## Practical reasoning for very expressive description logics (2000)

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Venue: | Journal of the Interest Group in Pure and Applied Logics 8 |

Citations: | 157 - 21 self |

### BibTeX

@INPROCEEDINGS{Horrocks00practicalreasoning,

author = {Ian Horrocks},

title = {Practical reasoning for very expressive description logics},

booktitle = {Journal of the Interest Group in Pure and Applied Logics 8},

year = {2000},

pages = {293--323}

}

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### Abstract

Description Logics (DLs) are a family of knowledge representation formalisms mainly characterised by constructors to build complex concepts and roles from atomic ones. Expressive role constructors are important in many applications, but can be computationally problematical. We present an algorithm that decides satisfiability of the DL ALC extended with transitive and inverse roles and functional restrictions with respect to general concept inclusion axioms and role hierarchies; early experiments indicate that this algorithm is well-suited for implementation. Additionally, we show that ALC extended with just transitive and inverse roles is still in PSpace. We investigate the limits of decidability for this family of DLs, showing that relaxing the constraints placed on the kinds of roles used in number restrictions leads to the undecidability of all inference problems. Finally, we describe a number of optimisation techniques that are crucial in obtaining implementations of the decision procedures, which, despite the hight worst-case complexity of the problem, exhibit good performance with real-life problems. 1

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Citation Context ...is problem can be dealt with by using a semantic branching technique adapted from the Davis-Putnam-Logemann-Loveland procedure (DPL) commonly used to solve propositional satisfiability (SAT) problems =-=[12; 21]-=-. Instead of choosing an unexpanded disjunction in (x), a single disjunct D is chosen from one of the unexpanded disjunctions in (x). The two possible sub-trees obtained by adding either D or ¬D to (x... |

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Citation Context ...hms is that their relative simplicity facilitates the application of a range of optimisation techniques. Several systems based on S logics have now been implemented (e.g., FaCT [30], DLP[40] and RACE =-=[23]-=-), and have demonstrated that suitable optimisation techniques can lead to a dramatic improvement in the performance of the algorithms when used in realistic applications. A system based on the SHIF l... |

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Citation Context ...{ I R , if i =1 = otherwise i∈ R I ◦ (R I ) i−1 , For both SH and ALC+, concept satisfiability is an Exptime-complete problem. This result is easily derived from the Exptime-hardness proof for PDL in =-=[18]-=- and from the proof that PDL is in Exptime in [41]. Nevertheless, implementations of algorithms for SH exhibit good performance in realistic applications [34] whereas, at the moment, this seems to be ... |

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Citation Context ...he completion tree. Summing up we get: Theorem 4.9 The modified algorithm is a PSpace decision procedure for satisfiability and subsumption of SI-concepts. 5 The Undecidability of Unrestricted SHN In =-=[28]-=- we describe an algorithm for SHIQ basedontheSHIF-algorithm already presented. Like earlier DLs that combine a hierarchy of (transitive and non-transitive) roles with some form of number restrictions ... |

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Citation Context ... variant of SHIF is still an open problem. For convenience, we will refer to SHN with arbitrary roles in number restrictions as SHN + . The undecidability proof uses a reduction of the domino problem =-=[7]-=- adapted from [4]. This problem asks if, for a set of domino types, there exists a tiling of an 2 grid such that each point of the grid is covered with one of the domino types, and adjacent dominoes a... |

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Citation Context ... on an extension of the well known DL ALC [45] to include transitively closed primitive roles [43]; we will call this logic S due to its relationship with the propositional (multi) modal logic S4 (m) =-=[44]-=-. 1 This basic DL is then extended in a variety of ways—see Figure 1 for an overview. Definition 2.1 Let NC be a set of concept names and R asetof role names with transitive role names R+ ⊆ R. The set... |

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Citation Context ...g the DL to transitive roles can lead to a lower complexity, and that transitive roles, even when combined with role hierarchies, allow for algorithms that behave quite well in realistic applications =-=[31]-=-. However, until now it has been unclear if this is still true when inverse roles are also present. In this paper we present various aspects of our research in this direction. Firstly, we motivate our... |

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Citation Context ...l-life problems. Keywords: description logic, modal logic, automated reasoning, tableaux algorithm 1 Motivation Description Logics (DLs) are a well-known family of knowledge representation formalisms =-=[17]-=-. They are based on the notion of concepts (unary predicates, classes) and roles (binary relations), and are mainly characterised by constructors that allow complex concepts and roles to be built from... |

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Citation Context ...tree with a label identical to that of x. The expansion process could then be repeated indefinitely. This problem can be dealt with by blocking: halting the expansion process when a cycle is detected =-=[3; 8]-=-. For logics without inverse roles, the general procedure is to check the label of each new node y, and if it is a subset [2] of the label of an ancestor node x, then no further expansion of y is perf... |

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Citation Context ...sumption and satisfiability, and both of these problems with respect to terminologies) and how they are interrelated. The logics we will discuss are all based on an extension of the well known DL ALC =-=[45]-=- to include transitively closed primitive roles [43]; we will call this logic S due to its relationship with the propositional (multi) modal logic S4 (m) [44]. 1 This basic DL is then extended in a va... |

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Citation Context ...x) that present only one expansion possibility and to detect a clash when a disjunction in (x) has no expansion possibilities. This simplification has been called boolean constraint propagation (BCP) =-=[20]-=-. In effect, the inference rule ¬C1,... ,¬Cn,C1 ⊔ ...⊔ Cn ⊔ D D Downloaded from http://jigpal.oxfordjournals.org at Serial Record on August 13, 2010 is being used to simplify the conjunctive concept r... |

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Citation Context ...eature of these algorithms is that their relative simplicity facilitates the application of a range of optimisation techniques. Several systems based on S logics have now been implemented (e.g., FaCT =-=[30]-=-, DLP[40] and RACE [23]), and have demonstrated that suitable optimisation techniques can lead to a dramatic improvement in the performance of the algorithms when used in realistic applications. A sys... |

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Citation Context ...ption and (un)satisfiability w.r.t. role hierarchies can be reduced to each other: C ⊑R D iff C ⊓¬D is unsatisfiable w.r.t. R, andCis unsatisfiable w.r.t. R iff C ⊑R A ⊓¬A for some concept name A. In =-=[37; 3; 44; 1]-=-, theinternalisation of terminological axioms is introduced, a technique that reduces reasoning with respect to a (possibly cyclic) terminology to satisfiability of concepts. In [31], we saw how role ... |

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Citation Context ...ption and (un)satisfiability w.r.t. role hierarchies can be reduced to each other: C ⊑R D iff C ⊓¬D is unsatisfiable w.r.t. R, andCis unsatisfiable w.r.t. R iff C ⊑R A ⊓¬A for some concept name A. In =-=[37; 3; 44; 1]-=-, theinternalisation of terminological axioms is introduced, a technique that reduces reasoning with respect to a (possibly cyclic) terminology to satisfiability of concepts. In [31], we saw how role ... |

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Citation Context ...sumption and satisfiability of concepts are known for a wide variety of DLs. Transitive and inverse roles play an important role not only in the adequate representation of complex, aggregated objects =-=[35]-=-, but also for reasoning with conceptual data models [9]. Moreover, defining concepts using general concept inclusion axioms seems natural and is crucial for representing conceptual data models. The r... |

117 |
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Citation Context ... points) being expanded before “newer” ones, and this led to more effective pruning if a clash was discovered. Using the heuristic disturbed this ordering and reduced the effectiveness of backjumping =-=[29]-=-. Moreover, MOMS-style heuristics are of little value themselves in description logic systems because they rely for their effectiveness on finding the same disjuncts recurring in multiple unexpanded d... |

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Citation Context ...s computed simply by counting the number of times C or its negation occur in minimally sized disjunctions. There are several variants of this heuristic, including the heuristic from Jeroslow and Wang =-=[36]-=-. The Jeroslow and Wang heuristic considers all occurrences of a disjunct, weighting them according to the size of the disjunction in which they occur. The heuristic then selects the disjunct with the... |

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Citation Context ...abel of a node x, syntactic branching works by choosing an unexpanded disjunction (C1 ⊔ ...⊔ Cn) in (x) and searching the different models obtained by adding each of the disjuncts C1, ... , Cn to (x) =-=[22]-=-. As the alternative branches of the search tree are not disjoint, there is nothing to prevent the recurrence of an unsatisfiable disjunct in different branches. The resulting wasted expansion could b... |

84 |
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Citation Context ...erarchy R. The subrole relation ⊑* is the transitive-reflexive closure of ⊑ over R∪{Inv(R) ⊑ Inv(S) | R ⊑ S ∈R}. SHIQ is obtained from SHI by allowing, additionally, for qualified number restrictions =-=[26]-=-, i.e., for concepts of the form �nR.C and �nR.C, whereRis a simple role, C is a concept, and n ∈ . A role is called simple iff it is neither transitive nor has transitive sub-roles. SHIN is the restr... |

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Citation Context ...For both SH and ALC+, concept satisfiability is an Exptime-complete problem. This result is easily derived from the Exptime-hardness proof for PDL in [18] and from the proof that PDL is in Exptime in =-=[41]-=-. Nevertheless, implementations of algorithms for SH exhibit good performance in realistic applications [34] whereas, at the moment, this seems to be more problematical for ALC+. We believe that the m... |

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Citation Context ...d from http://jigpal.oxfordjournals.org at Serial Record on August 13, 2010 3 Blocking The algorithms we are going to present for deciding satisfiability of SI-andSHIFconcepts use the tableaux method =-=[25]-=-, in which the satisfiability of a concept D is3. BLOCKING 243 tested by trying to construct a model of D. The model is represented by a tree in which nodes correspond to individuals and edges corres... |

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Citation Context ...sion axioms seems natural and is crucial for representing conceptual data models. The relevant inference problems for (an extension of) ALC augmented in the described manner are known to be decidable =-=[15]-=-, and worst-case optimal inference algorithms have been described [16]. However, to the best of our knowledge, nobody has found efficient means to deal with their high degree of non-determinism, which... |

70 | Fact and ifact
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Citation Context ... optimisation techniques can lead to a dramatic improvement in the performance of the algorithms when used in realistic applications. A system based on the SHIF logic has also been implemented (iFaCT =-=[32]-=-) and has been shown to be similarly amenable to optimisation. DL systems are typically used to classify a KB, and the optimisation techniques used in such systems can be divided into four categories ... |

63 | Rosati: “Data Integration in Data Warehousing
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Citation Context ...d to as iFaCT. iFaCT has been used to conduct some initial experiments with a terminology representing (fragments of) database schemata and inter schema assertions from a data warehousing application =-=[10]-=- (a slightly simplified version of the proposed encoding was used to generate SHIF terminologies). iFaCT is able to classify this terminology, which contains 19 concepts and 42 axioms, in less than 0.... |

61 | Cardinality restrictions on concepts
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Citation Context ...ith by blocking: halting the expansion process when a cycle is detected [3; 8]. For logics without inverse roles, the general procedure is to check the label of each new node y, and if it is a subset =-=[2]-=- of the label of an ancestor node x, then no further expansion of y is performed: x is said to block y. The resulting tree corresponds to a cyclical model in which y is identified with x. To deal with... |

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Citation Context ...ata models. The relevant inference problems for (an extension of) ALC augmented in the described manner are known to be decidable [15], and worst-case optimal inference algorithms have been described =-=[16]-=-. However, to the best of our knowledge, nobody has found efficient means to deal with their high degree of non-determinism, which so far prohibits their use in realistic applications. This is mainly ... |

55 | Optimising description logic subsumption
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Citation Context ...rom the Exptime-hardness proof for PDL in [18] and from the proof that PDL is in Exptime in [41]. Nevertheless, implementations of algorithms for SH exhibit good performance in realistic applications =-=[34]-=- whereas, at the moment, this seems to be more problematical for ALC+. We believe that the main reason for this discrepancy, at least in the case of tableau algorithm implementations, lies in the diff... |

48 | Intelligent Backtracking on Constraint Satisfaction Problems: Experimental and Theoretical Results
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Citation Context ... is illustrated in Figure 8. This problem can be addressed by adapting a form of dependency directed backtracking called backjumping, which has been used in solving constraint satisfiability problems =-=[5]-=- (a similar technique was also used in the HARP theorem prover [39]). Backjumping works by labelling each concept in a node label with a dependency set indicating the branching points on which it depe... |

48 | What’s in an aggregate: Foundations for description logics with tuples and sets
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Citation Context ...additional 84 axioms, but one which FaCT is unable to classify in 12 hours of CPU time. As discussed in Section 3, an extension of the embedding technique can be used to eliminate number restrictions =-=[14]-=-, but requires a target logic which supports the transitive closure of roles, i.e., converse-PDL. The even larger number of axioms that this embedding would introduce makes it unlikely that tractable ... |

48 | A concept language extended with different kinds of transitive roles
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Citation Context ...xford University Press240 Practical Reasoning for Very Expressive Description Logics these algorithms can handle not only transitive roles but also the transitive closure of roles. It has been shown =-=[43]-=- that restricting the DL to transitive roles can lead to a lower complexity, and that transitive roles, even when combined with role hierarchies, allow for algorithms that behave quite well in realist... |

45 | DLP system description
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Citation Context ... these algorithms is that their relative simplicity facilitates the application of a range of optimisation techniques. Several systems based on S logics have now been implemented (e.g., FaCT [30], DLP=-=[40]-=- and RACE [23]), and have demonstrated that suitable optimisation techniques can lead to a dramatic improvement in the performance of the algorithms when used in realistic applications. A system based... |

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Citation Context ...ly generated problems, particularly those that are over-constrained (likely to be unsatisfiable) [34]. It is also effective with some of the hand crafted problems from the Tableaux’98 benchmark suite =-=[24; 6]-=-. However it is of little benefit when classifying realistic KBs [33]. 6.2 Local Simplification Local simplification is another technique used to reduce the size of the search space resulting from the... |

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Citation Context ...ption and (un)satisfiability w.r.t. role hierarchies can be reduced to each other: C ⊑R D iff C ⊓¬D is unsatisfiable w.r.t. R, andCis unsatisfiable w.r.t. R iff C ⊑R A ⊓¬A for some concept name A. In =-=[37; 3; 44; 1]-=-, theinternalisation of terminological axioms is introduced, a technique that reduces reasoning with respect to a (possibly cyclic) terminology to satisfiability of concepts. In [31], we saw how role ... |

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Citation Context ...ting a form of dependency directed backtracking called backjumping, which has been used in solving constraint satisfiability problems [5] (a similar technique was also used in the HARP theorem prover =-=[39]-=-). Backjumping works by labelling each concept in a node label with a dependency set indicating the branching points on which it depends. A concept C ∈ (x) depends on a branching point if C was added ... |

37 |
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Citation Context ...ide variety of DLs. Transitive and inverse roles play an important role not only in the adequate representation of complex, aggregated objects [35], but also for reasoning with conceptual data models =-=[9]-=-. Moreover, defining concepts using general concept inclusion axioms seems natural and is crucial for representing conceptual data models. The relevant inference problems for (an extension of) ALC aug... |

33 | Goals for concept representation in the GALEN project
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Citation Context ... and thus a huge performance improvement. For example, when using either FaCT or DLP with backjumping disabled in order to classify a large (≈3,000 concept) KB derived from the European Galen project =-=[42]-=-, single satisfiability tests were encountered that could not be solved even after several weeks of CPU time. Classifying the same KB with backjumping enabled takes less than 100s of CPU time for eith... |

27 |
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Citation Context ...ly generated problems, particularly those that are over-constrained (likely to be unsatisfiable) [34]. It is also effective with some of the hand crafted problems from the Tableaux’98 benchmark suite =-=[24; 6]-=-. However it is of little benefit when classifying realistic KBs [33]. 6.2 Local Simplification Local simplification is another technique used to reduce the size of the search space resulting from the... |

25 |
Number restrictions on complex roles in description logics
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(Show Context)
Citation Context ...is still an open problem. For convenience, we will refer to SHN with arbitrary roles in number restrictions as SHN + . The undecidability proof uses a reduction of the domino problem [7] adapted from =-=[4]-=-. This problem asks if, for a set of domino types, there exists a tiling of an 2 grid such that each point of the grid is covered with one of the domino types, and adjacent dominoes are “compatible” w... |

21 | Comparing subsumption optimizations
- Horrocks, Patel-Schneider
- 1998
(Show Context)
Citation Context ...ely to be unsatisfiable) [34]. It is also effective with some of the hand crafted problems from the Tableaux’98 benchmark suite [24; 6]. However it is of little benefit when classifying realistic KBs =-=[33]-=-. 6.2 Local Simplification Local simplification is another technique used to reduce the size of the search space resulting from the application of non-deterministic expansion rules. Before any nondete... |

17 |
The complexity of propositional tense logics
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Citation Context ...algorithm, we employ a refined blocking strategy which necessitates a second label � for each node. This blocking technique, while discovered independently, is based on ideas similar to those used in =-=[46]-=- to show that satisfiability for K4t can be decided in PSpace. 3 In the following, we will describe and motivate this blocking technique; detailed proofs as well as a similar result for SIN can be fou... |

16 | Description logics with inverse roles, functional restrictions, and n-ary relations
- Giacomo, Lenzerini
- 1994
(Show Context)
Citation Context ... known algorithm that is able to deal with the combination of transitive closure, inverse roles, and functional restrictions on roles relies on an elaborate polynomial reduction to a CPDL terminology =-=[13]-=-, and the capability of CPDL to internalise the resulting general terminological axioms. The large number and the nature of the axioms generated by this reduction make it very unlikely that an impleme... |

16 |
random 3-SAT problems and the DavisPutnam procedure
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(Show Context)
Citation Context ...is problem can be dealt with by using a semantic branching technique adapted from the Davis-Putnam-Logemann-Loveland procedure (DPL) commonly used to solve propositional satisfiability (SAT) problems =-=[12; 21]-=-. Instead of choosing an unexpanded disjunction in (x), a single disjunct D is chosen from one of the unexpanded disjunctions in (x). The two possible sub-trees obtained by adding either D or ¬D to (x... |

12 | R.: A note on encoding inverse roles and functional restrictions in alc knowledge bases
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(Show Context)
Citation Context ...aCT is able to classify this terminology, which contains 19 concepts and 42 axioms, in less than 0.1s of (266MHz Pentium) CPU time. In contrast, eliminating inverse roles using an embedding technique =-=[11]-=- gives an equisatisfiable FaCT terminology with an additional 84 axioms, but one which FaCT is unable to classify in 12 hours of CPU time. As discussed in Section 3, an extension of the embedding tech... |

4 |
A PSpace-algorithm for deciding ALCI R + -satisfiability
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- 1998
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Citation Context ...ubsumption) of SI-concepts in Pspace. In this paper we only sketch most of the proofs. For details on the SHIF-algorithm, please refer to [35], for details on the SI-andSIN-algorithm, please refer to =-=[27]-=-. The correctness of the algorithms can be proved by showing that they create a tableau for a concept iff it is satisfiable. For ease of construction, we assume all concepts to be in negation normal f... |