An automatic sequence (or k-automatic sequence) is an infinite sequence of terms characterized by a finite automaton. The n-th term of the sequence is a mapping of the final state of the automaton when its input is the digits of n in some fixed base k. A k-automatic set is a set of non-negative integers for which the sequence of values of its characteristic function is an automatic sequence: that is, membership of n in the set can be determined by a finite state automaton on the digits of n in base k.
An automaton reading the base k digits from the most significant is said to be direct reading, and from the least signficant is reverse reading. However the two directions lead to the same class of sequences.
Every automatic sequence is a morphic word.
Read more about Automatic Sequence: Automaton Point of View, Substitution Point of View, Decimation, 1-automatic Sequences, Properties, Examples, Automatic Real Number
Famous quotes containing the words automatic and/or sequence:
“Predictions of the future are never anything but projections of present automatic processes and procedures, that is, of occurrences that are likely to come to pass if men do not act and if nothing unexpected happens; every action, for better or worse, and every accident necessarily destroys the whole pattern in whose frame the prediction moves and where it finds its evidence.”
—Hannah Arendt (1906–1975)
“We have defined a story as a narrative of events arranged in their time-sequence. A plot is also a narrative of events, the emphasis falling on causality. “The king died and then the queen died” is a story. “The king died, and then the queen died of grief” is a plot. The time sequence is preserved, but the sense of causality overshadows it.”
—E.M. (Edward Morgan)