### Abstract

Let L_{k,m} be the set of formulas of first order logic containing only variables from x_{1}, x_{2}, ... x_{k} and having quantifier depth at most m. Let C_{k,m} be the extension of L _{k,m} obtained by allowing counting quantifiers meaning that there are at least i distinct xj 's. It is shown that for constants h ≥ 1, there are pairs of graphs such that h-dimensional Weisfeiler-Lehman refinement (h-dim W-L) can distinguish the two graphs, but requires at least a linear number of iterations. Despite of this slow progress, 2h-dim W-L only requires O(n) iterations, and 3h-1-dim W-L only requires O(log n) iterations. In terms of logic, this means that there is a c > 0 and a class of non-isomorphic pairs (G^{h}_{n}H^{h}_{n}) of graphs with G ^{h}_{n} and H^{h}_{n} having O(n) vertices such that the same sentences of L_{h+1}cn and C_{h+1}cn hold (h + 1 variables, depth cn), even though G^{h}_{n} and H ^{h}_{n} can already be distinguished by a sentence of L _{k,m} and thus C_{km} for some k > h and m = O(log n).

Original language | English (US) |
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Title of host publication | Automata, Languages and Programming - 28th International Colloquium, ICALP 2001, Proceedings |

Editors | Fernando Orejas, Paul G. Spirakis, Jan van Leeuwen |

Publisher | Springer Verlag |

Pages | 322-333 |

Number of pages | 12 |

ISBN (Print) | 3540422870, 9783540422877 |

DOIs | |

State | Published - 2001 |

Event | 28th International Colloquium on Automata, Languages and Programming, ICALP 2001 - Crete, Greece Duration: Jul 8 2001 → Jul 12 2001 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 2076 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 28th International Colloquium on Automata, Languages and Programming, ICALP 2001 |
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Country | Greece |

City | Crete |

Period | 7/8/01 → 7/12/01 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Automata, Languages and Programming - 28th International Colloquium, ICALP 2001, Proceedings*(pp. 322-333). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2076 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-48224-5_27