First note that we are not allowed to write an
infinitely long program.
Since we can only assign constants
it must be that 's' can only take on a finite set of
distinct values (less than the number of statements
in the code for example) ....... let us say N.
Recall that we cannot look ahead or
back up on
input ... So, we must read all the a's before reading
the first b. Then we read all the b's.
Now suppose there are K different
in the code containing a 'getchar()'
When a 'b' is finally read there are only NK possibilities:
One for each value of 's' and 'getchar()' combination.
But these are the only means of knowing
'a's were read (i.e. they are the only things that distinguish
one run of a program from another on different strings).
Since there can be many more than NK
a's in the string,
the program has no way of knowing how many b's it should see.
(pigeon hole principle)
There is a formal method of proof called
the pumping lemma
which can be used in more general contexts
and proves stronger results.