The complexity of information extraction

How difficult are decision problems based on natural data, such as pattern recognition? To answer this question, decision problems are characterized by introducing four measures defined on a Boolean function f of N variables: the implementation cost C(f), the randomness R(f), the deterministic entropy H(f), and the complexity K(f). The highlights and main results are roughly as follows. 1) C(f) = R(f) = H( f ) = K( f ), all measured in bits. 2) Decision problems based on natural data are partially random (in the Kolmogorov sense) and have low entropy with respect to their dimensionality, and the relations between the four measures translate to lower and upper bounds on the cost of solving these problems. 3) Allowing small errors in the implementation of f saves a lot in the low entropy case but saves nothing in the high-entropy case. If f is partially structured, the implementation cost is reduced substantially.