## Modal Logics, Description Logics and Arithmetic Reasoning (1999)

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Venue: | ARTIFICIAL INTELLIGENCE |

Citations: | 25 - 3 self |

### BibTeX

@ARTICLE{Ohlbach99modallogics,,

author = {Hans Jürgen Ohlbach and Jana Koehler},

title = {Modal Logics, Description Logics and Arithmetic Reasoning},

journal = {ARTIFICIAL INTELLIGENCE},

year = {1999},

volume = {109},

pages = {1--31}

}

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### Abstract

We introduce mathematical programming and atomic decomposition as the basic modal (T-Box) inference techniques for a large class of modal and description logics. The class of description logics suitable for the proposed methods is strong on the arithmetical side. In particular there may be complex arithmetical conditions on sets of accessible worlds (role fillers). The atomic decomposition technique can deal with set constructors for modal parameters (role terms) and parameter (role) hierarchies specied in full propositional logic. Besides the standard modal operators, a number of other constructors can be added in a relatively straightforward way. Examples are graded modalities (qualified number restrictions) and also generalized quantiers like `most', `n%', `more' and `many'.

### Citations

785 |
A framework for representing knowledge
- Minsky
- 1975
(Show Context)
Citation Context ...e, and formulae which are necessarily true. Therefore the modal 2operator was called the necessitation operator [11,12]. Description logics, on the other hand, are late descendants of Minski's frames =-=[18-=-] and Brachman's KL-ONE [6]. They come in a variety of dierent versions, e.g. ALC [24], CLASSIC [7], KRIS [2], LOOM [17] and even in class-based object oriented formalisms [10]. Common to most of them... |

634 |
Modal Logic: An Introduction
- Chellas
- 1980
(Show Context)
Citation Context ...odal logics were introduced to distinguish between formulae which are true just by chance, and formulae which are necessarily true. Therefore the modal 2operator was called the necessitation operator =-=[11,12-=-]. Description logics, on the other hand, are late descendants of Minski's frames [18] and Brachman's KL-ONE [6]. They come in a variety of dierent versions, e.g. ALC [24], CLASSIC [7], KRIS [2], LOOM... |

584 | An Overview of the KL-ONE Knowledge Representation System, in
- Brachman, Schmolze
- 1985
(Show Context)
Citation Context ...ecessarily true. Therefore the modal 2operator was called the necessitation operator [11,12]. Description logics, on the other hand, are late descendants of Minski's frames [18] and Brachman's KL-ONE =-=[6-=-]. They come in a variety of dierent versions, e.g. ALC [24], CLASSIC [7], KRIS [2], LOOM [17] and even in class-based object oriented formalisms [10]. Common to most of them is the separation of a de... |

261 | A Schema for Integrating Concrete Domains into Concept Languages
- Baader, Hanschke
- 1991
(Show Context)
Citation Context ... cubic capacity of 500 cc per cylinder.) If only numeric features occur in the arithmetic part then DL arc89 is almost like Baader and Hanschke's ALC(D), ALC with the concrete domain D = real numbers =-=-=-[3]. The dierence is our treatment of the existential quantier, which introduces a numeric constraint for the rolesllers. Therefore the consistency and subsumption checking algorithms are very dierent... |

255 | A correspondence theory for terminological logics: Preliminary report
- Schild
- 1991
(Show Context)
Citation Context ...e worlds (rolesllers). Although many modal logics and description logics are syntactic variants of each other { the description logic ALC, for example, corresponds exactly to the multi-modal logic Km =-=[22-=-] { their origin is completely dierent. Modal logics were introduced to distinguish between formulae which are true just by chance, and formulae which are necessarily true. Therefore the modal 2operat... |

228 | W.: The complexity of concept languages
- Donini, Lenzerini, et al.
- 1997
(Show Context)
Citation Context ...ssive than propositional logic. Much eort has been invested in recent years to explore the borderline between propositional logic and FOL by investigating various versions of description logics, see [=-=14]-=- for a good summary of recent results. Most methods for checking consistency of concept formulae and subsumption between concept formulae are tableau algorithms. Starting with a tableau entry a : ' (t... |

220 | Living With Classic: When and How to Use a KL-ONE-Like Language
- Brachman, McGuiness, et al.
- 1991
(Show Context)
Citation Context ...on operator [11,12]. Description logics, on the other hand, are late descendants of Minski's frames [18] and Brachman's KL-ONE [6]. They come in a variety of dierent versions, e.g. ALC [24], CLASSIC [=-=7]-=-, KRIS [2], LOOM [17] and even in class-based object oriented formalisms [10]. Common to most of them is the separation of a description logic database into a so-called T-Box (terminological box) and ... |

201 | Reasoning and Revision in Hybrid Representation Systems
- Nebel
- 1990
(Show Context)
Citation Context ...orm jrj n or jrj n, where n is an integer, ' c is a conjunction of concept names, and role terms consist of role names only, and the role hierarchy AR is empty then this language is the logic T F [1=-=9,14]-=-. If the atomic decomposition of the role term r is fa 1 ; : : : ; a n g then 8 r:s, (8 a 1 :s^ : : : ^ 8 a n : ): (10) Therefore the ' 8 -part of a DL arc8 -concept formula can be normalized such tha... |

168 | Attributive Concept Descriptions with Complements - Schmidt-Schauß, Smolka - 1991 |

155 |
The theory of representations of Boolean algebras
- Stone
- 1936
(Show Context)
Citation Context ...ional axiomatization. It exploits the fact that the elements and connectives of Boolean algebras can always be interpreted as sets and the corresponding set operations (Stone's representation theorem =-=[25]-=-). With some elementary Boolean algebra theory one can show that the models of the propositional axiomatization correspond to the atoms of the Boolean algebra generated by the closure of the sets unde... |

147 |
Attributive concept descriptions with complements
- Schmidt-Schau��, Smolka
- 1991
(Show Context)
Citation Context ...he necessitation operator [11,12]. Description logics, on the other hand, are late descendants of Minski's frames [18] and Brachman's KL-ONE [6]. They come in a variety of dierent versions, e.g. ALC [=-=24]-=-, CLASSIC [7], KRIS [2], LOOM [17] and even in class-based object oriented formalisms [10]. Common to most of them is the separation of a description logic database into a so-called T-Box (terminologi... |

101 |
Terminological Cycles: Semantics and Computational Properties
- Nebel
- 1991
(Show Context)
Citation Context ...has-child:(person ^ jhas-childj 1): Truly recursive concept denitions, where the rewriting does not terminate, are possible, but they require a dierent approach to the one presented in this paper [20=-=,23]-=-. Therefore we always assume that the concept formulae are in expanded normal form. (Since the expanded normal form may be exponential, a clever implementation needs to avoid the expansion.) The atomi... |

68 | A description classifier for the predicate calculus - MacGregor - 1994 |

57 | H.J.: Quantifier elimination in second-order predicate logic - Gabbay, Ohlbach - 1992 |

41 |
Untersuchungen über das Eliminationsproblem der mathematischen Logik
- Ackermann
(Show Context)
Citation Context ...imination of the existentially quantied p amounts to generating all resolvents with p in the clause form of AR . The resolvents represent all consequences of p and therefore p is no longer necessary [=-=1]-=-. This way one can have large databases of Boolean axioms, but for the actual problem at hand, the atomic decomposition takes into account only the relevant Boolean variables. Factoring Principle A Bo... |

37 |
A unified framework for class based representation formalisms
- Calvanese, Lenzerini, et al.
- 1994
(Show Context)
Citation Context ...ants of Minski's frames [18] and Brachman's KL-ONE [6]. They come in a variety of dierent versions, e.g. ALC [24], CLASSIC [7], KRIS [2], LOOM [17] and even in class-based object oriented formalisms [=-=10-=-]. Common to most of them is the separation of a description logic database into a so-called T-Box (terminological box) and a so-called A-Box (assertional box). The T-Box contains specications of conc... |

36 |
Knowledge Representation and Inference System
- KRIS
- 1991
(Show Context)
Citation Context ...r [11,12]. Description logics, on the other hand, are late descendants of Minski's frames [18] and Brachman's KL-ONE [6]. They come in a variety of dierent versions, e.g. ALC [24], CLASSIC [7], KRIS [=-=2]-=-, LOOM [17] and even in class-based object oriented formalisms [10]. Common to most of them is the separation of a description logic database into a so-called T-Box (terminological box) and a so-calle... |

36 | Finite model reasoning in description logics, in
- Calvanese
- 1996
(Show Context)
Citation Context .... (We assume the underlying arithmetic algorithms can deal with negated formulae.) Atomic decomposition has previously been used to develop inference algorithms for description logics, for example in =-=[9,8-=-]. In their approach, the concepts themselves are decomposed, not the roles. Since the technique was applied to a dierent logic (with inverse roles and arbitrary terminological axioms) , one cannot co... |

33 | Tobies, “A Description Logic with Transitive and Converse Roles, Role Hierarchies and Qualifying Number Restrictions
- Horrocks, Sattler, et al.
- 1999
(Show Context)
Citation Context ...special properties (re exivity, symmetry, transitivity) are also of great interest for description logics. The has-part relation, for example, which is very useful in technical domains, is transitive =-=[16-=-]. The algorithms presented above depend very much on the fact that the quanti ers 8 r:' and 9 r:sover ordinary role terms with no special properties dene 33 levels of rolesllers, which correspond to ... |

31 |
Terminological cycles and the propositional -calculus
- Schild
- 1994
(Show Context)
Citation Context ...has-child:(person ^ jhas-childj 1): Truly recursive concept denitions, where the rewriting does not terminate, are possible, but they require a dierent approach to the one presented in this paper [20=-=,23]-=-. Therefore we always assume that the concept formulae are in expanded normal form. (Since the expanded normal form may be exponential, a clever implementation needs to avoid the expansion.) The atomi... |

29 | Description logics with concrete domains and aggregation
- Baader, Sattler
- 1998
(Show Context)
Citation Context ...ation of `the set of people with more grandchildren than children' requires an expression like jhas-child; has-childj jhas-childj or a kind of aggregation functions which can lead to undecidability [5]. 28 6 Concept Formulae with Existential Quantiers The existential quantier 9 r:spostulates the existence of an r-rolesller in the concepts. As already mentioned, this quantier is denable in our l... |

29 | Making object-oriented schemas more expressive, in
- Calvanese, Lenzerini
- 1994
(Show Context)
Citation Context .... (We assume the underlying arithmetic algorithms can deal with negated formulae.) Atomic decomposition has previously been used to develop inference algorithms for description logics, for example in =-=[9,8-=-]. In their approach, the concepts themselves are decomposed, not the roles. Since the technique was applied to a dierent logic (with inverse roles and arbitrary terminological axioms) , one cannot co... |

18 | Terminological cycles and the propositional µcalculus - Schild - 1994 |

10 | Description logics with symbolic number restrictions
- Baader, Sattler
- 1996
(Show Context)
Citation Context ...eature of the object's children. We have not yet investigated how this language extensionsts into our approach. It is certainly not straightforward because consistency is undecidable in this language =-=[-=-4]. 8.2 Other Functional Roles Functional roles which have exactly (or at most) one rolesller can be dened using arithmetic constraints and the universal quantier. jhas-namej = 1 ^ 8 has-name:name spe... |

9 |
How to extend a formal system with a Boolean Algebra component
- Ohlbach, Koehler
- 1998
(Show Context)
Citation Context ... Decomposition In the main part of the paper we show how the consistency and the subsumption problem of concept formulae can be mapped to equation solving problems. The atomic decomposition technique =-=[21]-=- plays a key role in this process. Since the technique is not widely known, we give a brief overview. Atomic decomposition exploits the possibility to decomposesnite sets of sets into mutually disjoin... |

5 |
Quanti elimination in second-order predicate logic
- Gabbay, Ohlbach
- 1992
(Show Context)
Citation Context ... in fact short for 9p AR if p does not occur in '. AR is a conjunction of propositional formulae, and therefore the existentially quantied p can be eliminated using a quantier elimination procedure [1=-=5-=-]. The result is some formula AR 0 which is equivalent to 9p AR , but does not contain p. In the propositional case, elimination of the existentially quantied p amounts to generating all resolvents wi... |

4 |
Hilbert’s tenth problem is undecidable
- Davis
- 1973
(Show Context)
Citation Context ...iplication with numbers is allowed. More advanced systems also allow for certain non-linear terms. The general non-linear Diophantine equation problem, however, is undecidable (Hilbert's 10th problem =-=[13-=-]). Therefore the arithmetic language should not be too expressive. There are only a few requirements about the arithmetic system, which are important for the purposes of this paper. In the basic mod... |

4 |
A Description Classi for the Predicate Calculus
- MacGregor
- 1994
(Show Context)
Citation Context ... Description logics, on the other hand, are late descendants of Minski's frames [18] and Brachman's KL-ONE [6]. They come in a variety of dierent versions, e.g. ALC [24], CLASSIC [7], KRIS [2], LOOM [=-=17]-=- and even in class-based object oriented formalisms [10]. Common to most of them is the separation of a description logic database into a so-called T-Box (terminological box) and a so-called A-Box (as... |

2 |
Modal Logic, volume 35 of Oxford Logic Guides
- Chargov, Zakharayaschev
- 1997
(Show Context)
Citation Context ...odal logics were introduced to distinguish between formulae which are true just by chance, and formulae which are necessarily true. Therefore the modal 2operator was called the necessitation operator =-=[11,12-=-]. Description logics, on the other hand, are late descendants of Minski's frames [18] and Brachman's KL-ONE [6]. They come in a variety of dierent versions, e.g. ALC [24], CLASSIC [7], KRIS [2], LOOM... |