Preface

Nataša Jonoska, Jarkko Kari
2009 Theoretical Computer Science  
Preface Since ancient times, people have had interest in tiles and tilings. Ornamental, symmetric coverings of the plane by polygons and other shapes have strong aesthetic appeal. Geometric questions dealing with symmetry and transitivity properties have been investigated for decades, even centuries. Modern applications in logic, computation theory, dynamical systems and self-assembly have evoked new interesting algorithmic questions concerning tiles and tilings. The ability to encode
more » ... n in tiles, realized half a century ago by Hao Wang, provides the foundation for these new approaches and applications. This special issue was initiated at the workshop "Tilings and self-assembly" held on July 7th, 2007, in Turku, Finland, as a satellite event of the annual conference "Developments in Language Theory". The workshop had prominent speakers on the fruitful connection of tilings to algorithmic studies of self-assembly. The topic of this special issue was chosen to be broader than just the self-assembly aspect of the workshop, and contributions were sought on all aspects of tilings. Another special issue "Aspects of molecular self-assembly" is edited in parallel, containing more self-assembly specific papers. All submitted papers went through a rigorous refereeing process. Five papers were selected for inclusion in the present issue, representing different aspects of modern tiling research. The first two papers concern picture languages, two-dimensional counterparts of classical string languages. The most natural two-dimensional analogies to regular languages are the so-called recognizable picture languages, defined as symbolto-symbol projections of Wang tiled rectangles. The paper by M. Anselmo and M. Madonia investigates the issues of unambiguity and determinism of recognizable picture languages over the unary alphabet. In the unary case only the dimensions of the tileable rectangle matter. Unambiguity and determinism play an important role in efficient parsing. The second paper, by R. Brijder and H.J. Hoogeboom, introduces a novel way to define picture languages by using Wang tilings where the tiling constraint is enforced only between consecutive tiles of some weaving snake path that visits every cell exactly once. It is shown that such snake tilings always define recognizable picture languages and, conversely, up to inflating the size by a constant, every recognizable picture language can be defined via snake tilings. The paper by F. Becker takes us to the Wang tile models of self-assembly. A geometric programming language is developed for designing self-assembling tile systems. Continuous signals along straight lines are used to help in designing self-assembling systems. As an example, it is demonstrated how the method can be applied to designing tiles that selfassemble into the shape of any desired convex polygon. The domino problem is the decision problem for determining whether a given set of Wang tiles admits a tiling of the infinite plane. Proved undecidable by R. Berger in the 60's, the domino problem has become an important tool for establishing undecidability results in various systems, including cellular automata. The paper by V. Lukkarila proves that the domino problem remains undecidable even under the additional constraint that the given Wang tile set is deterministic in the sense that tiles are uniquely determined by any two of the neighboring tiles. This result has implications for cellular automata, but potentially also for the self-healing tile set in self-assembly. The special issue ends with a paper by C. Goodman-Strauss where the concept of regular production systems for onedimensional strings is introduced as a general tool for analyzing tilings in Euclidean, hyperbolic or elliptic planes. Valid tilings are viewed as orbits in the corresponding regular production systems. As an application, simple conditions are given for determining whether a nearly arbitrary triangle admits a tiling of the hyperbolic plane. We thank all authors for their contributions in both special issues, and all the referees for their invaluable work in helping us to select the papers. Guest editors
doi:10.1016/j.tcs.2008.12.002 fatcat:ghm2oc3k35hrzncrp7flzothhq