Context insertions

In this paper we consider an operation of inserting contexts in a word controlled by a contextual scheme X which provides a selection criterion for contextual insertion. We say that a language L is k-stable w.r.t. a contextual scheme X if by making any k context insertions in a word of L we still obtain a word of L; L is k-anti-stable w.r.t. X if by making any k context insertions in a word of L we get a word not in L; L is called k-error-correctable w.r.t. X if by making any k context insertions in a word x of L we get either a word in L or a word not in L which cannot be also obtained by making k context insertions in a word z of L different from x. We prove that all these properties are decidable for regular languages. We then define a distance between two words that measures the minimal number of context insertions in one of the words in order to obtain the other. Some properties of this distance, which is actually a semimetric, are investigated. © 2011 Springer-Verlag Berlin Heidelberg.