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Multi-component grammars, known in the literature as “multiple context-free grammars” and “linear context-free rewriting systems”, describe the structure of a string by defining the properties of k-tuples of its substrings, in the same way as ordinary formal grammars (Chomsky’s “context-free”) define properties of substrings. It is shown that, for every fixed k, the family of languages described by k-component grammars is closed under the cyclic shift operation. On the other hand, the subfamily defined by well-nested k-component grammars is not closed under the cyclic shift, yet their cyclic shifts are always defined by well-nested (k+1)-component grammars.
Okhotin, A., & Sorokin, A. (2020). Cyclic shift on multi-component grammars. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 12038 LNCS, pp. 287–299). Springer. https://doi.org/10.1007/978-3-030-40608-0_20