February 20, 2017 Monday
Bedtime Story
Ackermann Function
Much to my surprise, it was that high school mathematics teacher
Wilhelm Ackermann who had first come up with a total computable function that
is not primitive recursive.
Now I have to tell you about primitive recursive functions.
Primitive recursive functions are a subclass of number-theoretic
functions.
These functions map natural numbers to natural numbers.
This essentially means that both the argument (input value) and
the function value (output value) of such a function would be a natural number.
So you see it that it is nothing great.
A simple function of addition would be primitive recursive.
So would be division, factorial and exponentiation.
If you ask why not subtraction and multiplication, this is simply
because subtraction is negative addition and multiplication is addition
multiple times.
Primitive recursive functions on the other hand are a subset of
total μ-recursive (partial) function.
I will not go very deep into this stuff since the point is to give
you a basic-level understanding of this concept.
Now let me show you what exactly Ackermann function is.
Ackerman denoted his function with the Greek letter phi or
.
It was a three-argument function stated as
(m,n,p).
The function was defined in a way such that for p with a value of
0, 1 and 2 it gave rise to the basic operations of additions, multiplication
and exponentiation.
Let us how he put it mathematically.
It is not so daunting even for an average ape like me so it is
worth writing them down:
With the values of p > 2, it becomes even more interesting.
Ackermann function with values 2 enters the realms of what in
mathematics is known as hyperoperations.
You can consider a hyperoperation as a way of compounding numbers
with a singular point being that the increase is dependent upon the iteration
of the previous hyperoperation.
So in that sense, even functions like successor, addition,
multiplication and exponentiation are also hyperoperations, successor operation
being the most primitive.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
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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