### 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) |
---|---|

Title of host publication | Automata, Languages and Programming - 28th International Colloquium, ICALP 2001, Proceedings |

Pages | 322-333 |

Number of pages | 12 |

State | Published - Dec 1 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) |
---|---|

Volume | 2076 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

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

Country | Greece |

City | Crete |

Period | 7/8/01 → 7/12/01 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### 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).

}

*Automata, Languages and Programming - 28th International Colloquium, ICALP 2001, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2076 LNCS, pp. 322-333, 28th International Colloquium on Automata, Languages and Programming, ICALP 2001, Crete, Greece, 7/8/01.

**Weisfeiler-lehman refinement requires at least a linear number of iterations.** / Furer, Martin.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Weisfeiler-lehman refinement requires at least a linear number of iterations

AU - Furer, Martin

PY - 2001/12/1

Y1 - 2001/12/1

N2 - Let Lk,m be the set of formulas of first order logic containing only variables from x1, x2, ... xk and having quantifier depth at most m. Let Ck,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 (GhnHhn) of graphs with G hn and Hhn having O(n) vertices such that the same sentences of Lh+1cn and Ch+1cn hold (h + 1 variables, depth cn), even though Ghn and H hn can already be distinguished by a sentence of L k,m and thus Ckm for some k > h and m = O(log n).

AB - Let Lk,m be the set of formulas of first order logic containing only variables from x1, x2, ... xk and having quantifier depth at most m. Let Ck,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 (GhnHhn) of graphs with G hn and Hhn having O(n) vertices such that the same sentences of Lh+1cn and Ch+1cn hold (h + 1 variables, depth cn), even though Ghn and H hn can already be distinguished by a sentence of L k,m and thus Ckm for some k > h and m = O(log n).

UR - http://www.scopus.com/inward/record.url?scp=77956227541&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956227541&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:77956227541

SN - 3540422870

SN - 9783540422877

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 322

EP - 333

BT - Automata, Languages and Programming - 28th International Colloquium, ICALP 2001, Proceedings

ER -