### Abstract

A free binoid Σ*(○,•) over a finite alphabet Σ is a free algebra generated by Σ with two independent associative operators, ○ and •. It has also the same identity λ to both operations. Any element of Σ*(○,•) is denoted uniquely by a sequence of symbols from the extended alphabet E(Σ)=Σ∪{○,•,(,)}, and any subset of a free binoid is called a binoid language. The set of regular binoid expressions are introduced so that all languages denoted by regular binoid expressions are those which contain finite binoid languages, and closed under five operations, ∪,○-concatenation, •-concatenation, ○-closure and •-closure. It is shown that for any regular (monoid) expression denoting a binoid language R, there exists a regular binoid expression denoting R. This result together with the main result in a previous paper implies that the class of binoid languages denoted by binoid regular expressions is the same as the class of binoid languages denoted by regular expressions over free binoids.

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

Pages (from-to) | 251-266 |

Number of pages | 16 |

Journal | Theoretical Computer Science |

Volume | 312 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - Jan 30 2004 |

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### Keywords

- Free binoid
- Regular (monoid) expression
- Regular binoid expression
- Regular language

### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Theoretical Computer Science*,

*312*(2-3), 251-266. https://doi.org/10.1016/j.tcs.2003.09.005

**Equivalence of regular binoid expressions and regular expressions denoting binoid languages over free binoids.** / Hashiguchi, Kosaburo; Sakakibara, Naoto; Jinbo, Shuji.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 312, no. 2-3, pp. 251-266. https://doi.org/10.1016/j.tcs.2003.09.005

}

TY - JOUR

T1 - Equivalence of regular binoid expressions and regular expressions denoting binoid languages over free binoids

AU - Hashiguchi, Kosaburo

AU - Sakakibara, Naoto

AU - Jinbo, Shuji

PY - 2004/1/30

Y1 - 2004/1/30

N2 - A free binoid Σ*(○,•) over a finite alphabet Σ is a free algebra generated by Σ with two independent associative operators, ○ and •. It has also the same identity λ to both operations. Any element of Σ*(○,•) is denoted uniquely by a sequence of symbols from the extended alphabet E(Σ)=Σ∪{○,•,(,)}, and any subset of a free binoid is called a binoid language. The set of regular binoid expressions are introduced so that all languages denoted by regular binoid expressions are those which contain finite binoid languages, and closed under five operations, ∪,○-concatenation, •-concatenation, ○-closure and •-closure. It is shown that for any regular (monoid) expression denoting a binoid language R, there exists a regular binoid expression denoting R. This result together with the main result in a previous paper implies that the class of binoid languages denoted by binoid regular expressions is the same as the class of binoid languages denoted by regular expressions over free binoids.

AB - A free binoid Σ*(○,•) over a finite alphabet Σ is a free algebra generated by Σ with two independent associative operators, ○ and •. It has also the same identity λ to both operations. Any element of Σ*(○,•) is denoted uniquely by a sequence of symbols from the extended alphabet E(Σ)=Σ∪{○,•,(,)}, and any subset of a free binoid is called a binoid language. The set of regular binoid expressions are introduced so that all languages denoted by regular binoid expressions are those which contain finite binoid languages, and closed under five operations, ∪,○-concatenation, •-concatenation, ○-closure and •-closure. It is shown that for any regular (monoid) expression denoting a binoid language R, there exists a regular binoid expression denoting R. This result together with the main result in a previous paper implies that the class of binoid languages denoted by binoid regular expressions is the same as the class of binoid languages denoted by regular expressions over free binoids.

KW - Free binoid

KW - Regular (monoid) expression

KW - Regular binoid expression

KW - Regular language

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U2 - 10.1016/j.tcs.2003.09.005

DO - 10.1016/j.tcs.2003.09.005

M3 - Article

AN - SCOPUS:0347600634

VL - 312

SP - 251

EP - 266

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 2-3

ER -