October 16, 2017 Monday
Bedtime Story
The Five Combinators I, K, S, and B and C
Last night I had told you that we will
consider the few types of combinators that exist in combinatory logic.
The first one is the I combinator or the
Identity combinator
It is written as
(I x) = x for all terms x
The second is the K combinator or the
constant combinator generates constant functions.
It is written (K x) and is such a function
that no matter what argument is fed into it, it will come out as x.
This is written as
((K x) y) = x for all terms x and y
Then there is the S combinator or the
Substitution combinator
First we will see how it is written and
then we will analyze what the combinator does.
(S x y z) = (x z(y z))
S combinator applies x to y after
substituting z for each of them.
Perhaps that may not make sense.
So let me clarify that for you.
Another way of putting the function of S
combinator would be to say that S applies x to y in the environment of z.
So now we have the three combinators I, K
and S.
It can be shown that I combinator can be
derived from K and S.
You can see how it can be done.
((S K K) x)
= (S K K x)
= (K x (K x))
= x
Since using just the combinators S and K we
managed to get x which was earlier defined by I, I can be defined using S and
K.
So ((S K K) x) = (I x) for any x.
There are two more combinators I would like
to introduce here as they are necessary from our story point of view.
These two combinators are B and C and
Schönfinkel introduced them in his paper published in 1924 at Gottingen that
was later discovered by Haskell Curry in 1927 at IAS Princeton.
The B and C combinators were shown with the
following reductions:
(B f g x) = (f (g x))
(C f x y) = (f y x)
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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