Efficient 3-SAT algorithms in the tile assembly model

Abstract

Self-assembly is a powerful process found in nature that guides simple objects assembling, on their own, into complex structures. Self-assembly is of interest to computer scientists because self-assembling systems can compute functions, assemble shapes, and guide distributed robotics systems. The tile assembly model is a formal mathematical model of self-assembly that allows the study of time and space complexities of self-assembling systems that lie at the heart of several molecular computer implementations and distributed computational software systems. These implementations and systems require efficient tile systems with small tilesets and fast execution times. The state of the art, however, requires vastly complex tile systems with large tilesets to implement fast algorithms. In this paper, I present S FS ; a tile system that decides 3-SAT by creating O H ð1:8393 n Þ nondeterministic assemblies in parallel, improving on the previous best known solution that requires Hð2 n Þ such assemblies. This solution directly improves the feasibility of building molecular 3-SAT solvers and efficiency of distributed software. I formally prove the correctness of the system, the number of required parallel assemblies, that the size of the system's tileset is 147 ¼ Hð1Þ; and that the assembly time is nondeterministic linear in the size of the input.

Entrusting Private Computation and Data to Untrusted Networks

“…We have developed multiple tile assemblies, specifically for sTile. These assemblies solve NP-complete problems [10], [11], [12] and factor integers [9]. In this paper, we use one of those assemblies to demonstrate how sTile can solve 3-SAT.…”

“…Tile assemblies are extensively studied mathematical objects [1], [6], [37], [41], [44]. Our own prior work has developed the notion of efficient computation with tile assemblies and constructed efficient assemblies to add and multiply integers [8], factor integers [9], and solve NP-complete problems [10], [11], [12]. More generally, the tile assembly model is Turing universal [6], [37].…”

“…It is possible to leverage more efficient algorithms that solve NP-complete problems to develop efficient tile assemblies. We have already designed one such assembly that solves 3-SAT in Oð1:8394 n Þ time, but do not describe it here because of its complexity [12]. We mention it here to make clear that tile assemblies can implement the same algorithms used on today's fastest systems that solve NP-complete problems, such as SAT solvers.…”

“…The compilation reduces [39] the problem for which the developer wishes to build a sTile-based system to a problem with a known tile assembly [10], [11], [12]. This compilation, as virtually all compilations, may result in less efficient systems than the direct approach of developing a tile assembly for each particular problem.…”

“…This analysis is specific to 3-SAT, but the running times for other NPcomplete problems will be very similar, since the fastest growing factor of 2 n will be the same. (Note that our analysis here assumes the naïve algorithm that runs in Oð2 n Þ time, but can be extended to more efficient algorithms, such as those used in today's SAT solvers [12]. We discuss the reasoning and implications for our assumption further in Section 4.6.…”