Behavior of Topological Cellular Automata

We introduce a new type of cellular automaton, one in which the link structure is dynamically coupled to the site values. The automaton structures are altered using simple Boolean rules, while the sites themselves are assigned values based on the mod 2 rule. We compare these dynamics to those in which the link structure is altered randomly and find th at in th e former case stable structures of noninteger dimensionality emerge. Fully exploring this model, we observe the effects of value rule alteration , initial lat tice structure alter ation , and alteration in the initi al value seeding and observe patterns of selforganization, growth, decay, and periodicity. Finally, we comment on the relationship between this model and randomly generate d Kauffman nets . 1. Introduct ion In recent years it has been shown that cellular automat a prov ide means of modelling a wide range of phys ical systems [1]. Typically, automaton dynamics are determined by an initial value seeding (with site valu es sp ecified from a certain range) and a set of simple, local transit ion rul es. This behavior takes place within a fixed lat ti ce st ructure (e.g., each site is linked to two, four, six, or more neighbors) . Ilachinski has point ed out the limi tations of these structurally static automata and has suggested a scheme for structurally dyn amic models [2]. Here, the lat t ice structures are dynamically coupled to the local sit e value configurations. A preliminary st udy of some of these models was completed by Ilachinski and Halpern , yield ing evid ence of a wide range of behav ior [3]. In this paper we wish to explore a new sort of topological automaton (TA) model, one in which lattice dynamics are det ermined by simple Boolean ru les. In this approach, both the sit e values and th e underlying lattice structures are treated in a simi lar manner , creating a nonlinear "feedback" mechanism determining future automata st ates. Thus, one can use topological auto mata to search for geomet ric self-organization . © 1990 Complex Systems Publications, Inc. 624 Paul Halpern and Gaetano Caltagirone Alth ough we wish to present a purely mathematical mod el, possible physical applications for TAs are numerous. These include schemes for crystal growth , pattern formation, and types of neural network mod els. While one can readil y model crystal growth using convent ional "value" cellular automata, topological automata can more precisely define an amorphous crystal structure determined by local interactions. In addition, lat tice gas models might be const ructed by use of topological automata. In particular , schemes might be considered in which t he value st ructure and link structure of a topological auto maton represent two different interactin g subst ances [4J. Topological automata may also provide a means of describing th e "geometric exciton" program of Wheeler [5], in which all particl es are considered to be geometric dist urbances, with space itself seen as emerging from a "pregeometry" of indetermin ate dimensionality. Since "value st ructure" solitons have been found in convent ional cellular automaton mod els [6], "link st ructure" solitons might emerge in a TA scheme for geometrodynamics. TA models could then describe the generation, transmission, and int eracti on of topological disturbances. Finally, there appears to be some relat ionship between TAs and Kauffman nets used in evolut ionary genet ic theory [7] . We shall briefly comment on this lat ter applicat ion in our conclusion. Let us now form ally define a Boolean topological automaton. Consider a network of N sites. Each of these sites may have value 0 or 1. Therefore at a given t ime t we define th e value st ructure of an automaton by the matri x vt (i = 1, N) . We furt her define the link struct ure by the connect ivity matrix Itj (i = 1, N ; j = 1, N ). Then two sit es i , j can be described as linked if and only if Itj = 1 (oth erwise I;j = 0) . In this case the sites can be said to be neighbors . We may also define the nextnearest neighbors of each sit e by use of the second ary connectivity matrix m;j. T wo site s i , j are defined as next-nearest neighbors if and only mlj = 1 (otherwise mlj = 0). The matrix mlj is completely determined by I;j in the following manner: