Properties
For given k and r, a set is k-automatic if and only if it is kr-automatic. Otherwise, for h and k multiplicatively independent, then a set is both h-automatic and k-automatic if and only if it is 1-automatic, that is, ultimately periodic.
If u(n) is a k-automatic sequence then the sequences u(kn) and u(kn−1) are ultimately periodic. Conversely, if v(n) is ultimately periodic then the sequence u defined by u(kn) = v(n) and otherwise zero is k-automatic.
Let u(n) be a k-automatic sequence over the alphabet A. If f is a uniform morphism from A∗ to B∗ then the word f(u) is k-automatic sequence over the alphabet B.
Let u(n) be a sequence over the alphabet A and suppose that there is an injective function j from A to the finite field Fq. The associated formal power series is
The sequence u is q-automatic if and only if the power series fu is algebraic over the rational function field Fq(z).
Read more about this topic: Automatic Sequence
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)