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| The '''star height problem''' in [[formal language theory]] is the question whether all [[regular language]]s can be expressed using [[Regular_expression#Formal_language_theory|regular expression]]s of limited [[star height]], i.e. with a limited nesting depth of [[Kleene star]]s. Specifically, is a nesting depth of one always sufficient? If not, is there an [[algorithm]] to determine how many are required? The problem was raised by {{harvtxt|Eggan|1963}}. | | The writer's name is [http://Www.Sharkbayte.com/keyword/Herlinda Herlinda] but she doesn't like recognize use her full appoint. Managing people wherever my primary income hails from and it is something I love. For years I have been living in Iowa hence there is no don't work toward changing it. To solve puzzles is an [http://www.answers.com/topic/element element] that I'm totally addicted that will help. Check out the latest news on this website: http://Vimeo.com/91965972<br><br>Feel free to surf to my blog; bibi jones ([http://Vimeo.com/91965972 Read Significantly more]) |
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| ==Families of regular languages with unbounded star height==
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| The first question was answered in the negative when in 1963, Eggan gave examples of regular languages of [[star height]] ''n'' for every ''n''. Here, the star height ''h''(''L'') of a regular language ''L'' is defined as the minimum star height among all regular expressions representing ''L''. The first few languages found by {{harvtxt|Eggan|1963}} are described in the following, by means of giving a regular expression for each language:
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| :<math>\begin{alignat}{2} | |
| e_1 &= a_1^* \\
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| e_2 &= \left(a_1^*a_2^*a_3\right)^*\\
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| e_3 &= \left(\left(a_1^*a_2^*a_3\right)^*\left(a_4^*a_5^*a_6\right)^*a_7\right)^*\\
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| e_4 &= \left(
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| \left(\left(a_1^*a_2^*a_3\right)^*\left(a_4^*a_5^*a_6\right)^*a_7\right)^*
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| \left(\left(a_8^*a_9^*a_{10}\right)^*\left(a_{11}^*a_{12}^*a_{13}\right)^*a_{14}\right)^*
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| a_{15}\right)^*
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| \end{alignat}
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| </math>
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| The construction principle for these expressions is that expression <math>e_{n+1}</math> is obtained by concatenating two copies of <math>e_n</math>, appropriately renaming the letters of the second copy using fresh alphabet symbols, concatenating the result with another fresh alphabet symbol, and then by surrounding the resulting expression with a Kleene star. The remaining, more difficult part, is to prove that for <math>e_n</math> there is no equivalent regular expression of star height less than ''n''; a proof is given in {{harv|Eggan|1963}}.
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| However, Eggan's examples use a large [[Alphabet (computer science)|alphabet]], of size 2<sup>''n''</sup>-1 for the language with star height ''n''. He thus asked whether we can also find examples over binary alphabets. This was proved to be true shortly afterwards by {{harvtxt|Dejean|Schützenberger|1966}}.
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| Their examples can be described by an [[inductive definition|inductively defined]] family of regular expressions over the binary alphabet <math>\{a,b\}</math> as follows–cf. {{harvtxt|Salomaa|1981}}:
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|
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| :<math>\begin{alignat}{2}
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| e_1 & = (ab)^* \\
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| e_2 & = \left(aa(ab)^*bb(ab)^*\right)^* \\
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| e_3 & = \left(aaaa \left(aa(ab)^*bb(ab)^*\right)^* bbbb \left(aa(ab)^*bb(ab)^*\right)^*\right)^* \\
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| \, & \cdots \\
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| e_{n+1} & = (\,\underbrace{a\cdots a}_{2^n}\, \cdot \, e_n\, \cdot\, \underbrace{b\cdots b}_{2^n}\, \cdot\, e_n \,)^*
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| \end{alignat}
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| </math>
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| Again, a rigorous proof is needed for the fact that <math>e_n</math> does not admit an equivalent regular expression of lower star height. Proofs are given by {{harv|Dejean|Schützenberger|1966}} and by {{harv|Salomaa|1981}}.
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| ==Computing the star height of regular languages==
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| In contrast, the second question turned out to be much more difficult, and the question became a famous open problem in formal language theory for over two decades {{harv|Brzozowski|1980}}. For years, there was only little progress. The [[pure-group language]]s were the first interesting family of regular languages for which the star height problem was proved to be [[decidable]] {{harv|McNaughton|1967}}. But the general problem remained open for more than 25 years until it was settled by [[Kosaburo Hashiguchi|Hashiguchi]], who in 1988 published an algorithm to determine the [[star height]] of any regular language. The algorithm wasn't at all practical, being of non-[[ELEMENTARY|elementary]] complexity. To illustrate the immense resource consumptions of that algorithm, Lombardy and Sakarovitch (2002) give some actual numbers:
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| {{cquote|
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| [The procedure described by Hashiguchi] leads to computations that are by far impossible, even for very small examples. For instance, if ''L'' is accepted by a 4 state automaton of loop complexity 3 (and with a small 10 element transition monoid), then a ''very low minorant'' of the number of languages to be tested with ''L'' for equality is:
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| <math>\left(10^{10^{10}}\right)^{\left(10^{10^{10}}\right)^{\left(10^{10^{10}}\right)}}.</math>
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| |4=S. Lombardy and J. Sakarovitch
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| |5=''Star Height of Reversible Languages and Universal Automata'', LATIN 2002
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| }}
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| Notice that alone the number <math>10^{10^{10}}</math> has 10 billion zeros when written down in [[decimal notation]], and is already ''by far'' larger than the [[Observable_universe#Matter_content|number of atoms in the observable universe]].
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| A much more efficient algorithm than Hashiguchi's procedure was devised by Kirsten in 2005. This algorithm runs, for a given [[nondeterministic finite automaton]] as input, within double-[[EXPSPACE|exponential space]]. Yet the resource requirements of this algorithm still greatly exceed the margins of what is considered practically feasible.
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| ==See also==
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| *[[Generalized star height problem]]
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| ==References==
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| *{{cite journal |first=Lawrence C. |last=Eggan |title=Transition graphs and the star-height of regular events | journal=[[Michigan Mathematical Journal]] | volume=10 | issue=4 | pages=385–397 | year=1963 | doi=10.1307/mmj/1028998975 | zbl=0173.01504 }}
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| *{{cite journal |first=Françoise |last=Dejean |authorlink2=Marcel-Paul Schützenberger |first2=Marcel-Paul |last2=Schützenberger |title=On a Question of Eggan |journal=[[Information and Control]] |volume=9 |issue=1 |pages=23–25 |year=1966 |doi=10.1016/S0019-9958(66)90083-0 }}
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| * {{cite journal
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| | title = The Loop Complexity of Pure-Group Events
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| | year = 1967
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| | last = McNaughton |first=Robert
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| | journal = Information and Control
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| | pages = 167–176
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| | volume = 11
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| | issue = 1–2
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| | doi=10.1016/S0019-9958(67)90481-0
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| }}
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| *{{cite book |authorlink=Janusz Brzozowski (computer scientist) |first=Janusz A. |last=Brzozowski |chapter=Open problems about regular languages |editor-first=Ronald V. |editor-last=Book |title=Formal language theory—Perspectives and open problems |pages=23–47 |publisher=Academic Press |location=New York |year=1980 |isbn=0-12-115350-9 }} [https://www.cs.uwaterloo.ca/research/tr/1980/CS-80-03.pdf (technical report version)]
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| * {{cite book |title=Jewels of Formal Language Theory |last= Salomaa |first= Arto |authorlink=Arto Salomaa |year=1981 |publisher=Pitman Publishing |location=Melbourne |isbn=0-273-08522-0 |zbl=0487.68063 }}
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| *{{cite journal |first=Kosaburo |last=Hashiguchi |title=Regular languages of star height one |journal=Information and Control |volume=53 |issue=2 |pages=199–210 |year=1982 |doi=10.1016/S0019-9958(82)91028-2 }}
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| *{{cite journal |first=Kosaburo |last=Hashiguchi |title=Algorithms for Determining Relative Star Height and Star Height |journal=Information and Computation |volume=78 |issue=2 |pages=124–169 |year=1988 |doi=10.1016/0890-5401(88)90033-8 }}
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| *{{cite paper |first=Sylvain |last=Lombardy |first2=Jacques |last2=Sakarovitch |title=Star Height of Reversible Languages and Universal Automata |work=5th Latin American Symposium on Theoretical Informatics (LATIN) 2002, vol. 2286 of LNCS |publisher=Springer |url=http://www-igm.univ-mlv.fr/~lombardy/publi/LATIN.pdf |year=2002 }}
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| *{{cite journal |first=Daniel |last=Kirsten |title=Distance Desert Automata and the Star Height Problem |journal=RAIRO - Informatique Théorique et Applications |volume=39 |issue=3 |pages=455–509 |year=2005 |doi=10.1051/ita:2005027 }}
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| * {{cite book | last=Sakarovitch | first=Jacques | title=Elements of automata theory | others=Translated from the French by Reuben Thomas | location=Cambridge | publisher=[[Cambridge University Press]] | year=2009 | isbn=978-0-521-84425-3 | zbl=1188.68177 }}
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| [[Category:Automata theory]]
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| [[Category:Formal languages]]
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| [[Category:Theorems in discrete mathematics]]
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The writer's name is Herlinda but she doesn't like recognize use her full appoint. Managing people wherever my primary income hails from and it is something I love. For years I have been living in Iowa hence there is no don't work toward changing it. To solve puzzles is an element that I'm totally addicted that will help. Check out the latest news on this website: http://Vimeo.com/91965972
Feel free to surf to my blog; bibi jones (Read Significantly more)