A New n-Ary Existential Quantifier in Description Logics

Answering regular path queries in expressive Description Logics via alternating tree-automata

The least common subsumer (lcs) of a set of concept descriptions is the most specific concept description that subsumes all of the concept descriptions in the given set. By computing the lcs, commonalities between concept descriptions can be made explicit. This is an important inference task useful in several applications, including, for instance, the bottom-up construction of description logic knowledge bases. Previous work on the lcs has concentrated on description logics that either allow for number restrictions or for existential restrictions. Many applications, however, require to combine these constructors. In this work, we present an algorithm for computing the lcs in the description logic ALEN which comprises both constructors—number and existential restrictions—as well as concept conjunction, primitive negation, and value restrictions. To prove correctness of our lcs algorithm, we develop a structural characterization of subsumption in ALEN.

In applications of AI systems where exact definitions of the important notions of the application domain are hard to come by, the use of traditional logic-based knowledge representation languages such as Description Logics may lead to very large and unintuitive definitions, and high complexity of reasoning. To overcome this problem, we define new concept constructors that allow us to define concepts in an approximate way. To be more precise, we present a family τEL(m) of extensions of the lightweight Description Logic EL that use threshold constructors for this purpose. To define the semantics of these constructors we employ graded membership functions m, which for each individual in an interpretation and concept yield a number in the interval [0,1] expressing the degree to which the individual belongs to the concept in the interpretation. Threshold concepts C⋈t for ⋈∈{<,≤,>,≥} then collect all the individuals that belong to C with degree ⋈t. The logic τEL(m) extends EL with threshold concepts whose semantics is defined relative to a function m. To construct appropriate graded membership functions, we show how concept measures ∼ (which are graded generalizations of subsumption or equivalence between concepts) can be used to define graded membership functions m∼. Then we introduce a large class of concept measures, called simi-d, for which the logics τEL(m∼) have good algorithmic properties. Basically, we show that reasoning in τEL(m∼) is NP/coNP-complete without TBox, PSpace-complete w.r.t. acyclic TBoxes, and ExpTime-complete w.r.t. general TBoxes. The exception is the instance problem, which is already PSpace-complete without TBox w.r.t. combined complexity. While the upper bounds hold for all elements of simi-d, we could prove some of the hardness results only for a subclass of simi-d. This article considerably improves on and generalizes results we have shown in three previous conference papers and it provides detailed proofs of all our results.

Although being quite inexpressive, the description logic (DL) FL0, which provides only conjunction, value restriction and the top concept as concept constructors, has an intractable subsumption problem in the presence of terminologies (TBoxes): subsumption reasoning w.r.t. acyclic FL0 TBoxes is coNP-complete, and becomes even ExpTimecomplete in case general TBoxes are used. In the present paper, we use automata working on infinite trees to solve both standard and non-standard inferences in FL0 w.r.t. general TBoxes. First, we give an alternative proof of the ExpTime upper bound for subsumption in FL0 w.r.t. general TBoxes based on the use of looping tree automata. Second, we em- ploy parity tree automata to tackle non-standard inference problems such as computing the least common subsumer and the difference of FL0 concepts w.r.t. general TBoxes.

In the process of information semantic integration for product design and manufacturing, it is very important to express the product information semantic. Description logics (DLs) are a family of state-of-the-art knowledge representation languages, and a decidable subset of first-order logic. Firstly, by analyzing the characteristics of product information, proposed a semantic information description framework based on description logics for product information integration, which is divided into three levels: basic description logic, classic extended logic and unusual extended logic. Then, in the framework, reducer was taken for example to illustrate the application of their description logic in product design and manufacturing formal semantic information. Finally, based on description logic, discussed reasoning problems of product design and manufacturing information.

Standard Description Logics (DLs) can encode quantitative aspects of an application domain through either number restrictions, which constrain the number of individuals that are in a certain relationship with an individual, or concrete domains, which can be used to assign concrete values to individuals using so-called features. These two mechanisms have been extended towards very expressive DLs, for which reasoning nevertheless remains decidable. Number restrictions have been generalized to more powerful comparisons of sets of role successors in $$\mathcal {ALCSCC}$$ ALCSCC , while the comparison of feature values of different individuals in $$\mathcal {ALC} (\mathfrak {D})$$ ALC ( D ) has been studied in the context of $$\omega $$ ω -admissible concrete domains $$\mathfrak {D}$$ D . In this paper, we combine both formalisms and investigate the complexity of reasoning in the thus obtained DL $$\mathcal {ALCOSCC}(\mathfrak {D})$$ ALCOSCC ( D ) , which additionally includes the ability to refer to specific individuals by name. We show that, in spite of its high expressivity, the consistency problem for this DL is ExpTime-complete, assuming that the constraint satisfaction problem of $$\mathfrak {D}$$ D is also decidable in exponential time. It is thus not higher than the complexity of the basic DL $$\mathcal {ALC}$$ ALC . At the same time, we show that many natural extensions to this DL, including a tighter integration of the concrete domain and number restrictions, lead to undecidability.

Description Logics (DLs) denote a family of knowledge representation formalisms that allow to represent the terminological knowledge of an application domain in a structured and well-defined way. On the one hand, the name Description Logics emphasizes that the basic elements of these logics are concept descriptions, i.e., expressions that are built from concept names (unary predicates) and role names (binary predicates) using the concept constructors provided by the DL. On the other hand, unlike some of their predecessor formalisms, DLs are equipped with a formal logic-based semantics, which is usually defined in a model-theoretic way; for some DLs it can also be declared by translating concept descriptions into first-order formulae.KeywordsDescription LogicExpressive PowerInference AlgorithmInference ProblemConcept DescriptionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Beth definability property, a well-known property from classical logic, is investigated in the context of description logics: if a general L-TBox implicitly defines an L-concept in terms of a given signature, where L is a description logic, then does there always exist over this signature an explicit definition in L for the concept? This property has been studied before and used to optimize reasoning in description logics. In this paper a complete classification of Beth definability is provided for extensions of the basic description logic ALC with transitive roles, inverse roles, role hierarchies, and/or functionality restrictions, both on arbitrary and on finite structures. Moreover, we present a tableau-based algorithm which computes explicit definitions of at most double exponential size. This algorithm is optimal because it is also shown that the smallest explicit definition of an implicitly defined concept may be double exponentially long in the size of the input TBox. Finally, if explicit definitions are allowed to be expressed in first-order logic, then we show how to compute them in single exponential time.

In this paper, we consider the expressive Description Logic (DL) $$\mathcal{ALCQPI}_{R^{+}}$$ having concept constructors called part restrictions that realize rational grading. Part restrictions are able to convey statements about a rational part of a set of successors, and they essentially enrich the expressive capabilities of DLs. That is why it is important to evaluate the cost of their use with respect to the reasoning complexity. We modify the tableaux algorithm used for $$\mathcal{ALCNI}_{R^{+}}$$ (a DL with just number restrictions instead of qualified number ones), and augment it with a specific technique for dealing with part restrictions to prove that concept satisfiability and subsumption in the considered DL are still in PSPACE.

Inspired by insights from theories of human reasoning and language, we propose additions to the Manchester OWL Syntax to improve comprehensibility. These additions cover: functional and inverse functional properties, negated conjunction, the definition of exceptions, and existential and universal restrictions. By means of an empirical study, we demonstrate the effectiveness of a number of these additions, in particular: the use of solely to clarify the uniqueness of the object in a functional property; the replacement of and with intersection in conjunction, which was particularly beneficial in negated conjunction; the use of except as a substitute for and not; and the replacement of some with including and only with noneOrOnly, which helped in certain situations to clarify the nature of these restrictions.

To represent and reason about contextualized knowledge often two-dimensional Description Logics (DLs) are employed, where one DL is used to describe contexts (or possible worlds) and the other DL is used to describe the objects, i.e. the relational structure of the specific contexts. Previous approaches for DLs of context that combined pairs of DLs resulted in undecidability in those cases where so-called rigid roles are admitted, i.e. if parts of the relational structure are the same in all contexts. In this paper, we present a novel combination of pairs of DLs and show that reasoning stays decidable even in the presence of rigid roles. We give complexity results for various combinations of DLs including \(\mathcal{ALC}\), \(\mathcal{SHOQ}\), and \(\mathcal{EL}\).KeywordsDescription LogicReasoning TaskConsistency ProblemObject LogicBinary CounterThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Computing the least common subsumer (lcs) in description logics is an inference task first introduced for sublanguages of CLASSIC. Roughly speaking, the lcs of a set of concept descriptions is the most specific concept description that subsumes all of the input descriptions. As such, the lcs allows to extract the commonalities from given concept descriptions, a task essential for several applications like, e.g., inductive learning, information retrieval, or the bottom-up construction of KR-knowledge bases. Previous work on the lcs has concentrated on description logics that either allow for number restrictions or for existential restrictions. Many applications, however, require to combine these constructors. In this work, we present an lcs algorithm for the description logic ALEN, which allows for both constructors (as well as concept conjunction, primitive negation, and value restrictions). The proof of correctness of our lcs algorithm is based on an appropriate structural characterization of subsumption in ALEN also introduced in this paper.

Learning the CLASSIC Description Logic: Theoretical and Experimental Results

In this paper we present a weak and a strong intuitionistic calculus for query answering in Description Logics (DL). Given the standard model-theoretic semantics for DL, a complete query-answering calculus has to perform complex case analyses to cope with implicit disjunctions stemming from some of the concept-forming operators in DL. To avoid this complexity we propose an intuitionistic approach to query answering based on the Sequent-Calculus-style axiomatization of DL we have developed in [20] and [21]. By taking into account only the intuitionistic inference schemata of this axiomatization, we obtain a strong intuitionistic query-answering calculus. An additional restriction to reasoning about explicit objects allows a further simplification of the proof theory and yields a weak intuitionistic calculus.We prove completeness of these calculi wrt axiomatic semantics based on the Intuitionistic Sequent Calculus. For the weak calculus we also give a least fixed point semantics as known from Deductive Databases and Logic Programming.KeywordsDescription LogicsIntuitionistic Sequent CalculusLeast Fixed Point SemanticsQuery Answering

Classical Description Logics (DLs) are not suitable to represent vague pieces of information. The attempts to achieve a solution have lead to the birth of fuzzy DLs and rough DLs. In this work, we provide a simple solution to join these two formalisms and define a fuzzy rough DL. We also show how to extend two reasoning algorithms for fuzzy DLs, which are implemented in the fuzzy DL reasoners fuzzyDL and DeLorean.KeywordsDescription LogicVague ConceptReasoning AlgorithmIndiscernibility RelationLukasiewicz LogicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.