### Abstract

We consider unsatisfiable Boolean formulas in conjunctive normal form. It is known that unsatisfiability can be shown by a regular resolution proof in time polynomial in the number of variables n, and exponential in the tree-width w. It is also known that satisfiability for bounded tree-width can actually be decided in time linear in the length of the formula and exponential in the tree-width w. We investigate the complexities of resolution proofs and arbitrary proofs in more detail depending on the number of variables n and the tree-width w. We present two very traditional algorithms, one based on resolution and the other based on truth tables. Maybe surprisingly, we point out that the two algorithms turn out to be basically the same algorithm with different interpretations. Similar results holds for a bound w' on the tree-width of the incidence graph for a somewhat extended notion of a resolution proof. The length of any proper resolution proof might be quadratic in n, but if we allow to introduce abbreviations for frequently occurring subclauses, then the proof length and running time are again linear in n.

Original language | English (US) |
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Title of host publication | LATIN 2012 |

Subtitle of host publication | Theoretical Informatics - 10th Latin American Symposium, Proceedings |

Pages | 387-398 |

Number of pages | 12 |

DOIs | |

State | Published - May 15 2012 |

Event | 10th Latin American Symposiumon Theoretical Informatics, LATIN 2012 - Arequipa, Peru Duration: Apr 16 2012 → Apr 20 2012 |

### 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 | 7256 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 10th Latin American Symposiumon Theoretical Informatics, LATIN 2012 |
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Country | Peru |

City | Arequipa |

Period | 4/16/12 → 4/20/12 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*LATIN 2012: Theoretical Informatics - 10th Latin American Symposium, Proceedings*(pp. 387-398). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7256 LNCS). https://doi.org/10.1007/978-3-642-29344-3_33