem arises when we decide that we'd like a variant of
our mathematical language with the negation operator. We'd like
to be able to evaluate this extension of our mathematical
language, but we're not concerned with turning negations into
strings. There's no straightforward way of achieving this using
ADTs  we're forced to add another data constructor to
node
, which may not be possible if we don't own
the original source code. We also add the appropriate case to
evaluate
.
datatype node
= LiteralNode of real
 AddNode of node * node
 NegateNode of node
fun evaluate (LiteralNode value) = value
 evaluate (AddNode (left, right)) = (evaluate left) + (evaluate right)
 evaluate (NegateNode term) = ~(evaluate term)
fun nodeToString (LiteralNode value) = Real.toString value
 nodeToString (AddNode (left, right)) =
"(" ^ (nodeToString left) ^ " + " ^ (nodeToString right) ^ ")"
Even if we can modify our definition of node
, we
still have a problem: we can no longer safely create functions
that operate over our original language. Consider the function
nodeToString
: since it no longer exhaustively
matches all possible instances of node
, it's not
typesafe. To restore type safety, we're forced to update it
to handle the case of NegateNode
:
datatype node
= LiteralNode of real
 AddNode of node * node
 NegateNode of node
fun evaluate (LiteralNode value) = value
 evaluate (AddNode (left, right)) = (evaluate left) + (evaluate right)
 evaluate (NegateNode term) = ~(evaluate term)
fun nodeToString (LiteralNode value) = Real.toString value
 nodeToString (AddNode (left, right)) =
"(" ^ (nodeToString left) ^ " + " ^ (nodeToString right) ^ ")"
 nodeToString (NegateNode term) = "" ^ (nodeToString term)
In general, ADTs make it easy to add extra functions that operate over existing data types, but difficult to safely extend those data types. Now, let's take a look at the same problem in an objectorientated language, specifically Java.
Objectorientation: interfaces and classes
We begin by defining the interface Node
, along
with two implementations, AddNode
and
LiteralNode
:
public interface Node {
}
public class LiteralNode implements Node {
private final double value;
public LiteralNode(double value) {
this.value = value;
}
}
public class AddNode implements Node {
private final Node left;
private final Node right;
public AddNode(Node left, Node right) {
this.left = left;
this.right = right;
}
}
For the sake of readability, let's leave out the constructors:
public interface Node {
}
public class LiteralNode implements Node {
private final double value;
}
public class AddNode implements Node {
private final Node left;
private final Node right;
}
Next, we want to evaluate each node to a single number.
We add an evaluate
method to the interface, and
add appropriate implementations to the concrete classes.
public interface Node {
double evaluate();
}
public class LiteralNode implements Node {
private final double value;
public double evaluate() {
return value;
}
}
public class AddNode implements Node {
private final Node left;
private final Node right;
public double evaluate() {
return left.evaluate() + right.evaluate();
}
}
Unlike our approach with ADTs in ML, extending our language to
support negation is straightforward. We simply add another
implementation of Node
, which doesn't require any
modification of the original source code.
public interface Node {
double evaluate();
}
public class NegateNode implements Node {
private final Node term;
public double evaluate() {
return term.evaluate();
}
}
public class LiteralNode implements Node {
private final double value;
public double evaluate() {
return value;
}
}
public class AddNode implements Node {
private final Node left;
private final Node right;
public double evaluate() {
return left.evaluate() + right.evaluate();
}
}
Unfortunately, safely adding another operation on nodes
requires us to modify the original source code for
Node
, which may not always be possible. In our
case, we want to be able to turn our original language of
add and literal nodes into strings, so we need to add a
nodeToString
method on both the Node
interface and the classes AddNode
and
LiteralNode
:
public interface Node {
double evaluate();
String nodeToString();
}
public class NegateNode implements Node {
private final Node term;
public double evaluate() {
return term.evaluate();
}
}
public class LiteralNode implements Node {
private final double value;
public double evaluate() {
return value;
}
public String nodeToString() {
return Double.toString(value);
}
}
public class AddNode implements Node {
private final Node left;
private final Node right;
public double evaluate() {
return left.evaluate() + right.evaluate();
}
public String nodeToString() {
return "(" + left.nodeToString() + " + " +
right.nodeToString() + ")";
}
}
Even if we can modify the original source code, by modifying
the interface, we've forced all implementations of
Node
to implement nodeToString
even
though we only ever wanted to use such an operation on our
original add and literal nodes. In particular, we're forced to
add nodeToString
to NegateNode
:
public interface Node {
double evaluate();
String nodeToString();
}
public class NegateNode implements Node {
private final Node term;
public double evaluate() {
return term.evaluate();
}
public String nodeToString() {
return "" + term.nodeToString();
}
}
public class LiteralNode implements Node {
private final double value;
public double evaluate() {
return value;
}
public String nodeToString() {
return Double.toString(value);
}
}
public class AddNode implements Node {
private final Node left;
private final Node right;
public double evaluate() {
return left.evaluate() + right.evaluate();
}
public String nodeToString() {
return "(" + left.nodeToString() + " + " +
right.nodeToString() + ")";
}
}
By using methods on interfaces, we have the opposite problem to ADTs: adding additional types of nodes without modifying or affecting existing code is straightforward, while it's difficult to safely add additional operations over those nodes.
Summary
In this particular example, our ideal solution would let us:

define
AddNode
andLiteralNode
, and an operationevaluate
over both of them. 
add a third type of node,
NegateNode
, whichevaluate
can be performed on, without modification of the original source code. 
add a second operation
nodeToString
over the original set of nodes,AddNode
andLiteralNode
, without modification of the original source code. 
not be forced to implement
nodeToString
forNegateNode
.
We can express these properties more generally as being able to:
 define a set of data types and operations over those data types
 add additional data types that can have the same operations applied to them, without modification of the original source code.
 add additional operations over those data types, without modification of the original source code.
 add these additional data types and operations independently. That is, if an extension ExtData adds a data type D, and another extension ExtOp adds an operation Op, we should be able to safely use both extensions without implementing the operation Op for the data type D, although we may choose to do so if we want to apply Op to D.
all while preserving typesafety.
Topics: Language design