Editing Star height problem

{{Short description|Problem in formal language theory}}

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 first introduced by Eggan in 1963.{{sfn|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 Eggan 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 {{harvtxt|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 Dejean and Schützenberger in 1966.{{sfn|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&ndash;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 {{harvtxt|Dejean|Schützenberger|1966}} and by {{harvtxt|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.{{sfn|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 language|decidable]].{{sfn|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.{{sfn|Hashiguchi|1988}} The algorithm wasn't at all practical, being of non-[[ELEMENTARY|elementary]] complexity. To illustrate the immense resource consumptions of that algorithm, {{harvtxt|Lombardy|Sakarovitch|2002}} give some actual numbers:

{{Quotation|text=[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>|author=S. Lombardy and J. Sakarovitch|title=''Star Height of Reversible Languages and Universal Automata''|source=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.{{sfn|Kirsten|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.

This algorithm has been optimized and generalized to trees by Colcombet and Löding in 2008,{{sfn|Colcombet|Löding|2008}} as part of the theory of regular cost functions.
It has been implemented in 2017 in the tool suite Stamina.{{sfn|Fijalkow|Gimbert|Kelmendi|Kuperberg|2017}}

==See also==
*[[Generalized star height problem]]
*[[Kleene's algorithm]] &mdash; computes a regular expression (usually of non-minimal star height) for a language given by a [[deterministic finite automaton]]

==Notes==
{{Reflist}}

==References==
* {{cite book |author-link=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=[https://archive.org/details/formallanguageth0000unse/page/23 23–47] |publisher=Academic Press |location=New York |year=1980 |isbn=978-0-12-115350-2 |chapter-url=https://archive.org/details/formallanguageth0000unse/page/23 }} [https://www.cs.uwaterloo.ca/research/tr/1980/CS-80-03.pdf (technical report version)]
* {{cite book|last1=Colcombet|first1=Thomas|last2=Löding|first2=Christof|title=Computer Science Logic |chapter=The Nesting-Depth of Disjunctive μ-Calculus for Tree Languages and the Limitedness Problem |series=Lecture Notes in Computer Science|volume=5213|year=2008|pages=416–430|issn=0302-9743|doi=10.1007/978-3-540-87531-4_30|isbn=978-3-540-87530-7}}
* {{cite journal |first1=Françoise |last1=Dejean |author-link2=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 |doi-access= }}
* {{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 | doi-access=free }}
* {{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
| doi-access = 
 }}
* {{cite book |title=Jewels of Formal Language Theory |last= Salomaa |first= Arto |author-link=Arto Salomaa |year=1981 |publisher=Pitman Publishing |location=Melbourne |isbn=978-0-273-08522-5 |zbl=0487.68063 }}

==Further reading==
* {{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 |doi-access= }}
* {{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 |doi-access=free }}
* {{cite book |first1=Sylvain |last1=Lombardy |first2=Jacques |last2=Sakarovitch |title=LATIN 2002: Theoretical Informatics |chapter=Star Height of Reversible Languages and Universal Automata |volume=2286 |series=Lecture Notes of Computer Science |publisher=Springer |url=http://www-igm.univ-mlv.fr/~lombardy/publi/LATIN.pdf |year=2002 |pages=76–90 |doi=10.1007/3-540-45995-2_12 |isbn=978-3-540-43400-9 }}
* {{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 |url=http://www.numdam.org/item/ITA_2005__39_3_455_0/ }}
* {{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 }}
* {{Cite conference |last=Fijalkow |first=Nathanaël |last2=Gimbert |first2=Hugo |last3=Kelmendi |first3=Edon |last4=Kuperberg |first4=Denis |date=2017 |editor-last=Carayol |editor-first=Arnaud |editor2-last=Nicaud |editor2-first=Cyril |title=Stamina: Stabilisation Monoids in Automata Theory |url=https://link.springer.com/chapter/10.1007/978-3-319-60134-2_9 |conference=CIAA |location=Cham |publisher=Springer International Publishing |pages=101–112 |doi=10.1007/978-3-319-60134-2_9 |isbn=978-3-319-60134-2}}

[[Category:Automata (computation)]]
[[Category:Formal languages]]
[[Category:Theorems in discrete mathematics]]