The emergence of global properties from local interactions: static properties and one-dimensional patterns

Groups of simple objects, interacting under local rules, can produce global properties of great complexity and rich functionality. In this paper, we address of whether a given global property can emerge from purely local interactions. We provide a necessary condition for the case of static properties, and provide complementary sufficient conditions for patterns on the special case of one-dimensional lattices. This work shows that two general considerations, local checkability and symmetry, are the main determinants of "emergeability," at least in these special cases. This work also shows that relatively unified constructions provide solutions to large classes of patterns.