## Finite model reasoning in description logics (1996)

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Venue: | In Proc. of the 5th Int. Conf. on the Principles of Knowledge Representation and Reasoning (KR-96 |

Citations: | 36 - 16 self |

### BibTeX

@INPROCEEDINGS{Calvanese96finitemodel,

author = {Diego Calvanese},

title = {Finite model reasoning in description logics},

booktitle = {In Proc. of the 5th Int. Conf. on the Principles of Knowledge Representation and Reasoning (KR-96},

year = {1996},

pages = {292--303},

publisher = {Morgan Kaufmann}

}

### Years of Citing Articles

### OpenURL

### Abstract

For the basic Description Logics reasoning with respect to finite models amounts to reasoning with respect to arbitrary ones, but finiteness of the domain needs to be considered if expressivity is increased and the finite model property fails. Procedures for reasoning with respect to arbitrary models in very expressive Description Logics have been developed, but these are not directly applicable in the finite case. We first show that we can nevertheless capture a restricted form of finiteness and represent finite modeling structures such as lists and trees, while still reasoning with respect to arbitrary models. The main result of this paper is a procedure to reason with respect to finite models in an expressive Description Logic equipped with inverse roles, cardinality constraints, and in which arbitrary inclusions between concepts can be specified without any restriction. This provides the necessary expressivity to go beyond most semantic and object-oriented Database models, and capture several useful extensions. 1

### Citations

477 |
Automata on infinite objects
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(Show Context)
Citation Context ...ty of rfcpdl is open. To show decidability of rfcpdl we reduce satisfiability of a formula to nonemptiness of the language accepted by a finite automaton on infinite objects derived from the formula (=-=Thomas, 1990-=-). This is a standard technique that provides tight upper bounds for various modal and temporal logics (Emerson & Jutla, 1989; Vardi & Wolper, 1994) and logics of programs (Vardi, 1985; Vardi & Wolper... |

334 | The Object Database Standard: ODMG-93 - Cattell, editor |

298 |
Results on the propositional µ-calculus
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Citation Context ...ted one to the next by means of role R. In terms of PDLs, wf (R) corresponds to a negated repeat formula ¬∆(R) (Streett, 1982), which is in turn equivalent to the least fixpoint expression 4 µX.∀R.X (=-=Kozen, 1983-=-). Thus we can use such constructor to express finite structures that admit a least-fixpoint definition. We illustrate this on the example of binary trees. Example 2 We can define a binary tree induct... |

260 | Results on the propositional -calculus - Kozen - 1983 |

255 | A correspondence theory for terminological logics: Preliminary report
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Citation Context ...ing with respect to arbitrary models in expressive DLs are based on a correspondence between DLs and Propositional Dynamic Logics (PDLs) and exploit the powerful reasoning methods developed for PDLs (=-=Schild, 1991-=-; De Giacomo & Lenzerini, 1994a). We show that the correspondence can in fact be extended to handle also well-foundedness, and present a reasoning technique for the resulting PDL. On the other hand, w... |

249 | Objects and views
- Abiteboul, Bonner
- 1991
(Show Context)
Citation Context ... have been proposed as extensions to existing models. Relevant examples are: • Classes defined through necessary and sufficient conditions, which correspond to so called views in object-oriented DBs (=-=Abiteboul & Bonner, 1991-=-). Using free assertions 5 it is also possible to specify only sufficient conditions for an object to be an instance of a class, and more general forms of integrity constraints. • Classes defined as t... |

228 | The complexity of concept languages
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(Show Context)
Citation Context ... the family of class-based formalisms we deal with. These formalisms exploit Description Logics for the definition of schemata, and the logics we define extend the well known logics of the AL-family (=-=Donini, Lenzerini, Nardi, & Nutt, 1991-=-a) by several expressive constructs. 2.1 SYNTAX AND SEMANTICS OF DESCRIPTION LOGICS The basic elements of Description Logics (DLs) are concepts and roles, which denote classes and binary relations, re... |

205 |
Automata-Theoretic Techniques for Modal Logics of Programs
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- 1986
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Citation Context ... (Thomas, 1990). This is a standard technique that provides tight upper bounds for various modal and temporal logics (Emerson & Jutla, 1989; Vardi & Wolper, 1994) and logics of programs (Vardi, 1985; =-=Vardi & Wolper, 1986-=-; Vardi & Stockmeyer, 1985), and we extend it to rfcpdl. It is based on the tree model property which holds also for rfcpdl. In particular, the automaton we obtain from a rfcpdl formula f is a hybrid ... |

178 | Description logics in data management
- Borgida
- 1995
(Show Context)
Citation Context ...y defining classes, which denote subsets of the domain, and by specifying various types of relations between classes, which establish their structural properties (Calvanese, Lenzerini, & Nardi, 1994; =-=Borgida, 1995-=-). In this context it is fundamental to provide methods for reasoning about such specifications in order to detect inconsistencies and hidden dependencies which arise from the specified constraints. T... |

147 | Terminological reasoning is inherently intractable
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- 1990
(Show Context)
Citation Context ...ma). Assumption 1 corresponds to the natural requirement that all the information concerning a class is contained in one place, which contributes to a better structuring of the represented knowledge (=-=Nebel, 1990-=-; Baader, 1990; Lecluse & Richard, 1989; Bergamaschi & Sartori, 1992). It may however be too restrictive, especially in those cases where one wants to state additional constraints that are not specifi... |

126 |
Logics of programs
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- 1990
(Show Context)
Citation Context ...ary models have been developed, which are based on a correspondence between DLs and Propositional Dynamic Logics (PDLs). PDLs are formalisms specifically designed for reasoning about program schemes (=-=Kozen & Tiuryn, 1990-=-), and their correspondence with DLs was described first in (Schild, 1991) and extended to more expressive logics in (De Giacomo & Lenzerini, 1994a). The correspondence allows one to reduce concept co... |

102 |
Terminological cycles: Semantics and computational properties
- Nebel
- 1991
(Show Context)
Citation Context ...schema are made, either implicitly or explicitly: 1. For each concept name there is at most one assertion in which that name appears in the left-hand side. 2. The schema contains no cycles 2 . 2 See (=-=Nebel, 1991-=-) for a formal definition of cycle in a schema.3. The schema contains only primitive concept specifications (primitive schema). Assumption 1 corresponds to the natural requirement that all the inform... |

96 | Part-Whole Relations in ObjectCentered Systems: an Overview
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- 1996
(Show Context)
Citation Context ...nverse roles. Another important use of the well-founded constructor is for the definition of well-founded relations, of which the part-of relation is a notable example with great practical relevance (=-=Artale, Franconi, Guarino, & Pazzi, 1996-=-; Sattler, 1995). Transitivity of the part-ofrelation can be captured by taking the transitive closure of a basic part of role, while asymmetricity and finiteness (i.e. well-foundedness) are imposed ... |

84 | Boosting the correspondence between description logics and propositional dynamic logics - Giacomo, Lenzerini - 1994 |

82 | TBox and ABox reasoning in expressive description logics - Giacomo, Lenzerini - 1996 |

64 |
Tractable concept languages
- Donini, Lenzerini, et al.
- 1991
(Show Context)
Citation Context ... the family of class-based formalisms we deal with. These formalisms exploit Description Logics for the definition of schemata, and the logics we define extend the well known logics of the AL-family (=-=Donini, Lenzerini, Nardi, & Nutt, 1991-=-a) by several expressive constructs. 2.1 SYNTAX AND SEMANTICS OF DESCRIPTION LOGICS The basic elements of Description Logics (DLs) are concepts and roles, which denote classes and binary relations, re... |

61 |
On taxonomic reasoning in conceptual design
- Bergamaschi, Sartori
- 1992
(Show Context)
Citation Context ...ment that all the information concerning a class is contained in one place, which contributes to a better structuring of the represented knowledge (Nebel, 1990; Baader, 1990; Lecluse & Richard, 1989; =-=Bergamaschi & Sartori, 1992-=-). It may however be too restrictive, especially in those cases where one wants to state additional constraints that are not specific to a certain class. Assumption 2 of acyclicity (which is meaningfu... |

57 |
Recurring dominoes;; making the highly undecidable highly understandable
- Harel
- 1983
(Show Context)
Citation Context ...in particular role-value maps, intersections of complex roles, and number restrictions on complex roles, make reasoning highly undecidable if they are used without restrictions (Schmidt-Schauß, 1989; =-=Harel, 1983-=-). Therefore, we syntactically restrict the use of these problematic constructors in a way similar to what done in (De Giacomo & Lenzerini, 1995) for the logic CAT S. The logic ALT , in which we disti... |

52 |
Modeling complex structures in object-oriented databases
- Lecluse, Richard
- 1989
(Show Context)
Citation Context ...ds to the natural requirement that all the information concerning a class is contained in one place, which contributes to a better structuring of the represented knowledge (Nebel, 1990; Baader, 1990; =-=Lecluse & Richard, 1989-=-; Bergamaschi & Sartori, 1992). It may however be too restrictive, especially in those cases where one wants to state additional constraints that are not specific to a certain class. Assumption 2 of a... |

49 | Structured objects: Modeling and reasoning - Calvanese, Giacomo, et al. - 1995 |

48 | What’s in an aggregate: Foundations for description logics with tuples and sets - Giacomo, Lenzerini - 1995 |

45 |
Terminological Cycles in KL-ONE-based Knowledge Representation Languages
- Baader
- 1990
(Show Context)
Citation Context ...on 1 corresponds to the natural requirement that all the information concerning a class is contained in one place, which contributes to a better structuring of the represented knowledge (Nebel, 1990; =-=Baader, 1990-=-; Lecluse & Richard, 1989; Bergamaschi & Sartori, 1992). It may however be too restrictive, especially in those cases where one wants to state additional constraints that are not specific to a certain... |

44 |
Propositional dynamic logic of looping and converse is elementary decidable
- Streett
- 1982
(Show Context)
Citation Context ...d by using wf (R), which expresses that there is no infinite sequence of objects, connected one to the next by means of role R. In terms of PDLs, wf (R) corresponds to a negated repeat formula ¬∆(R) (=-=Streett, 1982-=-), which is in turn equivalent to the least fixpoint expression 4 µX.∀R.X (Kozen, 1983). Thus we can use such constructor to express finite structures that admit a least-fixpoint definition. We illust... |

42 | Subsumption in KL-ONE is undecidable - Schmidt-Schau - 1989 |

37 |
A unified framework for class based representation formalisms
- Calvanese, Lenzerini, et al.
- 1994
(Show Context)
Citation Context ... is represented in form of a schema by defining classes, which denote subsets of the domain, and by specifying various types of relations between classes, which establish their structural properties (=-=Calvanese, Lenzerini, & Nardi, 1994-=-; Borgida, 1995). In this context it is fundamental to provide methods for reasoning about such specifications in order to detect inconsistencies and hidden dependencies which arise from the specified... |

37 | Concept language with number restrictions and fixpoints, and its relationship with µ-calculus - Giacomo, Lenzerini - 1994 |

36 | Reasoning with inclusion axioms in description logics: Algorithms and complexity
- Calvanese
- 1996
(Show Context)
Citation Context ...ly in a recursive way and thus through cycles. The reason for not admitting cycles is twofold. On one hand cycles greatly increase the computational complexity of reasoning on a schema (Baader, 1990; =-=Calvanese, 1996-=-a), and on the other hand, there is still no agreement in the field on which semantics to adopt in their presence. Besides descriptive semantics, which is the semantic specification described above, s... |

35 |
Improved upper and lower bounds for modal logics of programs
- VARDI, STOCKMEYER
- 1985
(Show Context)
Citation Context ...is a standard technique that provides tight upper bounds for various modal and temporal logics (Emerson & Jutla, 1989; Vardi & Wolper, 1994) and logics of programs (Vardi, 1985; Vardi & Wolper, 1986; =-=Vardi & Stockmeyer, 1985-=-), and we extend it to rfcpdl. It is based on the tree model property which holds also for rfcpdl. In particular, the automaton we obtain from a rfcpdl formula f is a hybrid automaton Hf := (Tf,Wf), t... |

31 | Terminological cycles and the propositional -calculus - Schild - 1994 |

28 | A concept language for an engineering application with part-whole relations
- Sattler
- 1995
(Show Context)
Citation Context ... well-founded constructor is for the definition of well-founded relations, of which the part-of relation is a notable example with great practical relevance (Artale, Franconi, Guarino, & Pazzi, 1996; =-=Sattler, 1995-=-). Transitivity of the part-ofrelation can be captured by taking the transitive closure of a basic part of role, while asymmetricity and finiteness (i.e. well-foundedness) are imposed by an assertion... |

27 | Unrestricted and Finite Model Reasoning in Class-Based Representation Formalisms. PhD thesis, Dipartimento di Informatica e Sistemistica, Università di Roma ”La Sapienza
- Calvanese
- 1996
(Show Context)
Citation Context ...ly in a recursive way and thus through cycles. The reason for not admitting cycles is twofold. On one hand cycles greatly increase the computational complexity of reasoning on a schema (Baader, 1990; =-=Calvanese, 1996-=-a), and on the other hand, there is still no agreement in the field on which semantics to adopt in their presence. Besides descriptive semantics, which is the semantic specification described above, s... |

24 |
Subsumption in KL-ONE is Undecidable
- Schmidt-Schauß
- 1989
(Show Context)
Citation Context ...ructors, however, and in particular role-value maps, intersections of complex roles, and number restrictions on complex roles, make reasoning highly undecidable if they are used without restrictions (=-=Schmidt-Schauß, 1989-=-; Harel, 1983). Therefore, we syntactically restrict the use of these problematic constructors in a way similar to what done in (De Giacomo & Lenzerini, 1995) for the logic CAT S. The logic ALT , in w... |

19 |
The taming of converse: Reasoning about two-way computations
- Vardi
- 1985
(Show Context)
Citation Context ...m the formula (Thomas, 1990). This is a standard technique that provides tight upper bounds for various modal and temporal logics (Emerson & Jutla, 1989; Vardi & Wolper, 1994) and logics of programs (=-=Vardi, 1985-=-; Vardi & Wolper, 1986; Vardi & Stockmeyer, 1985), and we extend it to rfcpdl. It is based on the tree model property which holds also for rfcpdl. In particular, the automaton we obtain from a rfcpdl ... |

18 |
Terminological cycles and the propositional µcalculus
- Schild
- 1994
(Show Context)
Citation Context ...tor together with descriptive semantics. We remark that recently there have been also proposals for an integration of the different types of semantics by including fixpoint constructors in the logic (=-=Schild, 1994-=-; De Giacomo & Lenzerini, 1994b) and interpreting cycles with descriptive semantics. Assumption 3 is sometimes made in concept-based KR systems (Buchheit et al., 1995) and it is usually implicit in DB... |

16 | A refined architecture for terminological systems: Terminology = schema+views
- Buchheit, Donini, et al.
- 1998
(Show Context)
Citation Context ... so called fixpoint semantics have been considered (Nebel, 1991). Descriptive semantics, which is the one we use, has been advocated to be the one which gives the most intuitive results (Nebel, 1991; =-=Buchheit, Donini, Nutt, & Schaerf, 1995-=-), and it is also the only one that can be adopted for the most general type of schemata, called free, in which none of the three conditions above is required to hold 3 . (Baader, 1990) argues that fr... |

11 | Deductive entity-relationship modeling - Battista, Lenzerini - 1993 |

9 | On simultaneously determinizing and complementing !-automata - Emerson, Jutla - 1989 |

6 | Logical and computational properties of the description logic MIRTL
- BUONGARZONI, MEGHINI, et al.
- 1995
(Show Context)
Citation Context ...eness of the domain, since it can always be assumed. Unfortunately, the finite model property does not hold for more expressive logics, as those studied for example in (De Giacomo & Lenzerini, 1994a; =-=Buongarzoni, Menghini, Salis, Sebastiani, & Straccia, 1995-=-). In particular, this happens also for the logics which include inverse roles and functionality of roles, and which therefore have sufficient expressivity to represent semantic and objectoriented DB ... |

3 |
On simultaneously determinizing and complementing ω-automata
- Emerson, Jutla
- 1989
(Show Context)
Citation Context ...nguage accepted by a finite automaton on infinite objects derived from the formula (Thomas, 1990). This is a standard technique that provides tight upper bounds for various modal and temporal logics (=-=Emerson & Jutla, 1989-=-; Vardi & Wolper, 1994) and logics of programs (Vardi, 1985; Vardi & Wolper, 1986; Vardi & Stockmeyer, 1985), and we extend it to rfcpdl. It is based on the tree model property which holds also for rf... |

1 | di Informatica e Sistemistica, Universit`a di Roma "La Sapienza - Borgida, Lenzerini, et al. - 1995 |