## Monday, May 9, 2011

### Why Eager Languages Don't Have Products and Lazy Languages Don't Have Sums

In The Point of Laziness, Bob Harper mentioned a core duality between lazy and eager languages[1]. Eager languages have sum types but not product types and lazy languages have product types but no sum types. I've seen a few questions about what he means. I thought I'd try to explain. It all has to do with non-termination, our old friend "bottom" which I'll write ⊥. In this entire article I'm focusing on languages that don't allow any side effects other than ⊥ - that way I don't have to deal with ordering print statements or whatever. Also, I'm using an MLish notation that hopefully isn't too confusing.

Update: This comment by Dan Doel does a better job of explaining the issues than my whole article.

### Product Types

A product type is a pair. The pair type (A,B) is the product of A and B, and is often written as A * B or A × B. A 3-tuple[2] (A,B,C) can be seen as another way of saying ((A,B),C), etc. so any n-Tuple is just a pretty way to write nested pairs. The simplest way to define exactly what a pair means is through projection functions "first" and "second" which pull out the first or second element. The equations we want are

```first (a,b) == a
second (a,b) == b```

Pretty straight forward right? Well, those equations hold just fine in a lazy language even when a or b is ⊥. But in an eager language, first (a,⊥) == ⊥. Same deal with second. Oops. Hence lazy languages have product types in the sense that they completely meet these equations where eager languages don't.

That has a practical consequence. A compiler for an eager language can't necessarily simplify first(a, someComplicatedExpression) unless it can prove that someComplicatedExpression is never ⊥, and in general it can't do that. A lazy language compiler would always be free to throw away someComplicatedExpression.

### Sum Types

A sum type is an alternative between two choices. A | B is the sum of the two things, A and B, and is frequently written A + B. Choices among three elements, like A | B | C can be considered as just a pretty form of (A | B) | C - a nesting of binary choices. The simplest sum type is good old Boolean, the choice between true or false. Here's one equation we might expect to hold for Booleans and something that examines booleans, the if expression[3]

`((if condition then x else y), z) == if condition then (x,z) else (y,z)`

Ignore the possibility of x, y, and z being ⊥ and focus on "condition". In an eager language, condition must be evaluated in either of the two forms above and either way you get ⊥ if condition is ⊥. But in a lazy language, the left side will compute (⊥, z) where he right side will compute ⊥. Pulling the second element from those two results gives differing results, proving they are unequal.

Again, this has consequences. An eager language can simplify the right side form to the left side form if it wants to. A lazy language can't.

#### Footnotes

1. Strictly speaking I should say strict and non-strict instead of eager and lazy, but I'll go with Dr. Harper's wording because the difference isn't that important to the point.
2. Or a record (aka a struct). A tuple can be seen as a record where the field labels are numbers or a record can be seen as a tuple with some convenient labels for the positions.
3. If you are more familiar with C like languages that treat "if/else" as statements instead of expressions then pretend I'm using a ternary operator "condition ? x : y" instead of "if condition then x else y"