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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 { a , b } {\displaystyle \{a,\,b\}} that do not have consecutive a's can be defined by c {\displaystyle ^{c}} , where X c {\displaystyle X^{c}} denotes the complement of a subset X {\displaystyle X} of { a , b } ∗ {\displaystyle \{a,\,b\}^{*}}. The condition is equivalent to having generalized star height zero.
An example of a regular language which is not star-free is { n ∣ n ≥ 0 } {\displaystyle \{^{n}\mid n\geq 0\}} , 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, 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 AC.