On the Equivalence of Cellular Automata and the Tile Assembly Model

Abstract

In this paper, we explore relationships between two models of systems which are governed by only the local interactions of large collections of simple components: cellular automata (CA) and the abstract Tile Assembly Model (aTAM). While sharing several similarities, the models have fundamental differences, most notably the dynamic nature of CA (in which every cell location is allowed to change state an infinite number of times) versus the static nature of the aTAM (in which tiles are static components that can never change or be removed once they attach to a growing assembly). We work with 2-dimensional systems in both models, and for our results we first define what it means for CA systems to simulate aTAM systems, and then for aTAM systems to simulate CA systems. We use notions of simulate which are similar to those used in the study of intrinsic universality since they are in some sense strict, but also intuitively natural notions of simulation. We then demonstrate a particular nondeterministic CA which can be configured so that it can simulate any arbitrary aTAM system, and finally an aTAM tile set which can be configured so that it can be used to simulate any arbitrary nondeterministic CA system which begins with a finite initial configuration

Local approach and the theory of lovozerite structures

“…This has led to the rapid development of theoretical models of self-assembly of complex structures, among which the abstract tile assembly model (ATAM), where the role of "tiles" is played by plane squares with labeled sides [4][5][6], has become highly popular. Another model of self-assembly of atomic-molecular structures is given by cellular automata (CAs) [7,8], whose structure is essentially identical to that of the ATAM [9]. The idea of constructing complex structures through relatively simple local interactions of particles is far from new in itself and was repeatedly suggested by crystallographers in one or another form.…”

Intrinsic universality and the computational power of self-assembly

“…As mentioned above, the Tile Assembly Model can be thought of as an asynchronous and nondeterministic cellular automaton (CA), that models the notion of a crystal growth frontier. Hendricks and Patitz [14] formally relate the abstract Tile Assembly Model and CA: they give a single CA that simulates any tile assembly system, as well as a single tile set that simulates any nondeterministic CA with a finite initial configuration. The methods of updating configurations in both models are quite different (CA are infinite, synchronous and deterministic, while tile assembly is finite, asynchronous and nondeterministic), and so their constructions need to handle this.…”

“…The topic of intrinsic universality, with its tight notion of simulation, has given rise to a rich theory in cellular automata [6,7,20,21,4,2] and Wang tiling [17,18]. This short survey attempts to show that we are beginning to see this in self-assembly too [11,12,8,9,19,13,14].…”

Dynamic Simulation of 1D Cellular Automata in the Active aTAM

“…16) In a stronger sense of simulation that is similar to intrinsic universality, Hendricks and Patitz designed an aTAM simulator for 2D nondeterministic CAs. 8) Due to the lack of a tile detachment mechanism in aTAM, these simulators increase assembled structures with each time step as the computation proceeds. For example, in aTAM, a simulation of the first t steps of a 1D CA starting from an input of length m needs an array of size O (( m + 2 t ) t ).…”