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The Hartree-Fock-like assumption is to define
both these quantities are positive numbers less than 1; if any numerator is zero, the entire fraction is taken to be zero, even in those cases where the denominator might also be zero.
In fact and are probabilities which can be used in various ways, such as constructing the Markov matrix for spatial correlations; in an -stage de Bruijn diagram they give the relative probabilities for the different entering or emerging symbol during a shift.
The self-consistency equations can now be written
which presents the appearance of being a system of linear equations if one overlooks the fact that the 's and 's are not constants, but depend upon the very same unknown probabilities for which one is solving. The function is a set-theoretic Kronecker delta, 1 when its arguments coincide, zero when they do not.
The inner sum defines a matrix (writing all indices as arguments, not subscripts):
mapping one vector of probabilities for the -blocks into another.
For symmetry we have linearized the equations of Gutowitz et.al. by taking p(WXY) as the vector component, but we could just as easily have taken one of the other numerator terms; the Kolmogorov conditions are flexible enough to permit and to retain the same form in either case. Should the occasion arise to do so, we could distinguish the three different definitions of by the adjectives left, central, or right.
To implement the theory of Gutowitz et.al. , recall the reduced evolution matrix . For n-chains of cells i and j, it is defined by
The probabilistic version of E is just defined above, at least for one particular way of estimating the probabilities of ancestors. In any event, many essential properties of are determined by E, since both matrices have the same block diagonal structure, no matter whether the nonzero matrix elements are determined self-consistently from the nonlinear local structure theory or otherwise. , which is always fairly crowded for low values of n, becomes sparser and sparser as n increases. One hopes that it tends toward a stable form which could be described analytically; and which might also describe the full matrix .