Scaling of Preimages In Cellular Automata

Abst ract. A ce llula r auto ma to n con sists of a lattice of sites whose values evo lve deter m inistically accord ing to a lo cal int eract ion ruJe. For a given ru le and ar bit ra ry sp atia l seq uence, t he prei rnage of th e seq ue nce is defined to be t he se t of t u ples tha t a re map ped by t he rule o nto t he seq ue nce. Recu rrence rela tio ns a re pro vided t hat ex press t he nu mber of pre images for a gener al spatial seq ue nce in te rms of t he number of pre im ages for its subseq uences. T hese rela t ion s ar e a pp lied to t he analy sis of a quantity O'n defined as t he tota l number of pre image s for spatial seq ue nces of length n whose probab ilit ies of occ urrence a re in a ce rtai n sense m inimal aft er one ite ration of t he rul e. In par tic-ular , the recurrence relations a re used to cha racte rize a utomata rules by parameters representin g th e a mou nt of info rma t ion abo ut a n a r-bitrary spa tia l seque nce needed to det erm ine th e values t ha t, when ap pende d to t he seque nce , minimi ze t he number of preima ges. O n t he basis of t his cha racte riza t ion, it is proved th at , for all nea rest-neighbo r automata rules on infinite latt ices, t he quantity ern sca les exactly wit h t he lengt h n of the spatial seq uence ; that is, ern = C. 2 n for n sufficiently lar ge , where c is a consta nt depe nd ing o n t he ru le. A sy mme t r ic result holds in t he case of max imal preima ges.