Rule matrices, degree vectors, and preimages for cellular automata.

摘要

Cellular automata are mathematical systems characterized by discreteness (in space, time, and state values), determinism, and local interaction. Few analytical techniques exist for such systems. The rule matrix and degree vectors of a cellular automaton -- both of which are determined a priori from the function defining the automaton, rather than a posteriori from simulations of its evolution -- are introduced here as tools for understanding certain qualitative features of automaton behavior. The rule matrix represents in convenient form the information contained in an automaton's rule table; the degree vectors are computed from the rule matrix, and reflect the extent to which the system is ''one-to-one'' versus ''many-to-one'' on restricted subspaces of the mapping. The rule matrix and degree vectors determine, for example, several aspects of the enumeration and ''prediction'' of preimages for spatial sequences evolving under the rule, where the preimages of a sequence S are defined to be the set of sequences mapped by the automaton rule onto S. 2 figs., 2 tabs.