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More refined theories

Dresden and Wong[35] showed how to write the rule of evolution of Conway's Life in an algebraic from to which the rules for combining probability distribution functions could be applied. Schulman and Seiden[102] proceeded to obtain an explicit form for the evolution of probability, finding the cummulants hard to deal with, but nevertheless worked out an approximation and applied it to Life.

Wilbur, Lipman, and Shamma[116] decided instead to work with the probabilities of chains or blocks of cells rather than individual cells, including a survey of the self-consistent probabilities of triples of cells according to Wolfram's thirty two ``legal'' automata in their article. They obtain self-consistency from estimating the probabilities of the different ancestors in the reduced evolution matrix. They use the the same chains they are studying to estimate the probabilities of the extensions required as ancestors, treating the extensions as though they were a Markov process.

That is, starting with n-block probabilities, they use the quotient

as the probability that the -block chain x can be extended to the n-block chain ax; in terms of the notation we have already introduced, they assume A similar result, is supposed to hold for right extensions. Their working hypothesis is that the same relation serves to extend an n-block probability to an -block probability; making two extensions, one on each side, to get the ancestor of a given block, their equations for self-consistent block probabilities read

All terms refer to n-block probabilities or their sums.

Harold V. McIntosh