LST for One-dimensional Lattices

The A -order LST is defined by the same infinite system of equations appearing in equation 5.66, but with = rn -order Bayesian extended probability 

The LST approximation itself resides solely in the determination of A l4( B +2r)( 0 the general combinatorial structure of the CA rule is retained (for blocks of size A or less) by the function 6 ( (B ),B) (defined in equation 5.67). 

The general LST aJgorithni for approximating the temporal evolution of A-block measures using An ) is therefore as follows  

Obtain the length (A - - 2r) inverse block images of each possible length A block under the range-r rule.  

Find the Bayesian extended set of probabilities of each of the inverse (A - -2r)-block images of all possible A-blocks. 

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The essence of the LST for one-dimensional lattices resides in the fact that an operator TtN->N+i could be constructed (equation 5.71), mapping iV-block probability functions to [N -f l)-block probabilities in a manner which satisfies the Kolmogorov consistency conditions (equation 5.68). A sequence of repeated applications of this operator allows us to define a set of Bayesian extended probability functions Pm, M > N, and thus a shift-invariant measure on the set of all one-dimensional configurations, F. Unfortunately, a simple generalization of this procedure to lattices with more than one dimension, does not, in general, produce a set of consistent block probability functions. Extensions must instead be made by using some other, approximate, method. We briefly sketch a heuristic outline of one approach below (details are worked out in [guto87b]). 

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