Additive Cellular Automata with External Inputs

In this paper we consider a form of cellular automata (CA) that allows for external input at each stage of it s evolut ion. The invest igat ion is carr ied out from an algebra ic viewp oint whereby the state st ructure of the CA, together with it s action, can be int erpret ed rin g-theoretically. Specifically, we conside r the st ruct ure of a t tract ors for different cont rols and the associate d tr ee st ructures of the states that approach the at t rac tor . In parti cular, we relate the invari ant sets that arise for cont rolled and un controlled CA and give pr operti es on the number of st ates in the invariant set . Algebraic proper ti es of the CA rin g are used to qualify pr op erties of cycle lengt hs and number of cycles. A qualit ative equivalence for CA is introduced and equivalence of attra ctors is chara cterized by the cycle set structure for different inputs. Sufficient condit ions for qualitative equivalence in the presence of distinct inputs are found, and necessary and sufficient condit ions for qu alit ative equivalence in the presence of distinct inputs are given when an ext ra condit ion holds. Finally, the algebraic pr oper ti es of CA associate d wit h pe riodic input are investi gated , and some generalizat ions are discussed .