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 that do not have consecutive a's can be defined by, where denotes the complement of a subset of. The condition is equivalent to having generalized star height zero.
An example of a regular language which is not star-free is.
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.
