|
|
| (One intermediate revision by one other user not shown) |
| 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}}.
| | == 蘭大人は非常に不吉なああ見て == |
|
| |
|
| ==Families of regular languages with unbounded star height==
| | あなたの資格が非常に良いではありません。蘭大人は非常に不吉なああ見て。 '<br>彼は秦ゆうの本体に行く前<br>シルバーフラワー祖母の話 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_12.php クリスチャンルブタンジャパン]。<br><br>秦Yuは肩を撫で、シルバーフラワー親切に祖母は言った：あなたはとても驚くほど練習スピードだ」の後にあなたが持続されている場合は、私は数万年、あなたはその後、驚くべき資格にあなたに神の領域に達することができるようになります......参照プラス蘭の大人のヘルプは、またはであること、そして子どもたちに一緒に立つことがあります。 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_9.php クリスチャンルブタン メンズ 通販] '<br><br>秦Yuはうなずいた [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン 店舗 東京]。<br><br>「素晴らしい資質が、私はまだあなたが広場でハイエンドに認定されていますか見てなかったのか？ 'シルバーフラワー祖母の心違法チャンネルを、「どのようにちょうど3000年の間、魂は、この手順を実行するだろうか？'<br><br>秦ゆう今回チャネル：「おばあちゃんが言った、秦ゆうが私の心には決して忘れない、私は知らない何秦ゆうを思い出させるおばあちゃん。 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_12.php 靴 クリスチャンルブタン] '<br>シルバーフラワー祖母への大きなスタンドを持つ子どものために<br>、秦Yuは非常に丁重にまだある。<br><br>シルバーフラワー祖母が突然言った： [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン通販] 'それは思い出させる」ことになると、私は非常に重要なことだと思います。「シルバーフラワー祖母はヤンを見て |
| 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:
| | 相关的主题文章： |
| | | <ul> |
| :<math>\begin{alignat}{2}
| | |
| e_1 &= a_1^* \\
| | <li>[http://tools.netbib.de/cgi-bin/mojo.cgi http://tools.netbib.de/cgi-bin/mojo.cgi]</li> |
| 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)^*\\
| | <li>[http://www.luoboni.com/home.php?mod=space&uid=1 http://www.luoboni.com/home.php?mod=space&uid=1]</li> |
| e_4 &= \left(
| | |
| \left(\left(a_1^*a_2^*a_3\right)^*\left(a_4^*a_5^*a_6\right)^*a_7\right)^*
| | <li>[http://www.dbbdqn.com/plus/feedback.php?aid=2514 http://www.dbbdqn.com/plus/feedback.php?aid=2514]</li> |
| \left(\left(a_8^*a_9^*a_{10}\right)^*\left(a_{11}^*a_{12}^*a_{13}\right)^*a_{14}\right)^*
| | |
| a_{15}\right)^*
| | </ul> |
| \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]]
| |
蘭大人は非常に不吉なああ見て
あなたの資格が非常に良いではありません。蘭大人は非常に不吉なああ見て。 '
彼は秦ゆうの本体に行く前
シルバーフラワー祖母の話 クリスチャンルブタンジャパン。
秦Yuは肩を撫で、シルバーフラワー親切に祖母は言った：あなたはとても驚くほど練習スピードだ」の後にあなたが持続されている場合は、私は数万年、あなたはその後、驚くべき資格にあなたに神の領域に達することができるようになります......参照プラス蘭の大人のヘルプは、またはであること、そして子どもたちに一緒に立つことがあります。 クリスチャンルブタン メンズ 通販 '
秦Yuはうなずいた クリスチャンルブタン 店舗 東京。
「素晴らしい資質が、私はまだあなたが広場でハイエンドに認定されていますか見てなかったのか？ 'シルバーフラワー祖母の心違法チャンネルを、「どのようにちょうど3000年の間、魂は、この手順を実行するだろうか？'
秦ゆう今回チャネル：「おばあちゃんが言った、秦ゆうが私の心には決して忘れない、私は知らない何秦ゆうを思い出させるおばあちゃん。 靴 クリスチャンルブタン '
シルバーフラワー祖母への大きなスタンドを持つ子どものために
、秦Yuは非常に丁重にまだある。
シルバーフラワー祖母が突然言った： クリスチャンルブタン通販 'それは思い出させる」ことになると、私は非常に重要なことだと思います。「シルバーフラワー祖母はヤンを見て
相关的主题文章：