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| 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''.<ref name=as1>Allouche & Shallit (2003) p.152</ref><ref name=BLRS78>Berstel et al (2009) p.78</ref> 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''.<ref>Allouche & Shallit (2003) p.168</ref><ref name=PF13/>
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| An automaton reading the base ''k'' digits from the most significant is said to be ''direct reading'', and from the least significant is ''reverse reading''.<ref name=PF13>Pytheas Fogg (2002) p.13</ref> However the two directions lead to the same class of sequences.<ref name=PF15>Pytheas Fogg (2002) p.15</ref>
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| Every automatic sequence is a [[morphic word]].<ref name=LotIII524>Lothaire (2005) p.524</ref>
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| ==Automaton point of view==
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| Let ''k'' be a positive [[integer]], and ''D'' = (''E'', φ, ''e'') be a deterministic automaton where
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| *''E'' is the finite [[Set (mathematics)|set]] of [[State (computer science)|state]]s
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| *φ : ''E''×[0,''k'' − 1] → ''E'' is the transition function
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| *<math>e\in E</math> is the initial state
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| also let ''A'' be a finite set, and π:''E'' → ''A'' a [[Projection (mathematics)|projection]] towards ''A''.
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| Extend the transition function φ from acting on single digits to acting on strings of digits by defining the action of φ on a string ''s'' consisting of digits ''s''<sub>1</sub>''s''<sub>2</sub>...''s''<sub>''t''</sub> as:
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| :<math>\phi(e, s) = \phi(\phi(e, s_1s_2...s_{t-1}), s_t)\, .</math>
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| Define a function ''m'' from the set of positive integers to the set ''A'' as follows:
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| :<math>m(n) = \pi(\phi(e,s(n)))\, ,</math>
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| where ''s''(''n'') is ''n'' written in base ''k''. Then the sequence ''m'' = ''m''(1)''m''(2)''m''(3)... is called a '''''k''-automatic sequence'''.<ref name=as1/>
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| ==Substitution point of view==
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| Let σ be a ''k''-[[uniform morphism]] of the [[free monoid]] ''E''<sup>∗</sup>, so that <math>\sigma(E)\subseteq E^k</math> and which is [[prolongable morphism|prolongable]]<ref name=AS212>Allouche & Shallit (2003) p.212</ref> on <math>e\in E</math>: that is, σ(''e'') begins with ''e''. Let ''A'' and π be defined as above. Then if ''w'' is a [[fixpoint]] of σ, that is to say ''w'' = σ(''w''), then ''m'' = π(''w'') is a ''k''-automatic sequence over ''A'':<ref name=AS175>Allouche & Shallit (2003) p.175</ref> this is '''Cobham's theorem'''.<ref name=BLRS78/> Conversely every ''k''-automatic sequence is obtained in this way.<ref name=PF13/>
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| ==Decimation==
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| Fix ''k'' > 1. For a sequence ''w'' we define the ''k''-decimations of ''w'' for ''r''=0,1,...,''k''-1 to be the subsequences consisting of the letters in positions congruent to ''r'' modulo ''k''. The decimation kernel of ''w'' consists of the set of words obtained by all possible repeated decimations of ''w''. A sequence is ''k''-automatic if an only if the ''k''-decimation kernel is finite.<ref name=AS185>Allouche & Shallitt (2003) p.185</ref><ref name=ApCOw527>Lothaire (2005) p.527</ref><ref name=BR91>Berstel & Reutenauer (2011) p.91</ref>
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| ==1-automatic sequences==
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| ''k''-automatic sequences are normally only defined for ''k'' ≥ 2.<ref name=as1/> The concept can be extended to ''k'' = 1 by defining a 1-automatic sequence to be a sequence whose ''n''-th term depends on the [[unary numeral system|unary notation]] for ''n'', that is (1)<sup>''n''</sup>. Since a finite state automaton must eventually return to a previously visited state, all 1-automatic sequences are eventually periodic.
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| ==Properties==
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| For given ''k'' and ''r'', a set is ''k''-automatic if and only if it is ''k''<sup>''r''</sup>-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.<ref name=as345>Allouche & Shallit (2003) pp.345-350</ref>
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| If ''u''(''n'') is a ''k''-automatic sequence then the sequences ''u''(''k''<sup>''n''</sup>) and ''u''(''k''<sup>''n''</sup>−1) are ultimately periodic.<ref name=ApCoW529>Lothaire (2005) p.529</ref> Conversely, if ''v''(''n'') is ultimately periodic then the sequence ''u'' defined by ''u''(''k''<sup>''n''</sup>) = ''v''(''n'') and otherwise zero is ''k''-automatic.<ref name=BR103>Berstel & Reutenauer (2011) p.103</ref>
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| Let ''u''(''n'') be a ''k''-automatic sequence over the alphabet ''A''. If ''f'' is a [[uniform morphism]] from ''A''<sup>∗</sub> to ''B''<sup>∗</sub> then the word ''f''(''u'') is ''k''-automatic sequence over the alphabet ''B''.<ref name=ApCoW532>Lothaire (2005) p.532</ref>
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| 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 '''F'''<sub>''q''</sub>. The associated [[formal power series]] is
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| :<math> f_u(z) = \sum_n j(u(n)) z^n \ . </math>
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| The sequence ''u'' is ''q''-automatic if and only if the power series ''f''<sub>''u''</sub> is algebraic over the rational function field '''F'''<sub>''q''</sub>(''z'').<ref name=BR93>Berstel & Reutenauer (2011) p.93</ref>
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| ==Examples==
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| The following sequences are automatic:
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| * [[Thue-Morse sequence]]: take ''E'' = ''A'' = {0, 1}, ''e'' = 0, π = id, and σ such that σ(0) = 01, σ(1) = 10; we get the fixpoint 01101001100101101001011001101001..., which is in fact the Thue-Morse word. The ''n''-th term is the [[Parity (mathematics)|parity]] of the [[Binary numeral system#Representation|base 2 representation]] of ''n'' and the sequence is thus 2-automatic.<ref name=as1/><ref name=BLRS78/><ref name=LotIII525>Lothaire (2005) p.525</ref><ref name=BR92>Berstel & Reutenauer (2011) p.92</ref> The 2-kernel consists of the sequence itself and its complement.<ref name=ApCoW528>Lothaire (2005) p.528</ref> The associated power series ''T''(''z'') satisfies
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| ::<math> (1+z)^3 T^2 + (1+z)^2 T + z = 0 \ </math>
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| :over the field '''F'''<sub>2</sub>(''z'')</sub>.<ref name=BR94>Berstel & Reutenauer (2011) p.94</ref>
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| * [[Rudin–Shapiro sequence]]<ref name=LotIII525/><ref name=AS154>Allouche & Shallit (2003) p.154</ref>
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| * [[Baum–Sweet sequence]]<ref name=AS156>Allouche & Shallit (2003) p.156</ref>
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| * [[Regular paperfolding sequence]]<ref name=BR92/><ref name=AS155>Allouche & Shallit (2003) p.155</ref><ref name=LotIII526>Lothaire (2005) p.526</ref> and a general paperfolding sequence with a periodic sequence of folds<ref name=AS183>Allouche & Shallit (2003) p.183</ref>
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| * The '''period-doubling sequence''', defined by the parity of the power of 2 dividing ''n''; it is the fixed point of the morphism 0 → 01, 1 → 00.<ref name=AS176>Allouche & Shallit (2003) p.176</ref>
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| ==Automatic real number==
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| An ''automatic real number'' is a [[real number]] for which the base-''b'' expansion is an automatic sequence.<ref name=hejhal556>Shallitt (1999) p.556</ref><ref name=AS379>Allouche & Shallit (2003) p.379</ref> All such numbers are either [[rational number|rational]] or [[transcendental number|transcendental]], but not a [[U-number]].<ref>{{citation | first1=Boris | last1=Adamczewski | first2=Yann | last2=Bugeaud | title=On the complexity of algebraic numbers. I. Expansions in integer bases | journal=[[Annals of Mathematics]] | volume=165 | number=2 | year=2007 | pages=547–565 | zbl=1195.11094 }}</ref><ref>{{citation | last=Bugeaud | first=Yann | title=Distribution modulo one and Diophantine approximation | series=Cambridge Tracts in Mathematics | volume=193 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-0-521-11169-0 | zbl=pre06066616 | pages=192–193 }}</ref> Rational numbers are ''k''-automatic in base ''b'' for all ''k'' and ''b''.<ref name=AS379/>
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| ==References==
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| {{reflist}}
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| *{{cite book | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}
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| * {{cite book | last1=Berstel | first1=Jean | last2=Lauve | first2=Aaron | last3=Reutenauer | first3=Christophe | last4=Saliola | first4=Franco V. | title=Combinatorics on words. Christoffel words and repetitions in words | series=CRM Monograph Series | volume=27 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2009 | isbn=978-0-8218-4480-9 | url=http://www.ams.org/bookpages/crmm-27 | zbl=1161.68043 }}
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| * {{cite book | last1=Berstel | first1=Jean | last2=Reutenauer | first2=Christophe | title=Noncommutative rational series with applications | series=Encyclopedia of Mathematics and Its Applications | volume=137 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2011 | isbn=978-0-521-19022-0 | zbl=1250.68007 }}
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| * {{cite book | editor1-last=Berthé | editor1-first=Valérie | editor2-last=Rigo | editor2-first=Michel | title=Combinatorics, automata, and number theory | series=Encyclopedia of Mathematics and its Applications | volume=135 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2010 | isbn=978-0-521-51597-9 | zbl=1197.68006 }}
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| *{{cite book | editor1-first=Dennis A. | editor1-last=Hejhal | editor1-link=Dennis Hejhal | editor2-last=Friedman | editor2-first=Joel | editor3-last=Gutzwiller | editor3-first=Martin C. | editor3-link=Martin Gutzwiller | editor4-last=Odlyzko | editor4-first=Andrew M. | editor4-link=Andrew Odlyzko | title=Emerging applications of number theory. Based on the proceedings of the IMA summer program, Minneapolis, MN, USA, July 15--26, 1996 | series=The IMA volumes in mathematics and its applications | volume=109 | publisher=[[Springer-Verlag]] | year=1999 | isbn=0-387-98824-6 | last=Shallit | first=Jeffrey | author1-link=Jeffrey Shallit | chapter=Number theory and formal languages | pages=547–570 }}
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| *{{cite book | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Applied combinatorics on words | others=A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, Gesine Reinert, Sophie Schbath, Michael Waterman, Philippe Jacquet, Wojciech Szpankowski, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé| series=Encyclopedia of Mathematics and Its Applications | volume=105 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2005 | isbn=0-521-84802-4 | zbl=1133.68067 }}
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| *{{cite book | first=J. H. | last=Loxton | chapter=13. Automata and transcendence | pages=215–228 | title=New Advances in Transcendence Theory | editor1-link=Alan Baker (mathematician) | editor1-first=A. | editor1-last=Baker | publisher=[[Cambridge University Press]] | year=1988 | isbn=0-521-33545-0 | zbl=0656.10032 }}
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| * {{cite book | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}
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| {{DEFAULTSORT:Automatic Sequence}}
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| [[Category:Combinatorics on words]]
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| [[Category:Automata theory]]
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Nice to satisfy you, my title is Numbers Held though I don't really like becoming known as like that. I utilized to be unemployed but now I am a librarian and the salary has been really satisfying. South Dakota is exactly where me and my spouse live. Doing ceramics is what my family members and I enjoy.
Here is my web site :: http://www.youronlinepublishers.com/