Decomposable graphs and definitions with no quantifier alternation
Let $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism in terms of the adjacency and the equality relations. Let $D_0(G)$ be a variant of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Using large graphs decomposable in...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2005-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/3423/pdf |
| _version_ | 1797270392532893696 |
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| author | Oleg Pikhurko Joel Spencer Oleg Verbitsky |
| author_facet | Oleg Pikhurko Joel Spencer Oleg Verbitsky |
| author_sort | Oleg Pikhurko |
| collection | DOAJ |
| description | Let $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism in terms of the adjacency and the equality relations. Let $D_0(G)$ be a variant of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Using large graphs decomposable in complement-connected components by a short sequence of serial and parallel decompositions, we show examples of $G$ on $n$ vertices with $D_0(G) \leq 2 \log^{\ast}n+O(1)$. On the other hand, we prove a lower bound $D_0(G) \geq \log^{\ast}n-\log^{\ast}\log^{\ast}n-O(1)$ for all $G$. Here $\log^{\ast}n$ is equal to the minimum number of iterations of the binary logarithm needed to bring $n$ below $1$. |
| first_indexed | 2024-04-25T02:03:32Z |
| format | Article |
| id | doaj.art-fb0c56348c134d5f968912580ececb1f |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:03:32Z |
| publishDate | 2005-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-fb0c56348c134d5f968912580ececb1f2024-03-07T14:41:15ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502005-01-01DMTCS Proceedings vol. AE,...Proceedings10.46298/dmtcs.34233423Decomposable graphs and definitions with no quantifier alternationOleg Pikhurko0https://orcid.org/0000-0002-9524-1901Joel Spencer1Oleg Verbitsky2https://orcid.org/0000-0002-9524-1901Department of Mathematical SciencesCourant Institute of Mathematical Sciences [New York]Institut fur InformatikLet $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism in terms of the adjacency and the equality relations. Let $D_0(G)$ be a variant of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Using large graphs decomposable in complement-connected components by a short sequence of serial and parallel decompositions, we show examples of $G$ on $n$ vertices with $D_0(G) \leq 2 \log^{\ast}n+O(1)$. On the other hand, we prove a lower bound $D_0(G) \geq \log^{\ast}n-\log^{\ast}\log^{\ast}n-O(1)$ for all $G$. Here $\log^{\ast}n$ is equal to the minimum number of iterations of the binary logarithm needed to bring $n$ below $1$.https://dmtcs.episciences.org/3423/pdfdescriptive complexity of graphsfirst order logicehrenfeucht game on graphsgraph decompositions[info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co] |
| spellingShingle | Oleg Pikhurko Joel Spencer Oleg Verbitsky Decomposable graphs and definitions with no quantifier alternation Discrete Mathematics & Theoretical Computer Science descriptive complexity of graphs first order logic ehrenfeucht game on graphs graph decompositions [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| title | Decomposable graphs and definitions with no quantifier alternation |
| title_full | Decomposable graphs and definitions with no quantifier alternation |
| title_fullStr | Decomposable graphs and definitions with no quantifier alternation |
| title_full_unstemmed | Decomposable graphs and definitions with no quantifier alternation |
| title_short | Decomposable graphs and definitions with no quantifier alternation |
| title_sort | decomposable graphs and definitions with no quantifier alternation |
| topic | descriptive complexity of graphs first order logic ehrenfeucht game on graphs graph decompositions [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/3423/pdf |
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