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

P.R.J. Asveld

Research output: Book/ReportReportProfessional

5 Citations (Scopus)
47 Downloads (Pure)

Abstract

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.
Original languageUndefined
Place of PublicationEnschede
PublisherCentrum voor Telematica en Informatie Technologie
Number of pages12
Publication statusPublished - 2007

Publication series

NameCTIT Technical Report Series
No.07-28
ISSN (Print)1381-3625

Keywords

  • HMI-SLT: Speech and Language Technology
  • IR-67078
  • METIS-245697
  • EWI-9722

Cite this

Asveld, P. R. J. (2007). Generating All Circular Shifts by Context-Free Grammars in Greibach Normal Form. (CTIT Technical Report Series; No. 07-28). Enschede: Centrum voor Telematica en Informatie Technologie.
Asveld, P.R.J. / Generating All Circular Shifts by Context-Free Grammars in Greibach Normal Form. Enschede : Centrum voor Telematica en Informatie Technologie, 2007. 12 p. (CTIT Technical Report Series; 07-28).
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abstract = "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.",
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Asveld, PRJ 2007, Generating All Circular Shifts by Context-Free Grammars in Greibach Normal Form. CTIT Technical Report Series, no. 07-28, Centrum voor Telematica en Informatie Technologie, Enschede.

Generating All Circular Shifts by Context-Free Grammars in Greibach Normal Form. / Asveld, P.R.J.

Enschede : Centrum voor Telematica en Informatie Technologie, 2007. 12 p. (CTIT Technical Report Series; No. 07-28).

Research output: Book/ReportReportProfessional

TY - BOOK

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PY - 2007

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N2 - 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.

AB - 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.

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KW - METIS-245697

KW - EWI-9722

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Asveld PRJ. Generating All Circular Shifts by Context-Free Grammars in Greibach Normal Form. Enschede: Centrum voor Telematica en Informatie Technologie, 2007. 12 p. (CTIT Technical Report Series; 07-28).