Abstract
For a language $L$, we consider its cyclic closure, and more generally the language $C^{k}(L)$, which consists of all words obtained by partitioning words from $L$ into $k$ factors and permuting them. We prove that the classes of ET0L and EDT0L languages are closed under the operators $C^k$. This both sharpens and generalises Brandstädt's result that if $L$ is context-free then $C^{k}(L)$ is context-sensitive and not context-free in general for $k \geq 3$. We also show that the cyclic closure of an indexed language is indexed.
Highlights
In this note we investigate closure properties of context-free, ET0L, EDT0L and indexed languages under the operation of permuting a finite number of factors
Brandstädt (1981) proved that regular, context-sensitive and recursively enumerable languages are closed under Ck, so our results extend this list to include ET0L and EDT0L
We prove in Proposition 2.9 that if L1 is an ET0L language where each word in L1 has two symbols a, b appearing exactly once, L2 = {uabwv | uavbw ∈ L1} is ET0L
Summary
In this note we investigate closure properties of context-free, ET0L, EDT0L and indexed languages under the operation of permuting a finite number of factors. We sharpen a result of Brandstädt (1981) who proved that if L is context-free (respectively one-counter, linear) the language. The cyclic closure of a language, as well as the generalization Ck, are natural operations on languages, which can prove useful in determining whether a language belongs to a certain class. These operations are relevant when studying languages attached to conjugacy in groups and semigroups (see Ciobanu et al (2016))
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