Star height problem

The star height problem in formal language theory is the question whether all regular languages can be expressed using regular expressions of limited star height, i.e. with a limited nesting depth of Kleene stars. 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.^[1]

The first question was answered in the negative when in 1963, Eggan gave examples of regular languages of star heightn 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:

${\displaystyle {\begin{alignedat}{2}e_{1}&=a_{1}^{*}\\e_{2}&=\left(a_{1}^{*}a_{2}^{*}a_{3}\right)^{*}\\e_{3}&=\left(\left(a_{1}^{*}a_{2}^{*}a_{3}\right)^{*}\left(a_{4}^{*}a_{5}^{*}a_{6}\right)^{*}a_{7}\right)^{*}\\e_{4}&=\left(\left(\left(a_{1}^{*}a_{2}^{*}a_{3}\right)^{*}\left(a_{4}^{*}a_{5}^{*}a_{6}\right)^{*}a_{7}\right)^{*}\left(\left(a_{8}^{*}a_{9}^{*}a_{10}\right)^{*}\left(a_{11}^{*}a_{12}^{*}a_{13}\right)^{*}a_{14}\right)^{*}a_{15}\right)^{*}\end{alignedat}}}$

The construction principle for these expressions is that expression ${\displaystyle e_{n+1}}$ is obtained by concatenating two copies of ${\displaystyle e_{n}}$, 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 ${\displaystyle e_{n}}$ there is no equivalent regular expression of star height less than n; a proof is given in Eggan (1963).

However, Eggan's examples use a large alphabet, of size 2ⁿ-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.^[2] Their examples can be described by an inductively defined family of regular expressions over the binary alphabet ${\displaystyle \{a,b\}}$ as follows–cf. Salomaa (1981):

${\displaystyle {\begin{alignedat}{2}e_{1}&=(ab)^{*}\\e_{2}&=\left(aa(ab)^{*}bb(ab)^{*}\right)^{*}\\e_{3}&=\left(aaaa\left(aa(ab)^{*}bb(ab)^{*}\right)^{*}bbbb\left(aa(ab)^{*}bb(ab)^{*}\right)^{*}\right)^{*}\\\,&\cdots \\e_{n+1}&=(\,\underbrace {a\cdots a} _{2^{n}}\,\cdot \,e_{n}\,\cdot \,\underbrace {b\cdots b} _{2^{n}}\,\cdot \,e_{n}\,)^{*}\end{alignedat}}}$

Again, a rigorous proof is needed for the fact that ${\displaystyle e_{n}}$ does not admit an equivalent regular expression of lower star height. Proofs are given by Dejean & Schützenberger (1966) and by Salomaa (1981).

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.^[3] For years, there was only little progress. The pure-group languages were the first interesting family of regular languages for which the star height problem was proved to be decidable.^[4] But the general problem remained open for more than 25 years until it was settled by Hashiguchi, who in 1988 published an algorithm to determine the star height of any regular language.^[5] The algorithm wasn't at all practical, being of non-elementary complexity. To illustrate the immense resource consumptions of that algorithm, Lombardy & Sakarovitch (2002) give some actual numbers:

[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: ${\displaystyle \left(10^{10^{10}}\right)^{\left(10^{10^{10}}\right)^{\left(10^{10^{10}}\right)}}.}$

Notice that alone the number ${\displaystyle 10^{10^{10}}}$ has 10 billion zeros when written down in decimal notation, and is already by far larger than the number of atoms in the observable universe.

A much more efficient algorithm than Hashiguchi's procedure was devised by Kirsten in 2005.^[6] This algorithm runs, for a given nondeterministic finite automaton as input, within double-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,^[7] as part of the theory of regular cost functions. It has been implemented in 2017 in the tool suite Stamina.^[8]

• Generalized star height problem
• Kleene's algorithm— computes a regular expression (usually of non-minimal star height) for a language given by a deterministic finite automaton

• Brzozowski, Janusz A. (1980). "Open problems about regular languages". In Book, Ronald V. (ed.). Formal language theory—Perspectives and open problems. New York: Academic Press. pp. 23–47. ISBN 978-0-12-115350-2.(technical report version)
• Colcombet, Thomas; Löding, Christof (2008). "The Nesting-Depth of Disjunctive μ-Calculus for Tree Languages and the Limitedness Problem". Computer Science Logic. Lecture Notes in Computer Science. Vol. 5213. pp. 416–430. doi:10.1007/978-3-540-87531-4_30. ISBN 978-3-540-87530-7. ISSN 0302-9743.
• Dejean, Françoise; Schützenberger, Marcel-Paul (1966). "On a Question of Eggan". Information and Control. 9 (1): 23–25. doi:10.1016/S0019-9958(66)90083-0.
• Eggan, Lawrence C. (1963). "Transition graphs and the star-height of regular events". Michigan Mathematical Journal. 10 (4): 385–397. doi:10.1307/mmj/1028998975. Zbl 0173.01504.
• McNaughton, Robert (1967). "The Loop Complexity of Pure-Group Events". Information and Control. 11 (1–2): 167–176. doi:10.1016/S0019-9958(67)90481-0.
• Salomaa, Arto (1981). Jewels of Formal Language Theory. Melbourne: Pitman Publishing. ISBN 978-0-273-08522-5. Zbl 0487.68063.

• Hashiguchi, Kosaburo (1982). "Regular languages of star height one". Information and Control. 53 (2): 199–210. doi:10.1016/S0019-9958(82)91028-2.
• Hashiguchi, Kosaburo (1988). "Algorithms for Determining Relative Star Height and Star Height". Information and Computation. 78 (2): 124–169. doi:10.1016/0890-5401(88)90033-8.
• Lombardy, Sylvain; Sakarovitch, Jacques (2002). "Star Height of Reversible Languages and Universal Automata". LATIN 2002: Theoretical Informatics(PDF). Lecture Notes of Computer Science. Vol. 2286. Springer. pp. 76–90. doi:10.1007/3-540-45995-2_12. ISBN 978-3-540-43400-9.
• Kirsten, Daniel (2005). "Distance Desert Automata and the Star Height Problem". RAIRO - Informatique Théorique et Applications. 39 (3): 455–509. doi:10.1051/ita:2005027.
• Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge: Cambridge University Press. ISBN 978-0-521-84425-3. Zbl 1188.68177.
• Fijalkow, Nathanaël; Gimbert, Hugo; Kelmendi, Edon; Kuperberg, Denis (2017). Carayol, Arnaud; Nicaud, Cyril (eds.). Stamina: Stabilisation Monoids in Automata Theory. CIAA. Cham: Springer International Publishing. pp. 101–112. doi:10.1007/978-3-319-60134-2_9. ISBN 978-3-319-60134-2.