Abstract
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 functionfofNvariables: the implementation costC(f), the randomnessR(f), the deterministic entropyH(f), and the complexityK(f). The highlights and main results are roughly as follows,l) C(f) approx R(f) H(f) approx 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 offsaves a lot in the iow entropy case but saves nothing in the high-entropy case. Iffis partially structured, the implementation cost is reduced substantially.
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