Complexes of on-line self assembly

The Tile Assembly Model (TAM) is a mathematical model of nanoscale self-assembly. In this paper we this model to define an on-line self assembly models called Fair Online Assembly(FOAF) and its variation called the Bounded Fair Online Assembly (FOAB). We show that these two models are not equivalent to each other. We also introduce the concepts of Binary and Trinary Complexes for a Tile Assembly System (TAS) and show if the complexes have a special property (called Frontier Turn Off Point-FTP) then the corresponding Self Assemblies are FOAF. Finally we argue that FOAF, FOAB and the size of the T-frontier at the frontier turn off point may be used to measure the complexity of the TAS.