A duplication is basic phenomenon that occurs through molecular evolution on a biological sequence. A duplication on a string copies any substring of the string. We define k-pseudo-duplication of a string w that consists, roughly speaking, of all strings obtained from w by inserting after a substring u another substring obtained from u by at most k edit operations. We consider three variants of duplication operations, duplication, k-pseudo-duplication and reverse-duplication. First, we give the necessary and sufficient number of states that a nondeterministic finite automaton needs to recognize duplications on a string. Then, we show that regular languages and context-free languages are not closed under the duplication, k-pseudo-duplication and reverse-duplication operations. Furthermore, we show that the class of context-sensitive languages is closed under duplication, pseudo-duplication and reverse-duplication.
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