In the meantime let me give you a glimpse of "programming".
Let me present you K and KI, the Tweedledum and Tweedledee of the Eliminators.
You know K. Well I hope, K is the first combinator we met. It is the kestrel and it obeys the following dynamical law, where x and y are any combinators:
Kxy = x
So, on the two inputs xy, K eliminates y. K can rightly be called the Elementary Right Eliminator.
And KI ?
Easy: KIxy = (KIx)y = Iy = y
So KI is the left eliminator. Applied on xy it gives y. It is the Elementary Left Eliminator.
What is the relation with programming? Well, it happens that K and KI makes it possible to implement in a incredible simple way the most important control structure in computer science: the IF...THEN...ELSE.
We want to implement IF A THEN B ELSE C; meaning that if the condition A is true, where A is represented by some combinator, then the combinator B will be trigged, and if A is false then C will be trigged.
For this we need to represent the constant boolean TRUE and FALSE, like in formal logic.
Well, everybody in the field agree that one of the nicer choice consists in representing TRUE by
K,
and FALSE by
KI.
Do you see why? Well, with that representation "IF A THEN B ELSE C" is represented simply by
ABC
Why? Because if A reduces (is equal) to TRUE, that is: is equal to K then
ABC = KBC = B, and so if A is true then B is trigged.
And if A is false, that is if A = KI, obviously
ABC = KIBC = C, so if A is false then C is trigged.
That will be useful when we will write program. For this we need to solve the last fixed point post question, and to define numbers with those birds ... (that means two posts. Perhaps one more on "curryfication"). Then we will come back to the measure problem, and the question of what is an "observer moment" to make (hopefully) more precise our older conversations.
No question so far? Someone asks me "is there a COMBINATORS VII". The answer is "no".
Bruno
http://iridia.ulb.ac.be/~marchal/
