From: MenTaLguY <mental@...>
Date: 2007-06-08T02:36:37+09:00
Subject: Re: [SOLUTION][QUIZ] FizzBuzz (#126) [solution #2]

On Thu, 7 Jun 2007 20:45:04 +0900, Brad Phelan <bradphelan@xtargets.com> wrote:
> Are they real lists as in Ruby Array which I doubt or abstract concepts like
> the Church Numerals.

In some sense, these things are only as abstract as you want them to be.  Instead of implementing Church numerals like this (expanded a bit for clarity):

 ZERO = LAMBDA2 { |f,x| x }
 ONE = LAMBDA2 { |f,x| f[x] }
 SUCC = LAMBDA { |n| LAMBDA2 { |f,x| f[n[f][x]] } }
 ADD = LAMBDA2 { |m,n| m[SUCC][n] }
 MULTIPLY = LAMBDA2 { |m,n| LAMBDA2 { |f,x| m[n[f]][x] } }
 POWER = LAMBDA2 { |m,n| n[m] }

I could also have done something like this:

 class ChurchNumeral
   attr_reader :value

   def initialize(value)
     @value = value
   end

   def call(f)
     LAMBDA { |x| @value.times { x = f[x] } ; x }
   end
   alias [] call
 end

 ZERO = ChurchNumeral.new 0
 ONE = ChurchNumeral.new 1
 SUCC = LAMBDA do |n|
   if ChurchNumeral === n
     ChurchNumeral.new n.value + 1
   else
     LAMBDA2 { |f,x| f[n[f][x]] }
   end
 end
 ADD = LAMBDA2 do |m,n|
   if ChurchNumeral === m and ChurchNumeral === n
     ChurchNumeral.new m.value + n.value
   else
     m[SUCC][n]
   end
 end
 MULTIPLY = LAMBDA2 do |m,n|
   if ChurchNumeral === m and ChurchNumeral === n
     ChurchNumeral.new m.value * n.value
   else
     LAMBDA2 { |f,x| m[n[f]][x] }
   end
 end
 POWER = LAMBDA2 do |m,n|
   if ChurchNumeral === m and ChurchNumeral === n
     ChurchNumeral.new m.value ** n.value
   else
     n[m]
   end
 end

As described in my previous email, I did take the former sort of approach for lists though.

-mental