Intrinsic universality in tile self-assembly requires cooperation

We prove a negative result on the power of a model of algorithmic self-assembly for which it has been notoriously difficult to find general techniques and results. Specifically, we prove that Winfree’s abstract Tile Assembly Model, when restricted to use noncooperative tile binding, is not intrinsically universal. This stands in stark contrast to the recent result that, via cooperative binding, the abstract Tile Assembly Model is indeed intrinsically universal. Noncooperative self-assembly, also known as “temperature 1”, is where tiles bind to each other if they match on one or more sides, whereas cooperative binding requires binding on multiple sides. Our result shows that the change from singleto multi-sided binding qualitatively improves the kinds of dynamics and behavior that these models of nanoscale self-assembly are capable of. Our lower bound on simulation power holds in both two and three dimensions; the latter being quite surprising given that three-dimensional noncooperative tile assembly systems simulate Turing machines. On the positive side, we exhibit a three-dimensional noncooperative self-assembly tile set capable of simulating any two-dimensional noncooperative self-assembly system. Our negative result can be interpreted to mean that Turing universal algorithmic behavior in self-assembly does not imply the ability to simulate arbitrary algorithmic self-assembly processes. ∗LAMA, Université de Savoie. pierreetienne.meunier@univ-savoie.fr. †Department of Computer Science and Computer Engineering, University of Arkansas, patitz@uark.edu Supported in part by National Science Foundation Grant CCF-1117672. ‡Department of Computer Science and Software Engineering, University of Wisconsin– Platteville, Platteville, WI 53818, USA. summerss@uwplatt.edu. §LAMA, CNRS and Université de Savoie, guillaume.theyssier@univ-savoie.fr Supported in part by grant ’Agence Nationale de la Recherche ANR-09-BLAN-0164’ ¶Department of Computer Science, Tufts University, awinslow@cs.tufts.edu. Supported in part by National Science Foundation grants CCF-0830734 and CBET-0941538. ‖Computer Science, California Institute of Technology, woods@caltech.edu. Supported by National Science Foundation grants 0832824 (The Molecular Programming Project), CCF-1219274, and CCF-1162589.