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 , i.e. the language of strings consisting of an even number of "a". Marcel-Paul Schützenberger characterized star-free languages as those with aperiodic syntactic monoids. They can also be characterized logically as languages definable in FO[