Saturday, June 11, 2016

What kind of strings have Kolmogorov complexity equal to their length ?

I have been doing some reading about Information Theory and Kolmogorov complexity lately. I just thought about the following problem. Consider a language of all C programs (strings) which print themselves, formally Let $$L_{self} = \{ w\,\, | w\,\, \mbox{is a C program which prints itself} \}$$.
Following is an example of a C program which prints itself (on a side-note the trick behind the construction is to use the ASCII values for quotes and newlines). Clearly the Kolmogorov complexity of each w is less than or equal to the length of w -- given that we have constructed a program of length same as the string w.
I'm wondering if this is indeed optimal ? -- that is given a w, if its possible to produce a program to generate w and size strictly less than |w|.

char qt=34;
char nl=10;
char *str="char qt=34;%cchar nl=10;%cchar *str=%c%s%c;%cint main() {%c printf(str,nl,nl,qt,str,qt,nl,nl,nl,nl);%c}%c";
int main() {

BTW, compile the program with
gcc -w 
to suppress warnings about header inclusion.