Fixed point combinator

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Fixed point combinator - Example
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Fixed point combinator, Fixed point combinator - Example, Fixed point combinator - Existence of fixed point combinators, Fixed point combinator - Other fixed point combinators, combinatory logic, untyped lambda calculus, typed lambda calculus, anonymous recursion

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Fixed point combinator - Example: Encyclopedia II - Fixed point combinator - Example

Consider the factorial function (under Church encoding). The usual recursive mathematical equation is fact(n) = if n=0 then 1 else n * fact(n-1) We can express a "single step" of this recursion in lambda calculus as F = λf. λx. (ISZERO x) 1 (MULT x (f (PRED x))), where "f" is a place-holder argument for the factorial function to be passed to itself. The function F performs a single step in the evaluation of the recursive formula. A ...

In certain formalizations of mathematics, such as the untyped lambda calculus and combinatorial calculus, every expression can be considered a higher-order function. In these formalizations, the existence of a fixed-point combinator means that every function has at least one fixed point; a function may have more than one distinct fixed point. In some other systems, for example the simply typed lambda calculus, a well-typed fixed-point combinator cannot be written -- in those systems any support for recursion must be explicitly ...

A version of the Y combinator that can be used in call-by-value (applicative-order) evaluation is given by η-expansion of part of the ordinary Y combinator: Z = λf. (λx. f (λy. x x y)) (λx. f (λy. x x y)) The Y combinator can be expressed in the SKI-calculus as Y = S (K (S I I)) (S (S (K S) K) (K (S I I))) The simplest fixed point combinator in the SK-calculus, found by John Tromp, is Y = S S K (S (K (S S (S (S S K)))) K) which corresponds to the lambda expres ...

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Fixed point combinator - Example

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