Thermodynamically Favorable Computation via Tile Self-assembly

The recently introduced Thermodynamic Binding Networks (TBN) model was developed with the purpose of studying self-assembling systems by focusing on their thermodynamically favorable final states, and ignoring the kinetic pathways through which they evolve. The model was intentionally developed to abstract away not only the notion of time, but also the constraints of geometry. Collections of monomers with binding domains which allow them to form polymers via complementary bonds are analyzed to determine their final, stable configurations, which are those which maximize the number of bonds formed (i.e. enthalpy) and the number of independent components (i.e. entropy). In this paper, we first develop a definition of what it means for a TBN to perform a computation, and then present a set of constructions which are capable of performing computations by simulating the behaviors of space-bounded Turing machines and boolean circuits. In contrast to previous TBN results, these constructions are robust to great variability in the counts of monomers existing in the systems and the numbers of polymers that form in parallel. Although the Turing machine simulating TBNs are inefficient in terms of the numbers of unique monomer types required, as compared to algorithmic self-assembling systems in the abstract Tile Assembly Model (aTAM), we then show that a general strategy of porting those aTAM system designs to TBNs produces TBNs which incorrectly simulate computations. Finally, we present a refinement of the TBN model which we call the Geometric Thermodynamic Binding Networks (GTBN) model in which monomers are defined with rigid geometries and form rigid bonds. ∗Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX, USA [email protected] This author’s research was supported in part by National Science Foundation Grants CCF-1618895 and CCF-1652824. †Department of Computer Science and Information Systems, University of Wisconsin River Falls, River Falls, WI, USA [email protected] ‡Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA [email protected] This author’s research was supported in part by National Science Foundation Grants CCF-1422152 and CAREER-1553166. §Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA. [email protected]. This author’s research was supported in part by National Science Foundation Grants CCF-1422152 and CAREER-1553166. 1 ar X iv :1 80 2. 02 68 6v 1 [ cs .E T ] 8 F eb 2 01 8 Utilizing the constraints imposed by geometry, we then provide a GTBN construction capable of simulating Turing machines as efficiently as in the aTAM.