Decimations of languages and state complexity

Abstract

Let the words of a language $L$ be arranged in increasing radix order: $L=\{w_0,w_1,w_2,\dots \}$. We consider transformations that extract terms from $L$ in an arithmetic progression. For example, two such transformations are $\mathrm{even}\left(L\right)=\{w_0,w_2,w_4\dots \}$ and $\mathrm{odd}\left(L\right)=\{w_1,w_3,w_5,\dots \}$. Lecomte and Rigo observed that if $L$ is regular, then so are $\mathrm{even}\left(L\right)$, $\mathrm{odd}\left(L\right)$, and analogous transformations of $L$. We find good upper and lower bounds on the state complexity of this transformation. We also give an example of a context-free language $L$ such that $\mathrm{even}\left(L\right)$ is not context-free.