TITLE: Is it possible to create a string with known Kolmogorov Complexity?
QUESTION [5 upvotes]: I wish to compare compressors using strings with known Kolmogorov Complexity, but I haven't got the theoretical background and tools to understand how to do that. I'm just starting in this area and any pointer to papers/textbooks is highly appreciated.
I imagine that to "tune" the KC of a string with length $n$, one has to use some random source, otherwise the same program used to generate several strings could be used as the measure of complexity for those strings. I idealized a state machine with two states (A and B, starting state = A) and a transition probability $p$ that would produce the following strings with different complexities for different $p$:
p = 0.0:    AAAAAAAAAAAAAAAAAAAAAAAAAA... 
KC ~ 1:     = A * n

p = 0.1:    AAAAAABBBBBAAAAAAAAAAAABBB... 
KC ~ n/10?: = (A * 6) ++ (B * 5) ++ (A * 12) ++ ...

p = 0.5:    AABBABABBBBBBAAABBABBBABBBA...
KC = n?

p = 0.9:    ABABAABABABABABBABABABABAA... 
KC ~ n/10?: = (AB * 2) ++ A ++ (AB * 5) ++ B ++ (AB * 4) ++ ...

p = 1.0:    ABABABABABABAABABABABABABA...
KC ~ 1:     = AB * (n/2)

However, this look like more some measure of entropy than proper KC. Are they equivalent for randomized processes?

REPLY [4 votes]: Is it possible to create a string with known Kolmogorov Complexity?

There cannot be a universal computable prefix-free function $U$ with that property that there is an r.e. sequence $\sigma_n$ of strings with $K_U(\sigma_n) = n$ for all $n$. So, in that sense, we cannot produce the desired string effectively.
Proof: Consider a computable prefix-free function $f$ such that $f(1^n0) = \sigma_{2^n}$ for each $n$. Then by universality of $U$ there is a fixed $w$ such that, for all $n$, $U(w1^n0) = \sigma_{2^n}$. Choosing $N$ large enough, $|w1^N0| = |w| + N + 1 < 2^N$, so $K_U(\sigma_{2^N}) < 2^N$, contradiction. 
This generalizes to show it is impossible to have an r.e. sequence of pairs $(\sigma_i, n_i)$ with $K_U(\sigma_i) = n_i$ such that the set of $n_i$ that appears in the sequence is unbounded. 
It is possible for an r.e. sequence of strings to contain strings of arbitrarily high complexity - the sequence that just enumerates all the strings must have that property - but it is not possible to correctly guess (in an r.e. way) the exactly complexities of a sequence of strings with arbitrarily high complexities. 
This idea is at the root of the Chaitin incompleteness theorem. The additional idea there is that if we have an effective theory of arithmetic, we can enumerate all the proofs of statements of the form "$K_U(\sigma) = n$" to form an r.e. sequence of pairs $(\sigma, n)$. Thus: either the system proves some statements of that form that are false, or there is an $M$ such that $n<M$ in every statement of that form that is provable in the system. This applies to any fixed effective axiomatic system that includes statements that can be interpreted as "$K_U(\sigma) = n$".