Dominoes is a popular and well-known game possibly dating back three millennia. Players are given a set of domino tiles, each with two labeled square faces, and take turns connecting them into a growing chain of dominoes by matching identical faces. We show that singleplayer dominoes is in P, while multiplayer dominoes is hard: when players cooperate, the game is NP-complete, and when players c...

We study some decision problems concerning the tiling of the plane with Wang tiles We present a proof for the undecidability of the tiling problem for the whole plane and also for the periodic tiling In these poofs an aperiodic tile set is constructed We are interested in the second part in the link between the cardinality of possible tilings of the plane with a given tile set and the existence...

Many complex signals, such as point distributions and textures, cannot efficiently be synthesized and stored. In this work we present tile-based methods to solve this problem. Instead of synthesizing a complex signal when needed, the signal is synthesized on forehand over a small set of tiles. Arbitrary large amounts of that signal can then efficiently be generated when needed by generating a s...

Reinforcement learning (RL) is a powerful abstraction of sequential decision making that has an established theoretical foundation and has proven effective in a variety of small, simulated domains. The success of RL on realworld problems with large, often continuous state and action spaces hinges on effective function approximation. Of the many function approximation schemes proposed, tile codi...

This paper presents a novel approach for the problem of generating tiled code for nested for-loops, transformed by a tiling transformation. Tiling or supernode transformation has been widely used to improve locality in multi-level memory hierarchies, as well as to efficiently execute loops onto parallel architectures. However, automatic code generation for tiled loops can be a very complex comp...

In this thesis we study self-assembly systems, in particular, we focus on the tile complexity of shapes. Using the standard tile assembly model proposed by Rothemund and Winfree, we give two lower bounds on the tile complexity of arbitrary shapes in terms of the radius and diameter of the shape. Applying our results to a square yields a partial answer to a problem of Rothemund and Winfree. We a...

Tiling is widely used by compilers and programmer to optimize scientific and engineering code for better performance. Many parallel programming languages support tile/tiling directly through first-class language constructs or library routines. However, the current OpenMP programming language is tile oblivious, although it is the de facto standard for writing parallel programs on shared memory s...

The limit set of a celullar automaton consists of all the configurations of the automaton that can appear after arbitrarily long computations. It is known that the limit set is never empty--it contains at least one homogeneous configuration. A CA is called nilpotent if its limit set contains just one configuration. The present work proves that it is algorithmically undecidable whether a given o...

This paper addresses the problem of compiling perfectly nested loops for multicomputers (distributed memory machines). The relatively high communication startup costs in these machines renders frequent communication very expensive. Motivated by this, we present a method of aggregating a number of loop iterations into tiles where the tiles execute atomically – a processor executing the iteration...

Majumder, Reif and Sahu presented in [11] a model of reversible, errorpermitting tile self-assembly, and showed that restricted classes of tile assembly systems achieved equilibrium in (expected) polynomial time. One open question they asked was how the model would change if it permitted multiple nucleation, i.e., independent groups of tiles growing before attaching to the original seed assembl...