The Meaning and Usefulness of Closures

In the previous post, I talked about the Fixed Point Combinator in λ-Calculus, and mentioned that combinator is another word for a closed lambda term, i.e., an expression with no free variables:

λx.x, λxy.x, λxy.y, etc.

Take for example the term λxy.y, which is really shorthand for λx.λy.y:

FV(λx.λy.y)   =   FV(λy.y) − {x}   =   FV(y) − {y} − {x}   =   {y} − {y} − {x}   =   Ø.

The opposite of a combinator – any expression that uses free variables, is called a closure. Even though the meaning of the term is slightly different in different contexts, the idea is basically the same:

Examining the term λx.(f x), we have the free variables FV(λx.f x) …

= FV(f x) − {x} = FV(f) ∪ FV(x) − {x} = {f} ∪ {x} − {x} = {f, x} − {x}
= {f}.

In Haskell, the definition for a function, func:

func f = \x -> f x

… uses the variable f, bound outside of the abstraction. It is therefore a closure.