February 19, 2017 Sunday
Bedtime Story
Computable Functions
A computable function is that wherein a well-defined algorithm
exists that can with precisely defined set of steps that can compute the output
of a function from all the elements of its input.
In even more simple terms, if there exists a finite procedure or
an algorithm to compute a given function, then that function is computable.
Herbert Enderton, a professor of mathematics at UCLA (now
deceased) has given the following three characteristics of the procedure or the
algorithm that applied to computable functions:
1. The algorithm or the procedure should work for a large number
of arguments (in the mathematical sense). (There is no limit to the number of
arguments that is applicable).
2. The procedure is supposed to halt after a finite number of
steps to produce an output. (Again, there is no limit to the number of steps
needed to produce that output nor any time constraint).
3. There is no limit to the amount of storage space that could be
needed to compute a function.
Besides these 3, there are some other interesting caveats about
computable functions.
If the input argument does not happen to fall in the domain of the
function, then it never halts or it gets stuck.
There is no need for any device that is carrying out the
computable function to check the correctness of the result since it is assumed
that if there is an output, it must be correct as the sanctity of the algorithm
is never a suspect.
Mathematicians and logicians have their own technical way of
writing down a computable function.
It is interesting and not complicated and hence it is worth
mentioning it.
A function f:N
k→N is computable if and only
if there is an effective procedure that, given any k-tuple x of natural
numbers, will produce the value f(x).
Now I will need to tell you about tuple.
In mathematics, a tuple is a list of finite ordered (in sequence)
elements.
A n-tuple is an ordered list of n elements where n is a
non-negative integer.
So a 0-tuple is an empty on a null tuple.
A sequence such as (5, 10, 15, 20, 25) would be a 5-tuple.
So now let us go back and search for the first such computable
function.
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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