Independence (probability theory) - Revision history

en>Qwertyus: attach ) to text

2014-12-28T14:04:07Z

attach ) to text

← Older revision		Revision as of 16:04, 28 December 2014
Line 1:		Line 1:
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en>Mindmatrix: Reverted edits by 92.7.136.195 (talk) to last version by 216.113.168.141

2014-02-09T18:51:21Z

Reverted edits by 92.7.136.195 (talk) to last version by 216.113.168.141

← Older revision		Revision as of 20:51, 9 February 2014
Line 1:		Line 1:
	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])

	~~==Families of regular languages with unbounded star height==~~
	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:

	:~~<math>\begin{alignat}{2}~~
	~~e_1 &= a_1^* \\~~
	~~e_2 &= \left(a_1^a_2^a_3\right)^*\\~~
	~~e_3 &= \left(\left(a_1^a_2^a_3\right)^\left(a_4^a_5^a_6\right)^a_7\right)^*\\~~
	~~e_4 &= \left(~~
	\left(\left(a_1^a_2^a_3\right)^\left(a_4^a_5^a_6\right)^a_7\right)^*
	\left(\left(a_8^a_9^a_{10}\right)^\left(a_{11}^a_{12}^a_{13}\right)^a_{14}\right)^*
	a_{15}\right)^*
	~~\end{alignat}~~
	</~~math>~~

	~~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}}.

	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}}.
	~~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}}:~~

	~~:<math>\begin{alignat}{2}~~
	~~e_1 & = (ab)^* \\~~
	~~e_2 & = \left(aa(ab)^bb(ab)^\right)^* \\~~
	~~e_3 & = \left(aaaa \left(aa(ab)^bb(ab)^\right)^* bbbb \left(aa(ab)^bb(ab)^\right)^\right)^ \\~~
	~~\, & \cdots \\~~
	e_{n+1} & = (\,\underbrace{a\cdots a}_{2^n}\, \cdot \, e_n\, \cdot\, \underbrace{b\cdots b}_{2^n}\, \cdot\, e_n \,)^*
	~~\end{alignat}~~
	</~~math>~~

	~~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}}.~~

	~~==Computing the star height of regular languages==~~
	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:~~

	~~{{cquote\|~~
	~~[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:

	~~<math>\left(10^{10^{10}}\right)^{\left(10^{10^{10}}\right)^{\left(10^{10^{10}}\right)}}.</math>~~
	~~\|4=S. Lombardy and J. Sakarovitch~~
	~~\|5=''Star Height of Reversible Languages and Universal Automata'', LATIN 2002~~
	}}
	~~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]]~~.

	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.~~

	~~==See also==~~
	*[~~[Generalized star height problem]]~~

	~~==References==~~
	*{{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 }}
	*{{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 }}
	* {{cite journal
	~~\| title = The Loop Complexity of Pure-Group Events~~
	~~\| year = 1967~~
	~~\| last = McNaughton \|first=Robert~~
	~~\| journal = Information and Control~~
	~~\| pages = 167–176~~
	~~\| volume = 11~~
	~~\| issue = 1–2~~
	~~\| doi=10.1016/S0019-9958(67)90481-0~~
	}}
	*{{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)~~]
	* {{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 }}~~
	*{{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 }}
	*{{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 }}
	*{{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 }}~~
	*{{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 }}~~
	* {{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 }}

	~~[[Category:Automata theory]]~~
	~~[[Category:Formal languages]]~~
	~~[[Category:Theorems in discrete mathematics]~~]

en>Robinh: /* Characteristic function */

2013-12-10T20:39:43Z

Characteristic function

← Older revision		Revision as of 22:39, 10 December 2013
Line 1:		Line 1:
	~~My name~~ is ~~Yasmin~~ (25 years ~~old~~) and ~~my hobbies are Fishing~~ and ~~Metal detecting~~.<br><br>~~my web~~-~~site~~ ... [http://~~juhyun420~~.~~dothome~~.co.kr/~~xe/?document_srl~~=~~115121 Nevada Singles Chat~~]		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}}.

			==Families of regular languages with unbounded star height==
			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:

			:<math>\begin{alignat}{2}
			e_1 &= a_1^* \\
			e_2 &= \left(a_1^a_2^a_3\right)^*\\
			e_3 &= \left(\left(a_1^a_2^a_3\right)^\left(a_4^a_5^a_6\right)^a_7\right)^*\\
			e_4 &= \left(
			\left(\left(a_1^a_2^a_3\right)^\left(a_4^a_5^a_6\right)^a_7\right)^*
			\left(\left(a_8^a_9^a_{10}\right)^\left(a_{11}^a_{12}^a_{13}\right)^a_{14}\right)^*
			a_{15}\right)^*
			\end{alignat}
			</math>

			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}}.

			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}}.
			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}}:

			:<math>\begin{alignat}{2}
			e_1 & = (ab)^* \\
			e_2 & = \left(aa(ab)^bb(ab)^\right)^* \\
			e_3 & = \left(aaaa \left(aa(ab)^bb(ab)^\right)^* bbbb \left(aa(ab)^bb(ab)^\right)^\right)^ \\
			\, & \cdots \\
			e_{n+1} & = (\,\underbrace{a\cdots a}_{2^n}\, \cdot \, e_n\, \cdot\, \underbrace{b\cdots b}_{2^n}\, \cdot\, e_n \,)^*
			\end{alignat}
			</math>

			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}}.

			==Computing the star height of regular languages==
			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:

			{{cquote\|
			[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:

			<math>\left(10^{10^{10}}\right)^{\left(10^{10^{10}}\right)^{\left(10^{10^{10}}\right)}}.</math>
			\|4=S. Lombardy and J. Sakarovitch
			\|5=''Star Height of Reversible Languages and Universal Automata'', LATIN 2002
			}}
			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]].

			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.

			==See also==
			*[[Generalized star height problem]]

			==References==
			*{{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 }}
			*{{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 }}
			* {{cite journal
			\| title = The Loop Complexity of Pure-Group Events
			\| year = 1967
			\| last = McNaughton \|first=Robert
			\| journal = Information and Control
			\| pages = 167–176
			\| volume = 11
			\| issue = 1–2
			\| doi=10.1016/S0019-9958(67)90481-0
			}}
			*{{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)]
			* {{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 }}
			*{{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 }}
			*{{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 }}
			*{{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 }}
			*{{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 }}
			* {{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 }}

			[[Category:Automata theory]]
			[[Category:Formal languages]]
			[[Category:Theorems in discrete mathematics]]

en>IPWAI: /* Independent random variables */

2012-08-27T06:50:43Z

Independent random variables

New page

My name is Yasmin (25 years old) and my hobbies are Fishing and Metal detecting.<br><br>my web-site ... [http://juhyun420.dothome.co.kr/xe/?document_srl=115121 Nevada Singles Chat]

← Older revision		Revision as of 20:51, 9 February 2014
Line 1:		Line 1:
	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])

	~~==Families of regular languages with unbounded star height==~~
	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:

	:~~<math>\begin{alignat}{2}~~
	~~e_1 &= a_1^* \\~~
	~~e_2 &= \left(a_1^a_2^a_3\right)^*\\~~
	~~e_3 &= \left(\left(a_1^a_2^a_3\right)^\left(a_4^a_5^a_6\right)^a_7\right)^*\\~~
	~~e_4 &= \left(~~
	\left(\left(a_1^a_2^a_3\right)^\left(a_4^a_5^a_6\right)^a_7\right)^*
	\left(\left(a_8^a_9^a_{10}\right)^\left(a_{11}^a_{12}^a_{13}\right)^a_{14}\right)^*
	a_{15}\right)^*
	~~\end{alignat}~~
	</~~math>~~

	~~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}}.

	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}}.
	~~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}}:~~

	~~:<math>\begin{alignat}{2}~~
	~~e_1 & = (ab)^* \\~~
	~~e_2 & = \left(aa(ab)^bb(ab)^\right)^* \\~~
	~~e_3 & = \left(aaaa \left(aa(ab)^bb(ab)^\right)^* bbbb \left(aa(ab)^bb(ab)^\right)^\right)^ \\~~
	~~\, & \cdots \\~~
	e_{n+1} & = (\,\underbrace{a\cdots a}_{2^n}\, \cdot \, e_n\, \cdot\, \underbrace{b\cdots b}_{2^n}\, \cdot\, e_n \,)^*
	~~\end{alignat}~~
	</~~math>~~

	~~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}}.~~

	~~==Computing the star height of regular languages==~~
	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:~~

	~~{{cquote\|~~
	~~[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:

	~~<math>\left(10^{10^{10}}\right)^{\left(10^{10^{10}}\right)^{\left(10^{10^{10}}\right)}}.</math>~~
	~~\|4=S. Lombardy and J. Sakarovitch~~
	~~\|5=''Star Height of Reversible Languages and Universal Automata'', LATIN 2002~~
	}}
	~~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]]~~.

	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.~~

	~~==See also==~~
	*[~~[Generalized star height problem]]~~

	~~==References==~~
	*{{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 }}
	*{{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 }}
	* {{cite journal
	~~\| title = The Loop Complexity of Pure-Group Events~~
	~~\| year = 1967~~
	~~\| last = McNaughton \|first=Robert~~
	~~\| journal = Information and Control~~
	~~\| pages = 167–176~~
	~~\| volume = 11~~
	~~\| issue = 1–2~~
	~~\| doi=10.1016/S0019-9958(67)90481-0~~
	}}
	*{{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)~~]
	* {{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 }}~~
	*{{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 }}
	*{{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 }}
	*{{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 }}~~
	*{{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 }}~~
	* {{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 }}

	~~[[Category:Automata theory]]~~
	~~[[Category:Formal languages]]~~
	~~[[Category:Theorems in discrete mathematics]~~]