The complexity of concept languages

The basic feature of Terminological Knowledge Representation Systems is to represent knowledge by means of taxonomies, here called terminologiesand to provide a specialized reasoning engine to do inferences on these structures. The taxonomy is built through a representation language called concept language (or description logic), which is given well-defined set-theoretic semantics. The efficiency of reasoning has often been advocated as a primary motivation for the use of such systems. Deduction methods and computational properties of reasoning problems in concept languages are the subject of this paper. 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

Subsumption algorithms for concept languages

We investigate the subsumption problem in logic-based knowledge representation languages of the KL-ONE family and give decision procedures. All our languages contain as a kernel the logical connectives conjunction, disjunction, and negation for concepts, as well as role quantification. The algorithms are rule-based and can be understood as variants of tableaux calculus with a special control strategy. In the first part of the paper, we add number restrictions and conjunction of roles to the kernel language. We show that subsumption in this language is decidable, and we investigate sublanguages for which the problem of deciding subsumption is PSPACE-complete. In the second part, we amalgamate the kernel language with feature descriptions as used in computational linguistics. We show that feature descriptions do not increase the complexity of the subsumption problem

Specifying role interaction in concept languages

The KL-ONE concept language provides role-value maps (RVMs) as a concept forming operator that compares sets of role fillers. This is a useful means to specify structural properties of concepts. Recently, it has been shown that concept languages providing RVMs together with some other common concept-forming operators induce an undecidable subsumption problem. Thus, RVMs have been restricted to chainings of functional roles as, for example, in CLASSIC. Although this restricted RVM is still a useful operator, one would like to have additional means to specify interaction of general roles. The present paper investigates two concept languages for that purpose. The first one provides concept forming operators that generalize the restricted RVM in a different direction. Unfortunately, it turns out that this language also has an undecidable subsumption problem. The second formalism allows to specify structural properties w.r.t. roles without using general equality and is equipped with (complete) decision procedures for its associated reasoning problems

Qualifying number restrictions in concept languages

We investigate the subsumption problem in logic-based knowledge representation languages of the KL-ONE family. The language presented in this paper provides the constructs for conjunction, disjunction, and negation of concepts, as well as qualifying number restrictions. The latter ones generalize the well-known role quantifications (such as value restrictions) and ordinary number restrictions, which are present in almost all KL-ONE based systems. Until now, only little attempts were made to integrate qualifying number restrictions into concept languages. It turns out that all known subsumption algorithms which try to handle these constructs are incomplete, and thus detecting only few subsumption relations between concepts. We present a subsumption algorithm for our language which is sound and complete. Subsequently we discuss why the subsumption problem in this language is rather hard from a computational point of view. This leads to an idea of how to recognize concepts which cause tractable problems

The complexity of existential quantification in concept languages

Much of the research on concept languages, also called terminological languages, has focused on the computational complexity of subsumption. The intractability results can be divided into two groups. First, it has been shown that extending the basic language FL- with constructs containing some form of logical disjunction leads to co-NP-hard subsumption problems. Second, adding negation to FL- makes subsumption PSPACE-complete. The main result of this paper is that extending FL- 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

Queries, rules and definitions as epistemic statements in concept languages

Concept languages have been studied in order to give a formal account of the basic features of frame-based languages. The focus of research in concept languages was initially on the semantical reconstruction of frame-based systems and the computational complexity of reasoning. More recently, attention has been paid to the formalization of other aspects of frame-based languages, such as non-monotonic reasoning and procedural rules, which are necessary in order to bring concept languages closer to implemented systems. In this paper we discuss the above issues in the framework of concept languages enriched with an epistemic operator. In particular, we show that the epistemic operator both introduces novel features in the language, such as sophisticated query formulation and closed world reasoning, and makes it possible to provide a formal account for some aspects of the existing systems, such as rules and definitions, that cannot be characterized in a standard first-order framework

Subsumption between queries to object-oriented databases

Most work on query optimization in relational and object-oriented databases has concentrated on tuning algebraic expressions and the physical access to the database contents. The attention to semantic query optimization, however, has been restricted due to its inherent complexity. We take a second look at semantic query optimization in object-oriented databases and find that reasoning techniques for concept languages developed in Artificial Intelligence apply to this problem because concept languages have been tailored for efficiency and their semantics is compatible with class and query definitions in object-oriented databases. We propose a query optimizer that recognizes subset relationships between a query and a view (a simpler query whose answer is stored) in polynomial time

The complexity of concept languages

The 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 concept language (or description logic), which is given well-defined set-theoretic semantics. The efficiency of reasoning has often been advocated as a primary motivation for the use of such systems. Deduction methods and computational properties of reasoning problems in concept languages are the subject of this paper. 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

Decidable Reasoning in Terminological Knowledge Representation Systems

Terminological knowledge representation systems (TKRSs) are tools for designing and using knowledge bases that make use of terminological languages (or concept languages). We analyze from a theoretical point of view a TKRS whose capabilities go beyond the ones of presently available TKRSs. The new features studied, often required in practical applications, can be summarized in three main points. First, we consider a highly expressive terminological language, called ALCNR, including general complements of concepts, number restrictions and role conjunction. Second, we allow to express inclusion statements between general concepts, and terminological cycles as a particular case. Third, we prove the decidability of a number of desirable TKRS-deduction services (like satisfiability, subsumption and instance checking) through a sound, complete and terminating calculus for reasoning in ALCNR-knowledge bases. Our calculus extends the general technique of constraint systems. As a byproduct of the proof, we get also the result that inclusion statements in ALCNR can be simulated by terminological cycles, if descriptive semantics is adopted.Comment: See http://www.jair.org/ for any accompanying file