## The Complexity of Concept Languages (1991)

Venue: | Information and Computation |

Citations: | 232 - 33 self |

### BibTeX

@INPROCEEDINGS{Donini91thecomplexity,

author = {Francesco M. Donini and Maurizio Lenzerini and Daniele Nardi and Werner Nutt},

title = {The Complexity of Concept Languages},

booktitle = {Information and Computation},

year = {1991},

pages = {151--162},

publisher = {Morgan Kaufmann}

}

### Years of Citing Articles

### OpenURL

### Abstract

A basic feature of Terminological Knowledge Representation Systems is to represent knowledge by means of taxonomies, here called terminologies, and to provide a specialized reasoning engine to do inferences on these structures. The taxonomy is built through a representation language called a concept language (or description logic), which is given a well-defined set-theoretic semantics. The efficiency of reasoning has often been advocated as a primary motivation for the use of such systems. The main contributions of the paper are: (1) a complexity analysis of concept satisfiability and subsumption for a wide class of concept languages; (2) the algorithms for these inferences that comply with the worst-case complexity of the reasoning task they perform. This is an extended and revised version of a paper presented at the 2nd Int. Conf. on Principles of Knowledge Representation and Reasoning, Cambridge, MA, 1991. 1 Introduction Among computer systems based on Artificial Intelligence ...

### Citations

619 | An Overview of the KL-ONE Knowledge Representation System
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- 1985
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Citation Context ...e to describe the classes of "individuals having a female child" and of "individuals whose children are all female" by the concepts 9has-child.Female and 8has-child.Female: In a kl=-=-one-like notation (Brachman and Schmolze, 1985-=-), the latter concept corresponds to a link via the role has-child, and a value restriction to Female of this link. Number restrictions on roles denote classes of individuals having at least or at mos... |

430 |
First-order Logic and Automated Theorem Proving (2nd ed
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Citation Context ...t-Schau�� and Smolka, 1991), but employs a much simpler notation that emphasizes its similarity to the tableaux calculus for first-order predicate logic (see (Smullyan, 1968; Bell and Machover, 19=-=77; Fitting, 1990-=-)). The calculus consists of inference rules that decompose complex concepts according to the top-level construct. Actually, the rules can be simulated by applications of several rules of the tableaux... |

372 |
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- 1990
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Citation Context ...concept languages are notational variants of certain propositional modal logics. Among the languages we consider, ALC has a modal-logic counterpart, namely the multimodal version of the logic K (see (=-=Halpern and Moses, 1992-=-)). Actually, ALC-concepts and formulas in multimodal K can immediately be translated into each other. Moreover, a concept is satisfiable if and only if the corresponding formula is satisfiable. Resea... |

325 |
A Catalog of Complexity Classes
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- 1990
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Citation Context ...esque, 1984). Deduction methods and computational properties of reasoning problems in concept languages are the subject of this paper. We use standard notions from complexity theory (see for example (=-=Johnson, 1990-=-)). In the introduction we first describe informally concept languages and the reasoning services of a TKRS, then we illustrate the motivations and main results of the paper. 1.1 Concepts as Structure... |

271 | A correspondence theory for terminological logics
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- 1991
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Citation Context ...ere are at least two other areas of research that have a relationship with concept description languages and the associated inference problems: modal logic and query containment in databases. Schild (=-=Schild, 1991-=-) pointed out that certain concept languages are notational variants of certain propositional modal logics. Among the languages we consider, ALC has a modal-logic counterpart, namely the multimodal ve... |

192 | The tractability of Subsumption in frame-based description languages
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- 1984
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Citation Context ...nsequences of the formalization of the representation language, there is the possibility of studying the computational properties of inference by means of a complexity analysis, as first proposed in (=-=Brachman and Levesque, 1984-=-). Deduction methods and computational properties of reasoning problems in concept languages are the subject of this paper. We use standard notions from complexity theory (see for example (Johnson, 19... |

191 |
The computational complexity of provability in systems of modal propositional logic
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- 1977
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Citation Context ...ormula is satisfiable. Research in the complexity of the satisfiability problem for modal propositional logics started a long time before the complexity of concept languages was investigated. Ladner (=-=Ladner, 1977-=-) showed that satisfiability in unimodal K---a logic whose counterpart is the set of all ALC-concepts with at most one role symbol--- is PSPACE-complete. Interestingly, in order to prove membership in... |

182 |
What's in a link: Foundations for semantic networks
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- 1975
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Citation Context ...achman, 1979). This novel paradigm was motivated by the shortcomings of semantic networks, which have been criticized for their lack of formal semantics which results in ambiguity and contradictions (=-=Woods, 1975-=-; Brachman, 1985). During the last ten years, this line of research has led to the development of various TKRSs (see for example (Woods and Schmolze, 1992; SIGART, 1991; Baader et al., 1994)). These s... |

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162 |
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Citation Context ... to provide a specialized reasoning engine to do inferences on these structures. Such TKRSs have their roots in Structured Inheritance Networks proposed by Brachman and realized in the kl-one system (=-=Brachman, 1979-=-). This novel paradigm was motivated by the shortcomings of semantic networks, which have been criticized for their lack of formal semantics which results in ambiguity and contradictions (Woods, 1975;... |

159 |
Foundations of a Functional Approach to Knowledge Representation
- Levesque
- 1984
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Citation Context ...cation between the KRS and the rest of the KBS is realized via queries and answers to queries. The type of language used to represent knowledge, and the inferences drawn from it characterize the KRS (=-=Levesque, 1984-=-). In this paper we are concerned with Terminological Knowledge Representation Systems (TKRSs), whose basic feature is to represent knowledge by means of taxonomies, here called terminologies, and to ... |

154 | Terminological reasoning is inherently intractable
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- 1990
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Citation Context ...r to the computational complexity in the worst case, which represents only one aspect to be taken into account when considering the practical use of concept languages. Notice that, as pointed out in (=-=Nebel, 1990-=-), another aspect that deserves further investigation is the characterization of the average cases occurring in practice. Second, the techniques used for the complexity analysis have provided the form... |

149 | Attributive concept descriptions with complements - Schmidt-Schau, Smolka - 1991 |

143 | An empirical analysis of optimization techniques for terminological representation systems, or: Making KRIS get a move on
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- 1994
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Citation Context ...nd contradictions (Woods, 1975; Brachman, 1985). During the last ten years, this line of research has led to the development of various TKRSs (see for example (Woods and Schmolze, 1992; SIGART, 1991; =-=Baader et al., 1994-=-)). These systems make a clear distinction between intensional knowledge, or general knowledge about the problem domain, and extensional knowledge, which is specific to a particular problem. Typically... |

135 |
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Citation Context ...is name comes historically from the use of concept names as terms in the language of the external KBS. The module of the KRS containing the terminology is known as Terminological Box, or simply TBox (=-=Brachman et al., 1983). As-=- an example, we discuss a terminology whose pictorial representation is given in Figure 1, where we have the concept Parent defined by Parent := Person u 9has-child.Person u 8has-child.Person: "A... |

127 | Expressiveness and tractability in knowledge representation and reasoning - Levesque, Brachman - 1987 |

112 |
Elements of Relational Database Theory
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- 1989
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Citation Context ...uery languages may create arbitrary coreferences between different parts of a query by using the same variable name. On the other hand, query containment in relational query languages is undecidable (=-=Kanellakis, 1990-=-, pg. 1083), while subsumption is decidable for most concept languages (see results in this paper). It is interesting to observe that subsumption is undecidable in concept languages with a construct e... |

93 |
First order logic
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Citation Context ...ced by the completion calculus in (Schmidt-Schau�� and Smolka, 1991), but employs a much simpler notation that emphasizes its similarity to the tableaux calculus for first-order predicate logic (s=-=ee (Smullyan, 1968-=-; Bell and Machover, 1977; Fitting, 1990)). The calculus consists of inference rules that decompose complex concepts according to the top-level construct. Actually, the rules can be simulated by appli... |

84 | Subsumption algorithms for concept description languages - Hollunder, Nutt, et al. - 1990 |

74 | A Feature Logic with Subsorts
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Citation Context ... are disjoint if C I " D I = ; for every interpretationsI. Using the complement of concepts, we can reduce subsumption, equivalence, and disjointness of concepts to satisfiability problems (see a=-=lso (Smolka, 1988-=-)): Proposition 2.1 Let C and D be concepts. Then: 1. C is subsumed by D iff C u :D is not satisfiable; 2. C and D are equivalent iff (C u :D) and (:C u D) are not satisfiable; 3. C and D are disjoint... |

67 |
Tractable concept languages
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Citation Context ...cal reasoning started with the goal to identify languages for which subsumption can be decided in polynomial time (Brachman and Levesque, 1984). After a deep analysis of the tractabilitity threshold (=-=Donini et al., 1991-=-), it turned out that many interesting constructs lead to concept languages whose reasoning problems are intractable. One might conclude from the results in this paper that terminological reasoning in... |

66 | Subsumption algorithms for concept languages
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- 1990
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Citation Context ...orithms for these inferences that comply with the worst-case complexity of the reasoning task they perform. For our work we employ a calculus of logical constraints that has first been introduced in (=-=Hollunder and Nutt, 1990) an-=-d that significantly extends the one in (Schmidt-Schau�� and Smolka, 1991). In fact, it covers a more expressive language, and it employs a more concise notation, which points out its similarity t... |

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52 |
A Course
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Citation Context ...etion calculus in (Schmidt-Schau�� and Smolka, 1991), but employs a much simpler notation that emphasizes its similarity to the tableaux calculus for first-order predicate logic (see (Smullyan, 19=-=68; Bell and Machover, 1977-=-; Fitting, 1990)). The calculus consists of inference rules that decompose complex concepts according to the top-level construct. Actually, the rules can be simulated by applications of several rules ... |

51 | The complexity of existential quantification in concept languages
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- 1992
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Citation Context ...f ALCconcepts. Subsumption can be computed by reducing it to unsatisfiability: C is subsumed by D if and only if the ALC-concept C u :D is unsatisfiable. Hence subsumption too is PSPACE-complete. In (=-=Donini et al., 1992-=-), it was shown that unsatisfiability and subsumption in the languagesALE, obtained by adding qualified existential quantification to AL, is NP-complete, thus proving that existential quantification i... |

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Citation Context ...nderstanding of the properties of the language constructs and their interaction. This is not only valuable from a theoretical viewpoint, but gives insights to the designer of a TKRS (see for example (=-=Brachman, 1992-=-)), with clear indications of the language constructs or their combinations that are difficult to deal with and general methods to cope with them. Second, the complexity results have been obtained by ... |

45 | Undecidability of subsumption in NIKL - Patel-Schneider - 1989 |

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- 1994
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Citation Context ...algorithms developed for implemented systems, and can be used in the definition of worst-case test sets for verifying implementations. For example, the comparison of implemented systems described in (=-=Heinsohn et al., 1994-=-) has benefited from the results of the complexity analysis. There are at least two other areas of research that have a relationship with concept description languages and the associated inference pro... |

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

40 |
Computational Complexity of Terminological Reasoning
- Nebel
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Citation Context ...limited language, called FL \Gamma , and showed that for the seemingly slightly more expressive language FL subsumption is co-NP-hard (for a definition of FL \Gamma and FL see Subsection 4.3). Nebel (=-=Nebel, 1988) id-=-entified other constructs that give rise to co-NP-hard subsumption problems. Other work identified languages with undecidable subsumption problem (PatelSchneider, 1989; Schmidt-Schau��, 1989; Schi... |

22 |
I Lied About the Trees
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Citation Context .... This novel paradigm was motivated by the shortcomings of semantic networks, which have been criticized for their lack of formal semantics which results in ambiguity and contradictions (Woods, 1975; =-=Brachman, 1985-=-). During the last ten years, this line of research has led to the development of various TKRSs (see for example (Woods and Schmolze, 1992; SIGART, 1991; Baader et al., 1994)). These systems make a cl... |

22 |
Computers and Intractability—A Guide to NPCompleteness
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Citation Context ...edicate is-true is interpreted over B in the obvious way. Now, P . M is said to be valid if P . M 0 is valid over B. Deciding the validity of quantified boolean formulas is a PSPACE-complete problem (=-=Garey and Johnson, 1979). T-=-he following reformulation of the problem is due to (Schmidt-Schau�� and Smolka, 1991). A literal is a nonzero integer. A clause is a nonempty finite set N of literals such that l 2 N implies \Gam... |

22 |
A KL-ONE Classifier
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Citation Context ...hms was to transform two input concepts into labeled graphs and test whether one could be embedded into the other; the embedded graph would correspond to the more general concept (the subsumer) (see (=-=Lipkis, 1982-=-)). This method is called structural comparison, and the relation between concepts it computes is called structural subsumption. However, a careful analysis of the algorithms for structural subsumptio... |

12 | Hans-Jurgen Profitlich. An empirical analysis of optimization techniques for terminological representation systems or: Making KRIS get a move on - Baader, Franconi, et al. - 1994 |

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Citation Context ...1988) identified other constructs that give rise to co-NP-hard subsumption problems. Other work identified languages with undecidable subsumption problem (PatelSchneider, 1989; Schmidt-Schau��, 19=-=89; Schild, 1988-=-). However, neither (Brachman and Levesque, 1984; Levesque and Brachman, 1987) nor (Nebel, 1988) give algorithms for the co-NP-hard languages. The first nonstructural complete subsumption algorithm wa... |

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Citation Context ..."persons that are not female" and the class of "individuals that are female or male" by the expressions Person u :Female and Female t Male: An alternative notation for the first co=-=ncept, proposed in (Patel-Schneider and Swartout, 1993), is (AND Pers-=-on (NOT Female)). Most languages provide existential and universal quantification over roles that allow one to describe the classes of "individuals having a female child" and of "indivi... |

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Citation Context ...hecking Satisfiability In this section we introduce a calculus for checking the satisfiability of concepts. The calculus appeared first in (Hollunder et al., 1990) and is discussed in full detail in (=-=Nutt, 1993). I-=-t is influenced by the completion calculus in (Schmidt-Schau�� and Smolka, 1991), but employs a much simpler notation that emphasizes its similarity to the tableaux calculus for first-order predic... |

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Citation Context ...in ambiguity and contradictions (Woods, 1975; Brachman, 1985). During the last ten years, this line of research has led to the development of various TKRSs (see for example (Woods and Schmolze, 1992; =-=SIGART, 1991-=-; Baader et al., 1994)). These systems make a clear distinction between intensional knowledge, or general knowledge about the problem domain, and extensional knowledge, which is specific to a particul... |

1 | de Rijke and Wiebe van der Hoek. Counting objects in generalized quantifier theory, modal logic, and knowledge representation - Marten - 1995 |