Fixpoint operator
In computer science's combinatory logic, a fixed-point combinator (or fixpoint combinator) is a higher-order function fix that, for any function f, returns a fixed point x of that function. A fixed point of a function is a value that, when the value is applied as the input of the function, it returns the same value as its output.
In words: fix, when applied to an arbitrary function f, yields the same result as f applied to the result of applying fix to f. It is so named because, by setting , it represents a solution to the fixed point equation,
The fixed-point combinator fix therefore satisfies the equation:
A fixed point of a function f is a value that doesn't change under the application of the function f.
Functions that satisfy the equation for fix expand as,
A particular implementation of fix is Curry's paradoxical combinator Y, represented in lambda calculus by
In functional programming, the Y combinator can be used to formally define recursive functions in a programming language that doesn't support recursion.
This combinator may be used in implementing Curry's paradox. The heart of Curry's paradox is that untyped lambda calculus is unsound as a deductive system, and the Y combinator demonstrates that by allowing an anonymous expression to represent zero, or even many values. This is inconsistent in mathematical logic.
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