lambdaspeech :: closure

_h1 closure

_p {b Closure} is not mandatory in functional programming provided functions accept {b partial application}. For instance in Scheme we could write the [cons,car,cdr] structure using a closure

{pre
(def cons (lambda (x y) (lambda (z) (z x y))))
(def car (lambda (z) (z (lambda (x y) x))))
(def cdr (lambda (z) (z (lambda (x y) y))))

(car (cons '♥ '♠)) -> ♥
(cdr (cons '♥ '♠)) -> ♠ 
}

_p The {b cons} function waits for 2 values and returns a function waiting for 1 function to be applied to the 2 values kept in a closure created by the {b cons} function.

_p In '{lambda speech} we use partiel application

{pre °°
{def cons {lambda {x y z} {z x y}}}
{def car {lambda {z} {z {lambda {x y} x}}}}
{def cdr {lambda {z} {z {lambda {x y} y}}}}

{car {cons ♥ ♠}} -> ♥
{cdr {cons ♥ ♠}} -> ♠ 
°°}

_p The {b cons} function waits for 3 values but will be feeded with only 2. The result will be a function waiting for 1 function to be applied to these values stored in the function's body. Let's trace the {b car} application to a pair, replacing names by their lambda expressions

{pre °°
.  {{lambda {z} {z {lambda {x y} x}}} {{lambda {x y z} {z x y}} ♥ ♠}} == ♥ 
-> {{lambda {z} {z {lambda {x y} x}}} {{lambda {y z} {z ♥ y}} ♠}}     == ♥
-> {{lambda {z} {z {lambda {x y} x}}} {{lambda {z} {z ♥ ♠}}}}         == ♥
-> {{lambda {z} {z {lambda {x y} x}}}  {lambda {z} {z ♥ ♠}} }         == ♥
-> {{lambda {z} {z ♥ ♠}} {lambda {x y} x}}                            == ♥
-> {{lambda {x y} x} ♥ ♠}                                             == ♥ 
-> ♥
°°}

_p It's just text substitutions.

_p Note that in Scheme words must be quoted and not in '{lambda speech}. Another property of lambdas in '{lambda speech} is that they accept a number of values greater than the number of arguments. Extra arguments are simply gathrered in the last one. A kind of variadicity.