Star-free language

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 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 compliment, 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 AC⁰.