Arithmetic computation in the tile assembly model: Addition and multiplication

Formalized study of self-assembly has led to the definition of the tile assembly model Erik Winfree, Algorithmic self-assembly of DNA, Ph.D. Thesis, Caltech, Pasadena, CA, June 1998; Paul Rothemund, Erik Winfree, The program-size complexity of self-assembled squares, in: ACM Symposium on Theory of Computing, STOC02, Montreal, Quebec, Canada, 2001, pp. 459–468]. Research has identified two issues at the heart of self-assembling systems: the number of steps it takes for an assembly to complete, assuming maximum parallelism, and the minimal number of tiles necessary to assemble a shape. In this paper, I define the notion of a tile assembly system that computes a function, and tackle these issues for systems that compute the sum and product of two numbers. I demonstrate constructions of such systems with optimal Θ(1)$\Theta \left(1\right)$ distinct tile types and prove the assembly time is linear in the size of the input.