Abstract
Wang tiles proved to be a convenient tool for the design of aperiodic tilings in computer graphics and in materials engineering. While there are several algorithms for generation of finite-sized tilings, they exploit the specific structure of individual tile sets, which prevents their general usage. In this contribution, we reformulate the NP-complete tiling generation problem as a binary linear program, together with its linear and semidefinite relaxations suitable for the branch and bound method. Finally, we assess the performance of the established formulations on generations of several aperiodic tilings reported in the literature, and conclude that the linear relaxation is better suited for the problem.
Highlights
Wang tiles, squares with colored edges, were invented by and named in honor of Hao Wang, originally serving as a tool for studying the ∀∃∀ decidability problem of the predicate calculus [1]
Wang showed that the decidability problem is equivalent to the domino problem: assume a set of non-rotatable unit-sized (Wang) tiles with edges colored according to the ∀∃∀ problem
This paper aims to overcome the shortcomings of the methods outlined in the previous section, and to develop an approach that handles tiling of finitesized areas using arbitrary tile sets, together with a straightforward approach to define edge- or tilebased boundary conditions
Summary
Squares with colored edges, were invented by and named in honor of Hao Wang, originally serving as a tool for studying the ∀∃∀ decidability problem of the predicate calculus [1]. Wang showed that the decidability problem is equivalent to the domino problem: assume a set of non-rotatable unit-sized (Wang) tiles with edges colored according to the ∀∃∀ problem. A year later, Kahr reduced the Turing machine halting problem [3, 4] into the origin-constrained domino problem [5], which implies the domino problem is undecidable. This can be illustrated by introducing a Turing machine for each tile set, halting only if the domino problem is unsolvable. There is an infinite number of such tile sets, making the domino problem undecidable as well, and forbidding existence of a general finite algorithm for the generation of infinite tilings
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