Causal graph dynamics

Abstract

International audienceWe extend the theory of cellular automata to arbitrary, time-varying graphs. In other words we formalise, and prove theorems about, the intuitive idea of a labelled graph which evolves in time -- but under the natural constraint that information can only ever be transmitted at a bounded speed, with respect to the distance given by the graph. The notion of translation-invariance is also generalised. The definition we provide for these 'causal graph dynamics' is simple and axiomatic. The theorems we provide also show that it is robust. For instance, causal graph dynamics are stable under composition and under restriction to radius one. In the finite case some fundamental facts of cellular automata theory carry through: causal graph dynamics admit a characterisation as continuous functions, and they are stable under inversion. The provided examples suggest a wide range of applications of this mathematical object, from complex systems science to theoretical physics

Cellular automata over generalized Cayley graphs

“…The above proof was done via the axiomatic characterization of causal dynamics, as this paper enjoys a more straightforward formalization than [1]. In [1] the same result is proven via the constructive approach to causal graph dynamics (localizability), which has the advantage of extra information about the composed function.…”

“…Definitions 1 to 4 are as in [1]. The first two are reminiscent of the many papers seeking to generalize Cellular Automata to arbitrary, bounded degree, fixed graphs [24,7,16,15,12,31,20,30,29,5,22,9,27,28].…”

“…The next two definitions are standard, see [5,22] and [1], although here again the vertices of G are given names in disjoint subsets of V (X).S for some X. Basically, we need a notion of union of graphs, and for this purpose we need a notion of consistency between the operands of the union: Definition 19 (Consistency).…”

“…For Cellular Automata over Cayley graphs a complete reference is [6]. For Causal Graph Dynamics [1], these implications had to be reproven by hand, due to the lack of a clear topology in the set of graphs that was considered. Here we are able rely on the topology of generalized Cayley graphs and reuse Heine's Theorem out-of-the-box, which makes the setting of generalized Cayley graphs a very attractive one in order to generalize CA.…”

“…The work [1] by Dowek and one of the authors already achieves an extension of Cellular Automata to arbitrary, bounded degree, timevarying graphs, also through a notion of continuity, with the same motivations. However, graphs in [1] lack a compact metric over graphs, which is left as an open question. As a consequence all the necessary facts about the topology of Cayley graphs get reproven.…”