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In [[symbolic dynamics]] and related branches of [[mathematics]], a '''shift space''' or '''subshift''' is a set of [[infinity|infinite]] [[String (computer science)|word]]s that represent the evolution of a [[discrete system]]. In fact, shift spaces and ''[[Symbolic dynamics|symbolic dynamical systems]]'' are often considered [[synonym]]s.
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== Notation ==
Let ''A'' be a finite set of states. An ''infinite'' (respectively ''bi-infinite'') ''word'' over ''A'' is a sequence <math>\mathbf x=(x_n)_{n\in M}</math>, where <math>M=\mathbb N</math> (resp. <math>M=\mathbb Z</math>) and <math>x_n</math> is in ''A'' for any integer ''n''.
The [[shift operator]] <math>\sigma</math> acts on an infinite or bi-infinite word by shifting all symbols to the left, i.e.,
:<math>(\sigma(\mathbf x))(n)=x_{n+1}</math> for all ''n''.
In the following we choose <math>M=\mathbb N</math> and thus speak of infinite words, but all definitions are naturally generalizable to the bi-infinite case.
 
== Definition ==
A set of infinite words over ''A'' is a ''shift space'' if it is [[Closed set|closed]] with respect to the natural [[product topology]] of <math>A^\mathbb N</math> and invariant under the shift operator. Thus a set <math>S\subseteq A^\mathbb N</math> is a subshift [[if and only if]]
# for any ([[Pointwise convergence|pointwise]]) [[convergent sequence]] <math>(\mathbf x_k)_{k\geq 0}</math> of elements of ''S'', the [[Limit of a sequence|limit]] <math>\lim_{k\to\infty}\mathbf x_k</math> also belongs to ''S''; and
# <math>\sigma(S)=S</math>.
 
A shift space ''S'' is sometimes denoted as <math>(S,\sigma)</math> to emphasize the role of the shift operator.
 
Some authors<ref>{{cite journal|author = Thomsen, K.|title = On the structure of a sofic shift space|url =http://www.ams.org/journals/tran/2004-356-09/S0002-9947-04-03437-3/S0002-9947-04-03437-3.pdf |format = PDF Reprint|accessdate = 2012-01-27|journal = [[Transactions of the American Mathematical Society]]|volume = 356|year = 2004|pages = 3557–3619|doi = 10.1090/S0002-9947-04-03437-3|issue = 9 }}</ref> use the term ''subshift'' for a set of infinite words that is just invariant under the shift, and reserve the term ''shift space'' for those that are also closed.
 
== Characterization and sofic subshifts ==
A subset ''S'' of <math>A^\mathbb N</math> is a shift space if and only if there exists a set ''X'' of [[String (computer science)|finite word]]s such that ''S'' coincides with the set of all infinite words over ''A'' having no [[Substring|factor]] in ''X''.
 
In particular, if ''X'' is finite then ''S'' is called a [[subshift of finite type]] and more generally when ''X'' is a [[regular language]], the corresponding subshift is called '''sofic'''.
The name "sofic" was coined by {{harvtxt|Weiss|1973}}, based on the [[Hebrew language|Hebrew]] word סופי meaning "finite", to refer to the fact that this is a generalization of a finiteness property.<ref>{{citation
| last = Weiss | first = Benjamin | authorlink = Benjamin Weiss
| journal = Monatsh. Math.
| mr = 0340556
| pages = 462–474
| title = Subshifts of finite type and sofic systems
| volume = 77
| year = 1973}}. Weiss does not describe the origin of the word other than calling it a neologism; however, its Hebrew origin is stated by [[MathSciNet]] reviewer R. L. Adler.</ref>
 
== Examples ==
The first trivial example of shift space (of finite type) is the ''full shift'' <math>A^\mathbb N</math>.
 
Let <math>A=\{a,b\}</math>. The set of all infinite words over ''A'' containing at most one ''b'' is a sofic subshift, not of finite type. The set of all infinite words over ''A'' whose ''b'' form blocks of prime length is not sofic (this can be shown by using the [[pumping lemma]]).
 
== Further reading ==
* {{cite book
| author = Lind, Douglas
| coauthors = Marcus, Brian
| title = An Introduction to Symbolic Dynamics and Coding
| year = 1995
| publisher = Cambridge University Press
| location = Cambridge UK
| isbn = 0-521-55900-6 }}
* {{cite book
| last = Lothaire | first = M. | authorlink = M. Lothaire
| title = Algebraic Combinatorics on Words
| url = http://www-igm.univ-mlv.fr/~berstel/Lothaire/AlgCWContents.html
| accessdate = 2008-01-29
| year = 2002
| publisher = Cambridge University Press
| location = Cambridge UK
| isbn = 0-521-81220-8
| chapter = Finite and Infinite Words
| chapterurl = http://www-igm.univ-mlv.fr/%7Eberstel/Lothaire/ChapitresACW/C1.ps }}
* {{cite journal
| last1 = Morse | first1 = Marston
| author1-link = Marston Morse
| last2 = Hedlund | first2 = Gustav A. | author2-link = Gustav A. Hedlund
| title = Symbolic Dynamics
| journal = [[American Journal of Mathematics]]
| volume = 60
| year = 1938
| pages = 815–866
| doi = 10.2307/2371264
| issue = 4
| jstor=2371264}}
* {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
 
== References ==
<references />
 
[[Category:Symbolic dynamics]]
[[Category:Dynamical systems]]
[[Category:Combinatorics on words]]

Latest revision as of 10:48, 28 July 2014

Even though we often keep law firms in higher esteem as occupations get in society, couple of us know what they really do. It's not till we wind up embedded in legal proceedings we know how complicated the world of rules could be! If you need legal advice, the following write-up will get you were only available in the best direction.

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