## E-connections of abstract description systems

Citations: | 98 - 26 self |

### BibTeX

@MISC{Kutz_e-connectionsof,

author = {Oliver Kutz and Carsten Lutz and Frank Wolter and Michael Zakharyaschev},

title = {E-connections of abstract description systems},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

Combining knowledge representation and reasoning formalisms is an important and challenging task. It is important because non-trivial AI applications often comprise different aspects of the world, thus requiring suitable combinations of available formalisms modeling each of these aspects. It is challenging because the computational behavior of the resulting hybrids is often much worse than the behavior of their components. In this paper, we propose a new combination method which is computationally robust in the sense that the combination of decidable formalisms is again decidable, and which, nonetheless, allows non-trivial interactions between the combined components. The new method, called E-connection, is defined in terms of abstract description systems (ADSs), a common generalization of description logics, many logics of time and space, as well as modal and epistemic logics. The basic idea of E-connections is that the interpretation domains of n combined systems are disjoint, and that these domains are connected by means of n-ary ‘link relations. ’ We define several natural variants of E-connections and study in-depth the transfer of decidability from the component systems to their E-connections. Key words: description logics, temporal logics, spatial logics, combining logics, decidability.

### Citations

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Citation Context ...s which require predefined predicates or temporal and spatial dimensions, combinations of DLs with concrete domains such as the natural numbers equipped with predicates like <, Allen’s interval logic =-=[1]-=-, or region connection calculus RCC-8 [29] have been proposed [7, 19]. The addition of a concrete domain to a DL is a rather sensitive operation as far as the preservation of its nice computational pr... |

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Citation Context ...temporal and spatial dimensions, combinations of DLs with concrete domains such as the natural numbers equipped with predicates like <, Allen’s interval logic [1], or region connection calculus RCC-8 =-=[29]-=- have been proposed [7, 19]. The addition of a concrete domain to a DL is a rather sensitive operation as far as the preservation of its nice computational properties is concerned: even ‘weak’ DLs com... |

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(Show Context)
Citation Context ...tion of any finite number of decidable ADSs is decidable as well. Here are three simple examples of E-connections; in more detail they will be discussed in Section 4. DL–SL: A DL L1 (say, ALC or SHIQ =-=[22]-=-) talks about a domain D1 of abstract objects. A spatial logic L2 (say, qualitative S4u [35, 12, 33, 16] or quantitative MS of [34, 24]) talks about a spatial domain D2. An obvious E-connection is giv... |

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(Show Context)
Citation Context ...ensions, combinations of DLs with concrete domains such as the natural numbers equipped with predicates like <, Allen’s interval logic [1], or region connection calculus RCC-8 [29] have been proposed =-=[7, 19]-=-. The addition of a concrete domain to a DL is a rather sensitive operation as far as the preservation of its nice computational properties is concerned: even ‘weak’ DLs combined with ‘weak’ concrete ... |

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Citation Context ... chains of functional roles) and ALC +,◦,⊔ (extending ALC with the transitive closure, composition, and union of roles). For both DLs, the subsumption of concept descriptions is known to be decidable =-=[21, 32, 6]-=-. However, the subsumption problem for their union ALCF +,◦,⊔ is undecidable [3]. Recently, fusions (or independent joints) have been proposed as a more robust way of combining DLs [5, 4, 36]. But eve... |

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Citation Context ...ensions, combinations of DLs with concrete domains such as the natural numbers equipped with predicates like <, Allen’s interval logic [1], or region connection calculus RCC-8 [29] have been proposed =-=[7, 19]-=-. The addition of a concrete domain to a DL is a rather sensitive operation as far as the preservation of its nice computational properties is concerned: even ‘weak’ DLs combined with ‘weak’ concrete ... |

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Citation Context ... of formalisms modeling each of these aspects. It is difficult because the computational behavior of the resulting hybrids is often much worse than the behavior of the combined components; see, e.g., =-=[11, 14, 16]-=-. To be more specific, consider three examples related to description logics. (i) Classical description logics (DLs) represent knowledge at a rather abstract logical level. To cope with applications w... |

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Citation Context ... chains of functional roles) and ALC +,◦,⊔ (extending ALC with the transitive closure, composition, and union of roles). For both DLs, the subsumption of concept descriptions is known to be decidable =-=[21, 32, 6]-=-. However, the subsumption problem for their union ALCF +,◦,⊔ is undecidable [3]. Recently, fusions (or independent joints) have been proposed as a more robust way of combining DLs [5, 4, 36]. But eve... |

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Citation Context ...n with X. The concept country ⊓ E 1 (X) would denote then the union of all alpine countries. DL–TL: Now, let L3 be a temporal logic (say, point-based PTL [15] or HalpernShoham’s logic of intervals HS =-=[20]-=-) and let D3 be a set of time points or, respectively, time intervals interpreting L3. In this case, a natural relation E ⊆ D1 × D3 is given by taking (x, y) ∈ E iff y belongs to the life-span of x. D... |

94 |
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Citation Context ...ted to sets Γ of object assertions will be called the A-satisfiability problem for S. The following theorem is an almost immediate consequence of the decidability of the corresponding logics, consult =-=[17]-=- for S4u, [34] for MS, and [15] for PTL. Theorem 5. The satisfiability problem is decidable for the ADSs S4 ♯ u , MS♯ , and PTL ♯ . It is also decidable for L ♯ whenever L is a DL with a decidable sat... |

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Citation Context ...nt static knowledge which is timeand agent-independent. To take into account the dynamic aspects of knowledge, DLs have been extended with temporal, dynamic, epistemic and other intentional operators =-=[25, 9, 10, 13, 2, 39, 41]-=-. The resulting formalisms become ‘many-dimensional’ and sometimes show rather nasty computational behavior: combinations of simple description logics (say, ALC) with simple temporal logics (say, prop... |

85 | Modal logics for qualitative spatial reasoning
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Citation Context ...ples of E-connections; in more detail they will be discussed in Section 4. DL–SL: A DL L1 (say, ALC or SHIQ [22]) talks about a domain D1 of abstract objects. A spatial logic L2 (say, qualitative S4u =-=[35, 12, 33, 16]-=- or quantitative MS of [34, 24]) talks about a spatial domain D2. An obvious E-connection is given by the relation E ⊆ D1 × D2 defined by taking (x, y) ∈ E iff y belongs to the spatial extension of x ... |

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Citation Context ...cal information system about Europe. One part of the task is easy. You take the description logic ALCO (extending ALC with nominals, i.e., concept names which have to be interpreted as singleton sets =-=[31, 23]-=-) and, using concepts Country, Treaty, etc., nominals EU, Schengen treaty, object names Spain, Luxembourg, UK, etc., and a role member, write Luxembourg : ∃member.EU ⊓ ∃member.Schengen treaty Iceland ... |

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Citation Context ... chains of functional roles) and ALC +,◦,⊔ (extending ALC with the transitive closure, composition, and union of roles). For both DLs, the subsumption of concept descriptions is known to be decidable =-=[21, 32, 6]-=-. However, the subsumption problem for their union ALCF +,◦,⊔ is undecidable [3]. Recently, fusions (or independent joints) have been proposed as a more robust way of combining DLs [5, 4, 36]. But eve... |

60 | Temporalizing Description Logics
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Citation Context ...nt static knowledge which is timeand agent-independent. To take into account the dynamic aspects of knowledge, DLs have been extended with temporal, dynamic, epistemic and other intentional operators =-=[25, 9, 10, 13, 2, 39, 41]-=-. The resulting formalisms become ‘many-dimensional’ and sometimes show rather nasty computational behavior: combinations of simple description logics (say, ALC) with simple temporal logics (say, prop... |

57 | Reasoning with concrete domains
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(Show Context)
Citation Context ... DLs combined with ‘weak’ concrete domains can become undecidable; see, e.g., [8, 18, 28]. In fact, to investigate DLs with concrete domains is rather hard and requires developing new techniques, cf. =-=[27]-=-. (ii) Standard DLs have been designed to represent static knowledge which is timeand agent-independent. To take into account the dynamic aspects of knowledge, DLs have been extended with temporal, dy... |

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53 | Fusions of description logics and abstract description systems
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Citation Context ...cidable [21, 32, 6]. However, the subsumption problem for their union ALCF +,◦,⊔ is undecidable [3]. Recently, fusions (or independent joints) have been proposed as a more robust way of combining DLs =-=[5, 4, 36]-=-. But even fusions behave badly if the class of models is not closed under disjoint unions, which is the case when nominals or the negation of roles are required [5, 4] (or if we combine logics of tim... |

46 | A canonical model of the region connection calculus
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Citation Context ...ber.EU France : Country Schengen treaty ⊑ Treaty ∃member.Schengen treaty ⊑ Country, After that you have to say something about the geography of Europe. To this end you can use the spatial logic RCC-8 =-=[29, 12, 30, 37]-=- or the more expressive formalism of S4u in which one can encode the topological meaning of the RCC-8 predicates—DC (disconnected), EQ (equal), EC (externally connected), NTPP (non-tangential proper e... |

46 | Fusions of modal logics revisited
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Citation Context ...cidable [21, 32, 6]. However, the subsumption problem for their union ALCF +,◦,⊔ is undecidable [3]. Recently, fusions (or independent joints) have been proposed as a more robust way of combining DLs =-=[5, 4, 36]-=-. But even fusions behave badly if the class of models is not closed under disjoint unions, which is the case when nominals or the negation of roles are required [5, 4] (or if we combine logics of tim... |

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Citation Context ...omposition, and union of roles). For both DLs, the subsumption of concept descriptions is known to be decidable [21, 32, 6]. However, the subsumption problem for their union ALCF +,◦,⊔ is undecidable =-=[3]-=-. Recently, fusions (or independent joints) have been proposed as a more robust way of combining DLs [5, 4, 36]. But even fusions behave badly if the class of models is not closed under disjoint union... |

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Citation Context ...nt static knowledge which is timeand agent-independent. To take into account the dynamic aspects of knowledge, DLs have been extended with temporal, dynamic, epistemic and other intentional operators =-=[25, 9, 10, 13, 2, 39, 41]-=-. The resulting formalisms become ‘many-dimensional’ and sometimes show rather nasty computational behavior: combinations of simple description logics (say, ALC) with simple temporal logics (say, prop... |

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Citation Context ... is a rather sensitive operation as far as the preservation of its nice computational properties is concerned: even ‘weak’ DLs combined with ‘weak’ concrete domains can become undecidable; see, e.g., =-=[8, 18, 28]-=-. In fact, to investigate DLs with concrete domains is rather hard and requires developing new techniques, cf. [27]. (ii) Standard DLs have been designed to represent static knowledge which is timeand... |

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Citation Context ...ples of E-connections; in more detail they will be discussed in Section 4. DL–SL: A DL L1 (say, ALC or SHIQ [22]) talks about a domain D1 of abstract objects. A spatial logic L2 (say, qualitative S4u =-=[35, 12, 33, 16]-=- or quantitative MS of [34, 24]) talks about a spatial domain D2. An obvious E-connection is given by the relation E ⊆ D1 × D2 defined by taking (x, y) ∈ E iff y belongs to the spatial extension of x ... |

24 |
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Citation Context ...ber.EU France : Country Schengen treaty ⊑ Treaty ∃member.Schengen treaty ⊑ Country, After that you have to say something about the geography of Europe. To this end you can use the spatial logic RCC-8 =-=[29, 12, 30, 37]-=- or the more expressive formalism of S4u in which one can encode the topological meaning of the RCC-8 predicates—DC (disconnected), EQ (equal), EC (externally connected), NTPP (non-tangential proper e... |

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Citation Context ...ples of E-connections; in more detail they will be discussed in Section 4. DL–SL: A DL L1 (say, ALC or SHIQ [22]) talks about a domain D1 of abstract objects. A spatial logic L2 (say, qualitative S4u =-=[35, 12, 33, 16]-=- or quantitative MS of [34, 24]) talks about a spatial domain D2. An obvious E-connection is given by the relation E ⊆ D1 × D2 defined by taking (x, y) ∈ E iff y belongs to the spatial extension of x ... |

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Citation Context ...e highly undecidable [10, 41, 40, 16]. These logics also require new approaches [41, 16], and it is still unclear whether practical reasoning systems can be developed for many-dimensional logics, cf. =-=[26]-=-. (iii) Often there is a need to combine two or more description logics: while one part of the application domain may require constructors of DL1, another part can only be represented using constructo... |

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Citation Context ... is a rather sensitive operation as far as the preservation of its nice computational properties is concerned: even ‘weak’ DLs combined with ‘weak’ concrete domains can become undecidable; see, e.g., =-=[8, 18, 28]-=-. In fact, to investigate DLs with concrete domains is rather hard and requires developing new techniques, cf. [27]. (ii) Standard DLs have been designed to represent static knowledge which is timeand... |

14 |
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Citation Context ...hey will be discussed in Section 4. DL–SL: A DL L1 (say, ALC or SHIQ [22]) talks about a domain D1 of abstract objects. A spatial logic L2 (say, qualitative S4u [35, 12, 33, 16] or quantitative MS of =-=[34, 24]-=-) talks about a spatial domain D2. An obvious E-connection is given by the relation E ⊆ D1 × D2 defined by taking (x, y) ∈ E iff y belongs to the spatial extension of x whenever x occupies some space.... |

10 |
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Citation Context ...LCO with, say, SHIQ of [22], having the required constructors, doesn’t help either because transfer results for fusions are available so far only for DLs whose models are closed under disjoint unions =-=[5, 4]-=-—which is not the case if nominals are allowed as concept constructors. It seems that a perspective way to attack this problem is to connect SHIQ with ALCO. 10Let E contain three binary relations bet... |

5 | editors, “Frontiers of Combining Systems - Baader, Schulz - 1996 |

2 |
Ontology reasoning in the SHIQ(d) description logic
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Citation Context ...cal information system about Europe. One part of the task is easy. You take the description logic ALCO (extending ALC with nominals, i.e., concept names which have to be interpreted as singleton sets =-=[31, 23]-=-) and, using concepts Country, Treaty, etc., nominals EU, Schengen treaty, object names Spain, Luxembourg, UK, etc., and a role member, write Luxembourg : ∃member.EU ⊓ ∃member.Schengen treaty Iceland ... |