## FIX

2. <business, protocol> Financial Information eXchange.

## fix

1. <mathematics> The fixed point combinator. Called Y in combinatory logic. Fix is a higher-order function which returns a fixed point of its argument (which is a function).

```	fix :: (a -> a) -> a
fix f = f (fix f)```

Which satisfies the equation

`	fix f = x such that f x = x.`

Somewhat surprisingly, fix can be defined as the non-recursive lambda abstraction:

`	fix = \ h . (\ x . h (x x)) (\ x . h (x x))`

Since this involves self-application, it has an infinite type. A function defined by

`	f x1 .. xN = E`

can be expressed as

```	f = fix (\ f . \ x1 ... \ xN . E)
= (\ f . \ x1 ... \xN . E)
(fix (\ f . \ x1 ... \ xN . E))
= let f = (fix (\ f . \ x1 ... \ xN . E))
in \ x1 ... \xN . E```

If f does not occur free in E (i.e. it is not recursive) then this reduces to simply

`	f = \ x1 ... \ xN . E`

In the case where N = 0 and f is free in E, this defines an infinite data object, e.g.

```	ones = fix (\ ones . 1 : ones)
= (\ ones . 1 : ones) (fix (\ ones . 1 : ones))
= 1 : (fix (\ ones . 1 : ones))
= 1 : 1 : ...```

Fix f is also sometimes written as mu f where mu is the Greek letter or alternatively, if f = \ x . E, written as mu x . E.

Compare quine.

[Jargon File]

2. bug fix. 