Note on dissecting power of regular languages
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912585090072576 |
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| author | Rukavicka, Josef |
| author_facet | Rukavicka, Josef |
| contents | Let $c>1$ be a real constant. We say that a language $L$ is $c$-\emph{constantly growing} if for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c+\vert u\vert$. We say that a language $L$ is $c$-\emph{geometrically growing} if for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c\vert u\vert$. Given a language $L$, we say that $L$ is $REG$-\emph{dissectible} if there is a regular language $R$ such that $\vert L\setminus R\vert=\infty$ and $\vert L\cap R\vert=\infty$. In 2013, it was shown that every $c$-constantly growing language $L$ is $REG$-dissectible. In 2023, the following open question has been presented: "Is the family of geometrically growing languages $REG$-dissectible?" We construct a $c$-geometrically growing language $L$ that is not $REG$-dissectible. Hence we answer negatively to the open question. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_14114 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Note on dissecting power of regular languages Rukavicka, Josef Formal Languages and Automata Theory Let $c>1$ be a real constant. We say that a language $L$ is $c$-\emph{constantly growing} if for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c+\vert u\vert$. We say that a language $L$ is $c$-\emph{geometrically growing} if for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c\vert u\vert$. Given a language $L$, we say that $L$ is $REG$-\emph{dissectible} if there is a regular language $R$ such that $\vert L\setminus R\vert=\infty$ and $\vert L\cap R\vert=\infty$. In 2013, it was shown that every $c$-constantly growing language $L$ is $REG$-dissectible. In 2023, the following open question has been presented: "Is the family of geometrically growing languages $REG$-dissectible?" We construct a $c$-geometrically growing language $L$ that is not $REG$-dissectible. Hence we answer negatively to the open question. |
| title | Note on dissecting power of regular languages |
| topic | Formal Languages and Automata Theory |
| url | https://arxiv.org/abs/2310.14114 |