  Drawing on various notions from theoretical computer science, we present a
novel numerical approach, motivated by the notion of algorithmic probability,
to the problem of approximating the Kolmogorov-Chaitin complexity of short
strings. The method is an alternative to the traditional lossless compression
algorithms, which it may complement, the two being serviceable for different
string lengths. We provide a thorough analysis for all $\sum_{n=1}^{11} 2^n$
binary strings of length $n<12$ and for most strings of length $12\leq n
\leq16$ by running all $\sim 2.5 \times 10^{13}$ Turing machines with 5 states
and 2 symbols ($8\times 22^9$ with reduction techniques) using the most
standard formalism of Turing machines, used in for example the Busy Beaver
problem. We address the question of stability and error estimation, the
sensitivity of the continued application of the method for wider coverage and
better accuracy, and provide statistical evidence suggesting robustness. As
with compression algorithms, this work promises to deliver a range of
applications, and to provide insight into the question of complexity
calculation of finite (and short) strings.
