In lectures, my professor discussed Kolmogorov complexity for 10 minutes but I have too many questions opened.
My professor claimed (and I was able to prove it myself) that $|K(X)| \leq |x|+1$. But why it's not equal to it? For example given $100100100100$ how can we build Turing Machine with $n$ states s.t $n < 13$?
My professor claimed there is no Turing Machine that calculates Kolmogorov complexity for every input $x$ (and I was able to prove this one too). But, he also claimed that there is a Turing Machine which calculates Kolmogorov complexity for a specific input $x$. Why is this correct at all? How the machine would work? for example what is $K(100100100100)$ from above? It's a mysterious value so how can the machine calculate it?