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. The condition is equivalent to having generalized star height zero.

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

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

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

All star-free languages are in uniform AC⁰.

See also

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

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.