Structural aspects of tilings

In this paper, we study the structure of the set of tilings produced by any given tile-set. For better understanding this structure, we address the set of finite patterns that each tiling contains. This set of patterns can be analyzed in two different contexts: the first one is combinatorial and the other topological. These two approaches have independent merits and, once combined, provide somehow surprising results. The particular case where the set of produced tilings is countable is deeply investigated while we prove that the uncountable case may have a completely different structure. We introduce a pattern preorder and also make use of Cantor-Bendixson rank. Our first main result is that a tile-set that produces only periodic tilings produces only a finite number of them. Our second main result exhibits a tiling with exactly one vector of periodicity in the countable case.Comment: 11 page

An Optimal Algorithm for Tiling the Plane with a Translated Polyomino

We give a $O(n)$-time algorithm for determining whether translations of a polyomino with $n$ edges can tile the plane. The algorithm is also a $O(n)$-time algorithm for enumerating all such tilings that are also regular, and we prove that at most $\Theta(n)$ such tilings exist.Comment: In proceedings of ISAAC 201

On Derivatives and Subpattern Orders of Countable Subshifts

We study the computational and structural aspects of countable two-dimensional SFTs and other subshifts. Our main focus is on the topological derivatives and subpattern posets of these objects, and our main results are constructions of two-dimensional countable subshifts with interesting properties. We present an SFT whose iterated derivatives are maximally complex from the computational point of view, a sofic shift whose subpattern poset contains an infinite descending chain, a family of SFTs whose finite subpattern posets contain arbitrary finite posets, and a natural example of an SFT with infinite Cantor-Bendixon rank.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

Experimental Progress in Computation by Self-Assembly of DNA Tilings

Approaches to DNA-based computing by self-assembly require the use of D. T A nanostructures, called tiles, that have efficient chemistries, expressive computational power: and convenient input and output (I/O) mechanisms. We have designed two new classes of DNA tiles: TAO and TAE, both of which contain three double-helices linked by strand exchange. Structural analysis of a TAO molecule has shown that the molecule assembles efficiently from its four component strands. Here we demonstrate a novel method for I/O whereby multiple tiles assemble around a single-stranded (input) scaffold strand. Computation by tiling theoretically results in the formation of structures that contain single-stranded (output) reported strands, which can then be isolated for subsequent steps of computation if necessary. We illustrate the advantages of TAO and TAE designs by detailing two examples of massively parallel arithmetic: construction of complete XOR and addition tables by linear assemblies of DNA tiles. The three helix structures provide flexibility for topological routing of strands in the computation: allowing the implementation of string tile models