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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]

(1995-04-13)

2. bug fix.

(1998-06-25)

FIX

1. <networking> Federal Information Exchange.

2. <business, protocol> Financial Information eXchange.

(2001-05-14)

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