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:
“Natural selection, the blind, unconscious, automatic process which Darwin discovered, and which we now know is the explanation for the existence and apparently purposeful form of all life, has no purpose in mind. It has no mind and no mind’s eye. It does not plan for the future. It has no vision, no foresight, no sight at all. If it can be said to play the role of the watchmaker in nature, it is the blind watchmaker.”
—Richard Dawkins (b. 1941)
“It isn’t that you subordinate your ideas to the force of the facts in autobiography but that you construct a sequence of stories to bind up the facts with a persuasive hypothesis that unravels your history’s meaning.”
—Philip Roth (b. 1933)