### Abstract

We define context-free grammars with Müller acceptance condition that generate languages of countable words. We establish several elementary properties of the class of Müller context-free languages including closure properties and others. We show that every Müller context-free grammar can be transformed into a normal form grammar in polynomial space, and then we show that many decision problems can be decided in polynomial time for Müller context-free grammars in normal form. These decision problems include deciding whether the language generated by a normal form grammar contains only well-ordered, scattered, or dense words. In a further result, we establish a limitedness property of Müller context-free grammars: if the language generated by a grammar contains only scattered words, then either there is an integer n such that each word of the language has Hausdorff rank at most n, or the language contains scattered words of arbitrarily large Hausdorff rank. We also show that it is decidable which of the two cases applies.

Original language | English |
---|---|

Pages (from-to) | 17-32 |

Number of pages | 16 |

Journal | Theoretical Computer Science |

Volume | 416 |

DOIs | |

Publication status | Published - Jan 27 2012 |

### Fingerprint

### Keywords

- Context-free languages of countable words
- Countable words
- Müller acceptance condition
- Quasi-dense and dense words
- Scattered
- Well-ordered

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*416*, 17-32. https://doi.org/10.1016/j.tcs.2011.10.012

**On Müller context-free grammars.** / Ésik, Z.; Iván, Szabolcs.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 416, pp. 17-32. https://doi.org/10.1016/j.tcs.2011.10.012

}

TY - JOUR

T1 - On Müller context-free grammars

AU - Ésik, Z.

AU - Iván, Szabolcs

PY - 2012/1/27

Y1 - 2012/1/27

N2 - We define context-free grammars with Müller acceptance condition that generate languages of countable words. We establish several elementary properties of the class of Müller context-free languages including closure properties and others. We show that every Müller context-free grammar can be transformed into a normal form grammar in polynomial space, and then we show that many decision problems can be decided in polynomial time for Müller context-free grammars in normal form. These decision problems include deciding whether the language generated by a normal form grammar contains only well-ordered, scattered, or dense words. In a further result, we establish a limitedness property of Müller context-free grammars: if the language generated by a grammar contains only scattered words, then either there is an integer n such that each word of the language has Hausdorff rank at most n, or the language contains scattered words of arbitrarily large Hausdorff rank. We also show that it is decidable which of the two cases applies.

AB - We define context-free grammars with Müller acceptance condition that generate languages of countable words. We establish several elementary properties of the class of Müller context-free languages including closure properties and others. We show that every Müller context-free grammar can be transformed into a normal form grammar in polynomial space, and then we show that many decision problems can be decided in polynomial time for Müller context-free grammars in normal form. These decision problems include deciding whether the language generated by a normal form grammar contains only well-ordered, scattered, or dense words. In a further result, we establish a limitedness property of Müller context-free grammars: if the language generated by a grammar contains only scattered words, then either there is an integer n such that each word of the language has Hausdorff rank at most n, or the language contains scattered words of arbitrarily large Hausdorff rank. We also show that it is decidable which of the two cases applies.

KW - Context-free languages of countable words

KW - Countable words

KW - Müller acceptance condition

KW - Quasi-dense and dense words

KW - Scattered

KW - Well-ordered

UR - http://www.scopus.com/inward/record.url?scp=84855428902&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84855428902&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2011.10.012

DO - 10.1016/j.tcs.2011.10.012

M3 - Article

AN - SCOPUS:84855428902

VL - 416

SP - 17

EP - 32

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -