Computing in Finite Time

AbstractIf a message can have n different values and all values are equally probable, then the entropy of the message is log(n). In the present paper, we discuss the expectation value of the entropy, for an arbitrary probability distribution. We introduce a mixture of all possible probability distributions. We assume that the mixing function is uniform either in flat probability space, i.e. the unitary n-dimensional hypertriangleor in Bhattacharyya’s spherical statistical space, i.e. the unitary n-dimensional hyperoctant. A computation is a manipulation of an incoming message, i.e. a mapping in probability space: either a reversible mapping, i.e. a symmetry operation (rotation or reflection) in n-dimen sional spaceor an irreversible mapping, i.e. a projection operation from n-dimensional to lower-dimensional space. During a reversible computation, no isentropic path in the probability space can be found. Therefore we have to conclude that a computation cannot be represented by a message which merely follows a path in n-dimensional probability space. Rather, the point representing the mixing function travels along a path in an infinite-dimensional Hilbert space.