It is shown that the (infinite) tiling problem byWang tiles is undecidable even if the given tile set is deterministic by all four corners, i.e. a tile is uniquely determined by the colors of any two adjacent edges. The reduction is done from the Turing machine halting problem and uses the aperiodic tile set of Kari and Papasoglu. © 2008 Elsevier B.V. All rights reserved.

Wang tiles are square tiles with colored edges. We construct an aperiodic set of Wang tiles that is strongly deterministic in the sense that any two adjacent edges of a tile determine the tile uniquely. Consequently, the tiling group of this set is not hyperbolic and it acts discretely and co-compactly on a CAT(0) space.

We consider the problem of characterizing small geometric graphs whose structure uniquely determines the order type of its vertex set. We describe a set of edges that prevent the order type from changing by continuous movement and identify properties of the resulting graphs.

In this paper I investigate the problem of tagging elements of a set, and the elements of those elements, uniquely, when they admit an order, and two boundary elements are tagged. A heuristic sorting algorithm is also investigated.

Patterned self-assembly tile set synthesis (PATS) aims at finding a minimum tile set to uniquely self-assemble a given rectangular color pattern. For k ≥ 1, k-PATS is a variant of PATS that restricts input patterns to those with at most k colors. We prove the NP-hardness of 29-PATS, where the best known is that of 60-PATS.

We extend the restrictions which induce unique parsability in Chomsky grammars to accepting grammar systems. It is shown that the accepting power of global RC-uniquely parsable accepting grammar systems equals the computational power of deterministic pushdown automata. More computational power, keeping the parsability without backtracking, is observed for local accepting grammar systems satisfy...

A graph is called to be uniquely list colorable, if it admits a list assignment which induces a unique list coloring. We study uniquely list colorable graphs with a restriction on the number of colors used. In this way we generalize a theorem which characterizes uniquely 2–list colorable graphs. We introduce the uniquely list chromatic number of a graph and make a conjecture about it which is a...

Modern semiotics is a branch of logics that formally defines symbol-based communication. In recent years, the semiotic classification of signs has been invoked to support the notion that symbols are uniquely human. Here we show that alarm-calls such as those used by African vervet monkeys (Cercopithecus aethiops), logically satisfy the semiotic definition of symbol. We also show that the acquis...

There is a tacit assumption in neuroscience from the genetic to the systemic level, which holds that the biological foundations of humans are essentially similar to those of nonhuman primates, and that even human language can be understood by extending experiments with monkeys and apes. In accordance with this prevailing view, it has been assumed that language processing is, in principle, not e...

In this paper, we study the geometry induced by Fisher-Rao metric on parameter space of Dirichlet distributions. We show that is geodesically complete and has everywhere negative sectional curvature. An important consequence curvature for applications Fr{\'e}chet mean a set distributions uniquely defined in geometry.