## Resolution-Based Methods for Modal Logics (2000)

Venue: | Logic J. IGPL |

Citations: | 41 - 19 self |

### BibTeX

@ARTICLE{Nivelle00resolution-basedmethods,

author = {Hans De Nivelle and Renate A. Schmidt and Ulrich Hustadt},

title = {Resolution-Based Methods for Modal Logics},

journal = {Logic J. IGPL},

year = {2000},

volume = {8},

pages = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we give an overview of resolution methods for extended propositional modal logics. We adopt the standard translation approach and consider different resolution refinements which provide decision procedures for the resulting clause sets. Our procedures are based on ordered resolution and selection-based resolution. The logics that we cover are multi-modal logics defined over relations closed under intersection, union, converse and possibly complementation.

### Citations

331 | The relative efficiency of propositional proof systems
- Cook, Reckhow
- 1979
(Show Context)
Citation Context ... system B iff there is a polynomial function g such that for any formula ϕ, g maps derivations from ϕ in A to derivations from ϕ in B. The first notion generalises the notion of p-simulation found =-=in [6]-=-, who are only concerned with the p-simulation of proofs (that is, successful derivations leading to a proof). Simulation of search is a relationship in the opposite direction. It implies that A does ... |

210 | Modal languages and bounded fragments of predicate logic
- Andréka, Benthem, et al.
- 1996
(Show Context)
Citation Context ...d Fragment In this section we use ordered resolution with selection as a decision procedure for the guarded fragment. The guarded fragment wassrst shown decidable by Andreka, Nemeti and Van Benthem [1]. Gradel [20] has shown that the satisability problem for the guarded fragment is DEXPTIME-complete. There it was also shown that the guard condition is necessary only for the universal quantiers, ... |

197 |
Rewrite-based equational theorem proving with selection and simplification
- Bachmair, Ganzinger
- 1994
(Show Context)
Citation Context ...esolution, extended with selection. A-ordered resolution is well-known and widely used in resolution decision procedures [12, 13, 5, 34, 30, 26]. It follows from the results in Bachmair and Ganzinger =-=[3, 4]-=- that A-ordered resolution can be combined with a selection function. This selection function can override the A-ordering, give preference to inferences with negative literals. A-ordered resolution wi... |

154 | Resolution theorem proving
- BACHMAIR, GANZINGER
(Show Context)
Citation Context ...esolution, extended with selection. A-ordered resolution is well-known and widely used in resolution decision procedures [12, 13, 5, 34, 30, 26]. It follows from the results in Bachmair and Ganzinger =-=[3, 4]-=- that A-ordered resolution can be combined with a selection function. This selection function can override the A-ordering, give preference to inferences with negative literals. A-ordered resolution wi... |

56 | Nivelle. A superposition decision procedure for the guarded fragment with equality
- Ganzinger, de
- 1999
(Show Context)
Citation Context ...sive modal and description logics, resolution provers provide decision procedures, and often the same renements decide alsosrst-order generalisations such as the guarded fragment or Maslov's class K [=-=14, 2-=-6]. This paper gives an overview of dierent resolution renements which provide decision procedures forsrst-order fragments corresponding to a variety of extended modal logics. We will focus on fragmen... |

49 |
The Relative Eciency of Propositional Proof systems
- Cook, Reckhow
- 1979
(Show Context)
Citation Context ...f a system B i there is a polynomial function g such that for any formula ', g maps derivations from ' in A to derivations from ' in B. Thesrst notion generalises the notion of p-simulation found in [=-=6]-=-, who are only concerned with the p-simulation of proofs (that is, successful derivations leading to a proof). Simulation of search is a relationship in the opposite direction. It implies that A does ... |

45 |
Resolution methods for the decision problem
- Fermüller, Leitsch, et al.
- 1993
(Show Context)
Citation Context ...sion procedures for very expressive logics, while if we are interested in generating models for satisable formulae selection-based renements (or hyperresolution) are more natural (Fermuller et al. [12=-=, 13]-=-, Leitsch [30], Hustadt and Schmidt [28, 29]). We will describe three resolution decision procedures: an ordered resolution decision procedure for a class of clauses induced by K (m) (\; [; ; `) (Sect... |

30 | Back and Forth between Modal Logic and Classical Logic
- Andr6ka, Benthem, et al.
(Show Context)
Citation Context ...was introduced in Hustadt and Schmidt [28] with the purpose of handling expressive description logics. The other one is the guarded fragment, which was introduced by Andreka, Van Benthem and Nemeti [2=-=-=-] as the `modal subset ofsrst-order logic'. Although it did not quite meet the ambitious goals, it is an important fragment, containing many modal logics. The class of DL -clauses is related to the c... |

21 |
A Resolution Decision Procedure for the Guarded Fragment
- Nivelle
- 1998
(Show Context)
Citation Context ...guard condition is necessary only for the universal quantiers, when the formula is in negation normal form. A resolution decision procedure for the guarded fragment wassrst established in de Nivelle [=-=9]-=-. In Ganzinger and de Nivelle [14] the method was adapted to the guarded fragment with equality. It is shown there that the complexity of the resolution decision procedure is consistent with the compl... |

17 | A resolution-based decision procedure for extensions of K4
- Ganzinger, Hustadt, et al.
- 2001
(Show Context)
Citation Context ...resented a decision procedure are modal logics with transitive modalities. To decide extensions of K4 one possibility is to modify the calculus and add ordered chaining rules for transitive relations =-=[15]-=-. Another possibility is to use the resolution procedures described in this paper but block further inferences with clauses containing terms in which the level of nesting exceeds a pre-computed term d... |

15 | Soft typing for ordered resolution
- Ganzinger, Meyer, et al.
- 1997
(Show Context)
Citation Context ...will be performed. To prevent this we have to provide the means by which the resolution procedure can recognise the redundancy of further inference steps. This may possibly be realised by soft typing =-=[16-=-] or some form of blocking which is analogous to loop checking [27]. 290 Resolution-Based Methods for Modal Logics In this section we have focussed on single-step prexed tableaux calculi, but this cho... |

10 |
Rijke. A system of dynamic modal logic
- de
- 1992
(Show Context)
Citation Context ...or example, Bliksem [10], SPASS [40], and Otter [34]). The translation approach is generic, it can handlesrst-order modal logics, undecidable modal logics, for example, de Rijke's dynamic modal logic =-=[11]-=-, and combinations of modal and non-modal logics. In all cases we can at least ensure soundness and completeness. For a large class of expressive modal and description logics, resolution provers provi... |

8 |
Symbolic Logic
- J
- 1957
(Show Context)
Citation Context ...t was introduced in Hustadt and Schmidt [28] with the purpose of handling expressive description logics. The other one is the guarded fragment, which was introduced by Andréka, Van Benthem and Német=-=i [2] as th-=-e ‘modal subset of first-order logic’. Although it did not quite meet the ambitious goals, it is an important fragment, containing many modal logics. The class of DL ∗ -clauses is related to the... |

6 |
Nivelle. Ordering refinements of resolution
- de
- 1994
(Show Context)
Citation Context ...L) =A. A-orderings are extended to literals by L ≻ L ′ iff at(L) ≻ at(L ′ ). If one uses orderings that do not ignore the negation sign (these are called L-orderings), one does not loose compl=-=eteness [7]. -=-However L-orderings cannot be combined with selection. Given an A-ordering ≻, we define the maximal literals in a clause in the standard way: A literal L in a clause C is maximal in C, if there is n... |

5 | Resolution theorem proving - Fermuller, Leitsch, et al. - 2000 |

2 |
Nivelle. Ordering Re of Resolution
- de
- 1995
(Show Context)
Citation Context ...L) = A. A-orderings are extended to literals by L L 0 i at(L) at(L 0 ). If one uses orderings that do not ignore the negation sign (these are called L-orderings), one does not loose completeness [7]. However L-orderings cannot be combined with selection. Given an A-ordering , we dene the maximal literals in a clause in the standard way: A literal L in a clause C is maximal in C, if there is no... |

1 |
Nivelle. An overview of resolution decision procedures
- de
- 1998
(Show Context)
Citation Context ...to the class S + in Fermuller et al. [12]. This class was introduced there as the clause fragment belonging to the two-variable fragment. The class S + can only be decided by a non-liftable ordering [=-=8]-=-, or by an A-ordering combined with a rule called monadisation [12]. Since we try to root our approach on the common basis of liftable orderings, we slightly restrict the class, so that it can be deci... |

1 |
Modal languages and bounded fragments of
- Andréka, Németi, et al.
(Show Context)
Citation Context ...d Fragment In this section we use ordered resolution with selection as a decision procedure for the guarded fragment. The guarded fragment was first shown decidable by Andréka, Németi and Van Benthe=-=m [1].-=- Grädel [20] has shown that the satisfiability problem for the guarded fragment is DEXPTIME-complete. There it was also shown that the guard condition is necessary only for the universal quantifiers,... |