OnSomeOpenProblems intheTheory ofCellular Automata

thevector addition ofxandti. A stepofcomputation consists ofastate transformation ofeachcell ofthen-D space. Atanyinstant, thestates ofallthecells givethe configuration. Thatis, aconfiguration ofann-Dautomaton withstate setSisgiven byafunction f:In-*S,where f(z) = qsignifies thatthecell zisinstate q.Theconfiguration before thefirst step iscalled theinitial orinput configuration. Initial configurations aredenoted byzero subscripts andconfigurations after k-steps aredenoted by k-subscripts. We adoptthefollowing usualconventions: p = magnitude ofp,LPJ= thelargest integer nolarger than p,[pl= theleast integer noless thanp.Thedistance between(ai,,a2, **-,a.) and(bl,b2, * - *,bn) along theith axisisai- bi1.x+ yisthevector addition ofxand y,andnx= (n- 1)x+ x. Forcompleteness wealsodefine thestandard Jkand Hikneighborhoods.