Star-free language

In theoretical computer science and formal language theory, a regular language is said to be star-free if it can be described by a regular expression constructed from the letters of the alphabet, the empty word, the empty set symbol, all boolean operators – including complementation – and concatenation but no Kleene star.[1] The condition is equivalent to having generalized star height zero.

For instance, the language ${\displaystyle \Sigma ^{*}}$ of all finite words over an alphabet ${\displaystyle \Sigma }$ can be shown to be star-free by taking the complement of the empty set, ${\displaystyle \Sigma ^{*}={\bar {\emptyset }}}$. Then, the language of words over the alphabet ${\displaystyle \{a,\,b\}}$ that do not have consecutive a's can be defined as ${\displaystyle {\overline {\Sigma ^{*}aa\Sigma ^{*}}}}$, first constructing the language of words consisting of ${\displaystyle aa}$ with an arbitrary prefix and suffix, and then taking its complement, which must be all words which do not contain the substring ${\displaystyle aa}$.

An example of a regular language which is not star-free is ${\displaystyle (aa)^{*}}$,[2] i.e. the language of strings consisting of an even number of "a". For ${\displaystyle (ab)^{*}}$ where ${\displaystyle a\neq b}$, the language can be defined as ${\displaystyle \Sigma ^{*}\setminus (b\Sigma ^{*}\cup \Sigma ^{*}a\cup \Sigma ^{*}aa\Sigma ^{*}\cup \Sigma ^{*}bb\Sigma ^{*})}$, taking the set of all words and removing from it words starting with ${\displaystyle b}$, ending in ${\displaystyle a}$ or containing ${\displaystyle aa}$ or ${\displaystyle bb}$. However, when ${\displaystyle a=b}$, this definition does not create ${\displaystyle (aa)^{*}}$.

Marcel-Paul Schützenberger characterized star-free languages as those with aperiodic syntactic monoids.[3][4] They can also be characterized logically as languages definable in FO[<], the first-order logic over the natural numbers with the less-than relation,[5] as the counter-free languages[6] and as languages definable in linear temporal logic.[7]

All star-free languages are in uniform AC^0.

See also

• Star height
• Star height problem
• Generalized star-height problem

Notes

1. Lawson (2004) p.235
2. Arto Salomaa (1981). Jewels of Formal Language Theory. Computer Science Press. p. 53. ISBN 978-0-914894-69-8.
3. Marcel-Paul Schützenberger (1965). "On finite monoids having only trivial subgroups" (PDF). Information and Computation. 8 (2): 190–194. doi:10.1016/s0019-9958(65)90108-7.
4. Lawson (2004) p.262
5. Straubing, Howard (1994). Finite automata, formal logic, and circuit complexity. Progress in Theoretical Computer Science. Basel: Birkhäuser. p. 79. ISBN 3-7643-3719-2. Zbl 0816.68086.
6. McNaughton, Robert; Papert, Seymour (1971). Counter-free Automata. Research Monograph. Vol. 65. With an appendix by William Henneman. MIT Press. ISBN 0-262-13076-9. Zbl 0232.94024.
7. Kamp, Johan Antony Willem (1968). Tense Logic and the Theory of Linear Order. University of California at Los Angeles (UCLA).

References

• Lawson, Mark V. (2004). Finite automata. Chapman and Hall/CRC. ISBN 1-58488-255-7. Zbl 1086.68074.
• Diekert, Volker; Gastin, Paul (2008). "First-order definable languages". In Jörg Flum; Erich Grädel; Thomas Wilke (eds.). Logic and automata: history and perspectives (PDF). Amsterdam University Press. ISBN 978-90-5356-576-6.