## A New Combination Procedure for the Word Problem that Generalizes Fusion Decidability Results in Modal Logics (2004)

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Venue: | In David A. Basin and Michaël Rusinowitch, editors, IJCAR ’04 |

Citations: | 15 - 8 self |

### BibTeX

@INPROCEEDINGS{Baader04anew,

author = {Franz Baader and Silvio Ghilardi and Cesare Tinelli},

title = {A New Combination Procedure for the Word Problem that Generalizes Fusion Decidability Results in Modal Logics},

booktitle = {In David A. Basin and Michaël Rusinowitch, editors, IJCAR ’04},

year = {2004},

pages = {183--197},

publisher = {Springer-Verlag}

}

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### Abstract

Previous results for combining decision procedures for the word problem in the non-disjoint case do not apply to equational theories induced by modal logics---whose combination is not disjoint since they share the theory of Boolean algebras. Conversely, decidability results for the fusion of modal logics are strongly tailored towards the special theories at hand, and thus do not generalize to other equational theories.

### Citations

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Citation Context ...e results presented here. The interested reader can ¯nd them in [2]. 2 Preliminaries In this paper we will use standard notions from equational logic, universal algebra and term rewriting (see, e.g., =-=[5]-=-). We consider only ¯rst-order theories (with equality ) over a functional signature. We use the letters §;, possibly with subscripts, to denote signatures. Throughout the paper, we ¯x a countablyin... |

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Citation Context ...ranslate the rewritten term-clauses into the corresponding modal formulae, and then check whether their complement is unsatis¯able in all Kripke structures with a re°exive accessibility relation (see =-=[10]-=-, Fig. 5.1). non-disjoint case, this approach has the known decidability transfer results for validity in the fusion of modal logics [15, 25] as consequences. Our combination result is however more ge... |

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Citation Context ...idity problem (Does the formula ' follow from the global assumption Ã?). There are strong combination results that show that in many cases decidability transfers from two modal logics to their fusion =-=[15, 23, 25, 4]-=-. Again, transfer results for the harder decision problem, relativized validity,2 are easier to show than for the simpler one, validity. In fact, for validity the results only apply to so-called norma... |

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Citation Context ...ption logics into modal logic notation) for which the validity problem is decidable, but the relativized validity problem is not. This is, e.g, the case for description logics with feature agreements =-=[1]-=- or with concrete domains [3]. Our new combination approach is orthogonal to the previous combination approaches for the word problem in equational theories over non-disjoint signatures [7, 12]. On th... |

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Citation Context ... notation) for which the validity problem is decidable, but the relativized validity problem is not. This is, e.g, the case for description logics with feature agreements [1] or with concrete domains =-=[3]-=-. Our new combination approach is orthogonal to the previous combination approaches for the word problem in equational theories over non-disjoint signatures [7, 12]. On the one hand, the previous resu... |

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Citation Context ...d problems (see [6] for a more detailed discussion). Both the Nelson-Oppen method and the methods for combining decision procedures for the word problem have been generalized to the non-disjoint case =-=[11, 24, 7, 12]-=-. The main restriction on the theories to be combined is that they share only so-called constructors. In modal logics, one is interested in whether properties (like decidability, ¯nite axiomatizabilit... |

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Citation Context ...d problems (see [6] for a more detailed discussion). Both the Nelson-Oppen method and the methods for combining decision procedures for the word problem have been generalized to the non-disjoint case =-=[11, 24, 7, 12]-=-. The main restriction on the theories to be combined is that they share only so-called constructors. In modal logics, one is interested in whether properties (like decidability, ¯nite axiomatizabilit... |

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(Show Context)
Citation Context ...riate restrictions on the theories to be combined. In automated deduction, the Nelson-Oppen combination procedure [17, 16] as well as the problem of combining decision procedures for the word problem =-=[19, 21, 18, 6]-=- have drawn considerable attention. The Nelson-Oppen method combines decision procedures for the validity of quanti¯er-free formulae in so-called stably in¯nite theories. If we restrict the attention ... |

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Citation Context ...The §-diagram ¢§X(A) of A (w.r.t. X) consists of all ground §(X)-literals that hold in A. We write just¢§(A) whenX coincides with the whole carrier of A. By a result known as Robinson's Diagram Lemma =-=[9]-=- embeddings and diagrams are related as follows. Lemma 1. Let A be a §-algebra generated by a set X, and let B be an - algebra for some ¶ §(X). Then A can be §(X)-embedded into B i® B is a model of... |

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