Elements of a theory of simulation II: sequential dynamical systems

We study a class of discrete dynamical systems that is motivated by the generic structure of simulations. The systems consist of the following data: (a) a ®nite graph Y with vertex set f1; . . . ; ng where each vertex has a binary state, (b) functions Fi : F2 ! F2 and (c) an update ordering p. The functions Fi update the binary state of vertex i as a function of the state of vertex i and its Y-neighbors and leave the states of all other vertices ®xed. The update ordering is a permutation of the Y-vertices. By composing the functions Fi in the order given by p one obtains the sequential dynamical system (SDS): ‰FY ; pŠ ˆ Yn iˆ1 Fp…i† : F2 ! F2: We derive a decomposition result, characterize invertible SDS and study ®xed points. In particular we analyse how many di€erent SDS that can be obtained by reordering a given multiset of update functions and give a criterion for when one can derive concentration results on this number. Finally, some speci®c SDS are investigated. Ó 2000 Elsevier Science Inc. All rights reserved.