Accessible and Deterministic Automata: Enumeration and Boltzmann Samplers
We present a bijection between the set $\mathcal{A}_n$ of deterministic and accessible automata with $n$ states on a $k$-letters alphabet and some diagrams, which can themselves be represented as partitions of the set $[\![ 1..(kn+1) ]\!]$ into $n$ non-empty parts. This combinatorial construction sh...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2006-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/3499/pdf |
| _version_ | 1797270433434697728 |
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| author | Frédérique Bassino Cyril Nicaud |
| author_facet | Frédérique Bassino Cyril Nicaud |
| author_sort | Frédérique Bassino |
| collection | DOAJ |
| description | We present a bijection between the set $\mathcal{A}_n$ of deterministic and accessible automata with $n$ states on a $k$-letters alphabet and some diagrams, which can themselves be represented as partitions of the set $[\![ 1..(kn+1) ]\!]$ into $n$ non-empty parts. This combinatorial construction shows that the asymptotic order of the cardinality of $\mathcal{A}_n$ is related to the Stirling number $\{^{kn}_n\}$. Our bijective approach also yields an efficient random sampler of automata with $n$ states, of complexity $O(n^{3/2})$, using the framework of Boltzmann samplers. |
| first_indexed | 2024-04-25T02:04:11Z |
| format | Article |
| id | doaj.art-9935d85d78fa466f9622ecfa0b2fc791 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:04:11Z |
| publishDate | 2006-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-9935d85d78fa466f9622ecfa0b2fc7912024-03-07T14:34:36ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502006-01-01DMTCS Proceedings vol. AG,...Proceedings10.46298/dmtcs.34993499Accessible and Deterministic Automata: Enumeration and Boltzmann SamplersFrédérique Bassino0https://orcid.org/0000-0002-0127-9352Cyril Nicaud1https://orcid.org/0000-0002-8770-0119Laboratoire d'Informatique Gaspard-MongeLaboratoire d'Informatique Gaspard-MongeWe present a bijection between the set $\mathcal{A}_n$ of deterministic and accessible automata with $n$ states on a $k$-letters alphabet and some diagrams, which can themselves be represented as partitions of the set $[\![ 1..(kn+1) ]\!]$ into $n$ non-empty parts. This combinatorial construction shows that the asymptotic order of the cardinality of $\mathcal{A}_n$ is related to the Stirling number $\{^{kn}_n\}$. Our bijective approach also yields an efficient random sampler of automata with $n$ states, of complexity $O(n^{3/2})$, using the framework of Boltzmann samplers.https://dmtcs.episciences.org/3499/pdffinite automatabijectionasymptotic enumerationrandom generationboltzmann samplers[info.info-ds] computer science [cs]/data structures and algorithms [cs.ds][info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co] |
| spellingShingle | Frédérique Bassino Cyril Nicaud Accessible and Deterministic Automata: Enumeration and Boltzmann Samplers Discrete Mathematics & Theoretical Computer Science finite automata bijection asymptotic enumeration random generation boltzmann samplers [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| title | Accessible and Deterministic Automata: Enumeration and Boltzmann Samplers |
| title_full | Accessible and Deterministic Automata: Enumeration and Boltzmann Samplers |
| title_fullStr | Accessible and Deterministic Automata: Enumeration and Boltzmann Samplers |
| title_full_unstemmed | Accessible and Deterministic Automata: Enumeration and Boltzmann Samplers |
| title_short | Accessible and Deterministic Automata: Enumeration and Boltzmann Samplers |
| title_sort | accessible and deterministic automata enumeration and boltzmann samplers |
| topic | finite automata bijection asymptotic enumeration random generation boltzmann samplers [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/3499/pdf |
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