Fault Tolerance in Cellular Automata at High Fault Rates

A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the case we treat in this paper. We are concerned with the degree (the number of neighboring cells on which the state transition function depends) needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We consider both the traditional transient fault model (where faults occur independently in time and space) and a recently introduced combined fault model which also includes manufacturing faults (which occur independently in space, but which affect cells for all time). We also consider both a purely probabilistic fault model (in which the states of cells are perturbed at exactly the fault rate) and an adversarial model (in which the occurrence of a fault gives control of the state to an omniscient adversary). We show that there are cellular automata that can tolerate a fault rate $1/2 - \xi$ (with $\xi>0$) with degree $O((1/\xi^2)\log(1/\xi))$, even with adversarial combined faults. The simplest such automata are based on infinite regular trees, but our results also apply to other structures (such as hyperbolic tessellations) that contain infinite regular trees. We also obtain a lower bound of $\Omega(1/\xi^2)$, even with purely probabilistic transient faults only

SRT Division Algorithms As Dynamical Systems

Sweeney--Robertson--Tocher (SRT) division, as it was discovered in the late 1950s, represented an important improvement in the speed of division algorithms for computers at the time. A variant of SRT division is still commonly implemented in computers today. Although some bounds on the performance of the original SRT division method were obtained, a great many questions remained unanswered. In this paper, the original version of SRT division is described as a dynamical system. This enables us to bring modern dynamical systems theory, a relatively new development in mathematics, to bear on an older problem. In doing so, we are able to show that SRT division is ergodic, and is even Bernoulli, for all real divisors and dividends. With the Bernoulli property, we are able to use entropy to prove that the natural extensions of SRT division are isomorphic by way of the Kolmogorov--Ornstein theorem. We demonstrate how our methods and results can be applied to a much larger class of division algorithms

Fault Tolerance in Cellular Automata at Low Fault Rates

A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the case we treat in this paper. The conceptually simplest mechanism for correcting errors in a cellular automaton is to determine the next state of a cell by taking a majority vote among its neighbors (including the cell itself, if necessary to break ties). We are interested in which regular two-dimensional tessellations can tolerate faults using this mechanism, when the fault rate is sufficiently low. We consider both the traditional transient fault model (where faults occur independently in time and space) and a recently introduced combined fault model which also includes manufacturing faults (which occur independently in space, but which affect cells for all time). We completely classify regular two-dimensional tessellations as to whether they can tolerate combined transient and manufacturing faults, transient faults but not manufacturing faults, or not even transient faults.Comment: i+26 p

Remineralization Of Enamel Lesions Proximal To Dentin Cavitated Lesions Restored With Resin Modified Glass Ionomer In The Primary Dentition

Poster presentation of research proposal addressing: the evaluation of dental hard tissue remineralization proximal to glass ionomer restorations. It is hypothesized that glass ionomer used in class II restorations will provide significantly more bioavailable fluoride and hard tissue remineralization on the proximal surface of adjacent teeth as compared to the same restoration completed using resin composite materials.https://dune.une.edu/cdm_studpost/1001/thumbnail.jp

We are good to grow: dynamic integration of cell wall architecture with the machinery of growth

Despite differences in cell wall composition between the type I cell walls of dicots and most monocots and the type II walls of commelinid monocots, all flowering plants respond to the same classes of growth regulators in the same tissue-specific way and exhibit the same growth physics. Substantial progress has been made in defining gene families and identifying mutants in cell wall-related genes, but our understanding of the biochemical basis of wall extensibility during growth is still rudimentary. In this review, we highlight insights into the physiological control of cell expansion emerging from genetic functional analyses, mostly in Arabidopsis and other dicots, and a few examples of genes of potential orthologous function in grass species. We discuss examples of cell wall architectural features that impact growth independent of composition, and progress in identifying proteins involved in transduction of growth signals and integrating their outputs in the molecular machinery of wall expansion