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uncombin
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There are 8 letters in UNCOMBIN ( B3C3I1M3N1O1U1 )
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In computer science's combinatory logic, a Fixed-point combinator (or fixpoint combinator) is a higher-order function fix that, for any function f that has an attractive fixed point, returns a fixed point x of that function. A fixed point of a function is a value that, when applied as the input of the function, returns the same value as its output. * In other 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 * * * * x * = * * * fix * * * * f * * * {\displaystyle x={\textsf {fix}}\ f} * , it represents a solution to the fixed point equation,* * * * x * = * f * * x * * * {\displaystyle x=f\ x} * The fixed-point combinator fix therefore satisfies the equation: * * * * * * * fix * * * * f * = * f * * ( * * * fix * * * * f * ) * * * * for all * * f * * * {\displaystyle {\textsf {fix}}\ f=f\ ({\textsf {fix}}\ f)\ \ {\text{ for all }}f} * A fixed point of a function f is a value that does not change under the application of the function f. * Functions that satisfy the equation for fix expand as, * * * * * * * fix * * * * f * = * f * * ( * f * * ( * … * f * * ( * * * fix * * * * f * ) * … * ) * ) * * * {\displaystyle {\textsf {fix}}\ f=f\ (f\ (\ldots f\ ({\textsf {fix}}\ f)\ldots ))} * A particular implementation of fix is Curry's paradoxical combinator Y, represented in lambda calculus by * * * * * * * Y * * * = * λ * f * . * ( * λ * x * . * f * * ( * x * * x * ) * ) * * ( * λ * x * . * f * * ( * x * * x * ) * ) * * * {\displaystyle {\textsf {Y}}=\lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))} * In functional programming, the Y combinator can be used to formally define recursive functions in a programming language that does not 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. * Applied to a function with one variable the Y combinator usuall... |