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

P.R.J. Asveld

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    Abstract

    Let $\{a_1,a_2,\ldots,a_n\}$ be an alphabet of $n$ symbols and let $C_n$ be the language of circular shifts of the word $a_1a_2\cdots a_n$; so $C_n = \{a_1a_2\cdots a_{n-1}a_n, a_2a_3\cdots a_na_1, \ldots,a_na_1\cdots a_{n-2}a_{n-1}\}$. We discuss a few families of context-free grammars $G_n$ ($n\geq 1$) in Chomsky normal form such that $G_n$ generates $C_n$. The grammars in these families are inverstigated with respect to their descriptional complexity, i.e., we determine the number of nonterminal symbols $\nu(n)$ and the number of rules $\pi(n)$ of $G_n$ as functions of $n$. These $\nu$ and $\pi$ 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 $\nu$ and $\pi$ are linear.
    Original languageUndefined
    Pages (from-to)147-159
    Number of pages13
    JournalJournal of automata, languages and combinatorics
    Volume11
    Issue number2
    Publication statusPublished - 2006

    Keywords

    • permutation
    • circular shift
    • unambiguous grammar
    • cyclic shift
    • descriptional complexity
    • Context-free grammar
    • HMI-SLT: Speech and Language Technology
    • METIS-248491
    • IR-62108
    • Chomsky normal form
    • EWI-11710

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