Permutations of context-free, ET0L and indexed languages

TL;DR

This paper investigates how certain language classes, including ET0L, EDT0L, and indexed languages, behave under permutation operations like cyclic closure and factor partitioning, revealing closure properties and generalizations.

Contribution

It proves that ET0L and EDT0L languages are closed under the operator C^k, extending previous results on context-free languages and showing cyclic closure preserves indexed languages.

Findings

ET0L and EDT0L languages are closed under C^k operations.

Cyclic closure of an indexed language remains indexed.

Generalizes Brandstädt's result on context-free languages.

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 Brandst\"adt'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.