We discuss the use of negative bases in automatic sequences. Recently the theorem-prover Walnut has been extended to allow the use of base (—k) to express variables, thus permitting quantification over ℤ instead of ℕ. This enables us to prove results about two-sided (bi-infinite) automatic sequences. We first explain the theory behind negative bases in Walnut. Next, we use this new version of Walnut to give a very simple proof of a strengthened version of a theorem of Shevelev. We use our ideas to resolve two open problems of Shevelev from 2017. We also reprove a 2000 result of Shut involving bi-infinite binary words.
Estimating the cycle time of every job in a semiconductor manufacturing factory is a
critical task to the factory. Many recent studies have shown that pre-classifying a job before
estimating the cycle time of the job was beneficial to the forecasting accuracy. However, most
pre-classification approaches applied in this field could not absolutely classify jobs. Besides, whether
the pre-classification approach combined with the subsequent forecasting approach was suitable for
the data was questionable. For tackling these problems, an artificial neural network (ANN) approach
that equally divides and post-classifies jobs is proposed in this study in which a job is post-classified
by a BPN instead after the forecasting error is generated. In this novel way, only jobs which cycle time
forecasts are the same accurate will be clustered into the same category, and the classification
algorithm becomes tailored to the forecasting approach. For evaluating the effectiveness of the
proposed methodology and to make comparison with some existing approaches, some data were
collected from an actual semiconductor manufacturing factory. According to experimental results, the
forecasting accuracy (measured with root mean squared error (RMSE)) of the proposed methodology
was significantly better than those of the other approaches in most cases by achieving a 16%~44%
(and an average of 29%) reduction in RMSE over the comparison basis – multiple-factor linear
combination (MFLC). The effect of post-classification was also evident.
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