Effective complexity measures the information content of the regularities of
an object. It has been introduced by M. Gell-Mann and S. Lloyd to avoid some of
the disadvantages of Kolmogorov complexity, also known as algorithmic
information content. In this paper, we give a precise formal definition of
effective complexity and rigorous proofs of its basic properties. In
particular, we show that incompressible binary strings are effectively simple,
and we prove the existence of strings that have effective complexity close to
their lengths. Furthermore, we show that effective complexity is related to
Bennett's logical depth: If the effective complexity of a string x exceeds a
certain explicit threshold then that string must have astronomically large
depth; otherwise, the depth can be arbitrarily small.Comment: 14 pages, 2 figure