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. For instance, the language of words over the alphabet formula_1 that do not have consecutive a's can be defined by formula_2, where formula_3 denotes the complement of a subset formula_4 of formula_5. The condition is equivalent to having generalized star height zero.
An example of a regular language which is not star-free is formula_6.
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.