https://ris.utwente.nl/ws/files/6480056/Asveld06generating.pdf

Generating all permutations by context-free grammars in Chomsky normal form

http://www.cs.toronto.edu/~yuvalf/CFG-LB.pdf

Lower bounds for context-free grammars

https://core.ac.uk/download/pdf/11475198.pdf

Permuting operations on strings and their relation to prime numbers

For each alphabet $\Sigma_n=\{a_1,a_2,\ldots,a_n\}$, linearly ordered by $a_1<a_2<\cdots<a_n$, let $C_n$ be the language of circular or cyclic shifts over $\Sigma_n$, i.e., $C_n=\{a_1a_2\cdots a_{n-1}a_n,$$a_2a_3\cdots a_na_1,\ldots,a_na_1\cdots a_{n-2}a_{n-1}\}$. We study a few families of context-free grammars $G_n$ ($n\geq1$) in Greibach normal form such that $G_n$ generates $C_n$. The members of these grammar families are investigated with respect to the following descriptional complexity measures: the number of nonterminals $\nu(n)$, the number of rules $\pi(n)$ and the number of leftmost derivations $\delta(n)$ of $G_n$. As in the case of Chomsky normal form, these $\nu$, $\pi$ and $\delta$ are functions bounded by low-degree polynomials. However, the question whether there exists a family of grammars that is minimal with respect to all these measures remains open.

/pdf/generating-all-circular-shifts-by-context-free-grammars-in-183fu701a8.pdf

Generating All Circular Shifts by Context-Free Grammars in Greibach Normal Form

https://ris.utwente.nl/ws/files/5102304/2007CTIT-07-83.pdf

Generating all permutations by context-free grammars in Greibach normal form

Let {a1, a2,..., an} be an alphabet of n symbols and let Cn be the language of circular or cyclic shifts of the word a1a2 ... an; so Cn = {a1a2 ... an-1an, a2a3 ... ana1, ..., ana1..., an-2an-1}. We discuss a few families of context-free grammars Gn (n ≥ 1) in Chomsky normal form such that Gn generates Cn. The grammars in these families are investigated with respect to their descriptional complexity, i.e., we determine the number of nonterminal symbols ν(n) and the number of rules π(n) of Gn as functions of n. These ν and π happen to be functions bounded by low-degree polynomials, particularly when we focus our attention to unambiguous grammars. Finally, we introduce a family of minimal unambiguous grammars for which ν and π are linear.

/pdf/generating-all-circular-shifts-by-context-free-grammars-in-od30gg2e46.pdf

Peter R. J. Asveld

Papers

Generating all circular shifts by context-free grammars in Chomsky normal form