Reducing Tileset Size: 3-SAT and Beyond

In self-assembly research, reducing the number of distinct tiles necessary to compute functions can make it feasible to implement tile systems to solve complex problems. Existing methods for solving 3-SAT , a well-known NP-complete problem, in the tile assembly model involve either using Θ(n) distinct tiles to nondeterministically decide whether an n-variable Boolean formula is satisfiable or simulating a cellular automata system simulating a Turing machine, which requires a constant but large number of tiles to deterministically make the decision. Here, I propose a system that solves k-SAT nondeterministically, for all k ∈ N, in time linear in the input size using only 64 distinct tiles. Further, I propose a mechanism for converting the tilesets of tile systems for many NP-complete and other problems from tilesets whose size depends on the input into Θ(1)-size tilesets.