ABSTRACT
Theoretically, the distinctive characteristics of cellular automata pose a number of challenging problems in the analysis of discrete dynamical systems. Practically, as the evidence of lattice gases indicates, cellular automata-based algorithms may provide alternative and efficient techniques for solving certain partial differential equations. Problems in cellular automata research pose difficulties since they often fall outside the purview of traditional continuous mathematics; cellular automata problems typically reflect, in both their formulation and their solution, the features of discreteness and local interaction that make these systems distinctive. The analysis of preimages will depend upon tools known as the "rule matrix" and the "degree vectors" of an automaton. The rule matrix represents the rule table for the automaton in a convenient matrix form; increasing powers of the matrix capture the effect of evolution under the rule table for spatial sequences of increasing length. The chapter explores the implications of degree vectors for a particular "forecasting" problem for spatial sequences generated by the rule.
