The complexity of existential quantification in concept languages

Abstract

Much of the research on concept languages, also called terminological languages, has focused on t he computational complexity of subsumption. The intractability results can be divided into two groups. First, it has been shown that extending the basic language F £with constructs containing some form of logical disjunction leads to co-NPhard subsumption problems. Second, adding negation to F £makes subsumption PSPACE-complete.The main result of this paper is that extending F £with unrestricted existential quantification makes subsumption NP-complete. This is the first proof of intractability for a concept language containing no construct expressing disjunction-whether explicitly or implicitly. Unrestricted existential quantification is therefore, alongside disjunction, a source of computational complexity in concept languages.

Fusions of Description Logics and Abstract Description Systems

“…If the imposed restrictions are too severe, however, then the important notions of the application domain can no longer be expressed. Investigating this trade-off between the expressivity of DLs and the complexity of their inference problems has thus been one of the most important issues in DL research (see, e.g., Levesque & Brachman, 1987;Nebel, 1988;Schmidt-Schauß, 1989;Schmidt-Schauß & Smolka, 1991;Nebel, 1990;Donini, Lenzerini, Nardi, & Nutt, 1991, 1997Donini, Hollunder, Lenzerini, Spaccamela, Nardi, & Nutt, 1992;Schaerf, 1993;Donini, Lenzerini, Nardi, & Schaerf, 1994;De Giacomo & Lenzerini, 1994a, 1994bCalvanese, De Giacomo, & Lenzerini, 1999;Lutz, 1999;Horrocks, Sattler, & Tobies, 2000). This paper investigates an approach for extending the expressivity of DLs that (in many cases) guarantees that reasoning remains decidable: the fusion of DLs.…”

Nonstandard Inferences in Description Logics: The Story So Far

“…introduced in [50]. As shown in [50], the propagation of value restrictions on existential restrictions may lead to an exponential blow-up (see the concept descriptions C n introduced below Theorem 15).…”

“…introduced in [50]. As shown in [50], the propagation of value restrictions on existential restrictions may lead to an exponential blow-up (see the concept descriptions C n introduced below Theorem 15). Consequently, the size of the normal forms, and thus also of the description trees, may grow exponentially in the size of the original ALE concept descriptions.…”

Reasoning in Expressive Description Logics

“…Subsequent investigations showed that constraint systems can be seen as specialized forms of tableaux. Many results on algorithms for reasoning on concept expressions, and their complexity were then derived using tableau-based techniques [Donini et al 1991, Buchheit et al 1993, Donini et al 1996, Donini et al 1997, Horrocks 1998. Such techniques, besides being intuitively appealing, provided a useful framework for modularizing the problem of designing reasoning algorithms for languages formed by different sets on constructs.…”